We consider a Gaussian branching random walk under the constraint that the values at all leaves at generation n are non-negative. We obtain a remarkably precise description of the conditional law and the conditional field. The conditioning leads to an upward shift of the whole field. We obtain sharp estimates on this upward shift (up to o(1) terms). We show that the properly rescaled maximum converges to a Gumbel distribution (without a random shift!), and the rescaled minimum is exponentially distributed. We use tools from DGFFs on general graphs and estimates on random walks that are weakly attracted to zero either through a pinning potential or a drift. The talk is based on joint work with M. Fels (Technion) and O. Louidor(Technion).
In this talk, we consider diffusion problems with multiscale coefficients subject to a certain random model describing defects e.g. in production processes. We are interested in numerically simulating the solutions for any given sample. Due to the multiscale character of the problem, computational multiscale methods based upon problem-adapted spaces should be used. To reduce the computational complexity of calculating a multiscale basis for each sample, we propose an offline-online strategy. We determine local stiffness matrices for a cleverly chosen set of coefficients offline and can then assemble the system matrix for any sample quickly in the online phase. We discuss the error analysis of the approach as well as some recent further developments and connections.
We will discuss the large-scale behavior of diffusions with space-time stationary and ergodic, divergence-free drift. Such processes provide an approximation of transport in rough, incompressible flows and have been used to describe passive advected quantities such as temperature in several contexts. The primary purpose of this talk will be to show that, unlike in the time-independent or time-dependent periodic settings, the time-dependent stochastic case exhibits a fundamentally different behavior. On large-scales, the evolution is characterized by the solution to a stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogeneous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the limit. The results apply to uniformly elliptic, stationary and ergodic environments in which the drift admits a suitably defined stationary and square-integrable stream matrix.
The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we consider the stationary distribution of asymmetric simple exclusion processes with open boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow with the length of the underlying segment. Depending on the boundary parameters for the exclusion process, we provide sufficient conditions such that the projected stationary distribution is close in total variation distance to a product measure. This talk is based on joint work with Evita Nestoridi.
Random phase approximation (RPA) was proposed by Bohm and Pines to describe the correction to the Hartree-Fock theory. The heuristic derivation of the RPA has included notable work by theoretical physicists since the 1950s but the mathematical understanding remains very challenging. In this talk, I will discuss a rigorous bosonization method where suitable pairs of fermions are described by a quasi-bosonic Bogoliubov Hamiltonian, leading to a precise leading order formula of the correlation energy of electrons in the mean-field regime. This is joint work with M. R. Christiansen and C. Hainzl.
Magnetic skyrmions are nanoscale, topological solitons observed in thin magnetic films. Due to possible data storage applications, they are of great interest to the physics community. Typically thought of as stabilized by the so-called Dzyaloshinskii-Moriya interaction occurring in extremely thin films, it has been demonstrated that they may also arise due to stray field effects.
Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion Process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters. The interplay between these parameters is powered by the Yang-Baxter equation, which brings new intriguing results to this well-traveled territory. In particular, I will discuss new Lax-type equations for the Markov semigroups of the TASEP and its relatives. Based on a joint work with Axel Saenz.
We consider effective properties of suspensions of inertialess, rigid, anisotropic, Brownian particles in Stokes flows. Recent years have seen tremendous progress regarding the rigorous justification of effective fluid equations for non-Brownian suspensions, where the complex fluid can be described in terms of an effective viscosity. In contrast to this (quasi-)Newtonian behavior, anisotropic Brownian particles cause an additional elastic stress on the fluid which leads to a variety of fascinating phenomena. A rigorous derivation of viscoelastic fluid systems starting from particle models is completely missing so far. In this talk I will present first results in this direction starting from a simplified microscopic model where the particles evolve only due to rotational Brownian motion and cause a Brownian torque on the fluid. In the limit of infinitely many small particles with very small particle volume fraction, we rigorously obtain an elastic stress on the fluid in terms of the particle density that is given as the solution to an (in-)stationary Fokker-Planck equation.
Joint work with Marta Leocata (Scuola Normale Superiore Pisa) and Amina Mecherbet (Université Paris Cité)
Ginzburg-Landau fields are a class of models from statistical mechanics that describe the behavior of interfaces. The so-called Helffer-Sjöstrand representation relates them to a random walk in a time-dependent random environment. In the talk I will introduce these objects and survey some of the known results. I will then describe joint work with Wei Wu and Ofer Zeitouni on the asymptotics of the maximum of the Ginzburg-Landau fields in two dimensions.
We develop a theory of optimal transport for stationary random measures with a particular focus on stationary point processes. This provides us with a notion of geodesic distance between distributions of stationary random measures and induces a natural displacement interpolation between them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t.~the Poisson process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes. This is joint work with Martin Huesmann, Jonas Jalowy and Bastian Müller
Globally lipschitz transport maps have found many applications in the study of probabilistic functional inequalities such as logarithmic Sobolev and Poincaré inequalities, by transporting an inequality from a nice reference measure to another one. For example, a theorem of Caffarelli states that optimal transport maps from the standard Gaussian measure onto uniformly log-concave measures are $1$-lipschitz. This then recovers the sharp bounds of Bakry and Emery on the logarithmic Sobolev constant of such measures.In this talk, I will discuss a construction of non-optimal transport maps using the heat flow, due to Kim and Milman, and explain how it allows to get dimension-free lipschitz maps in new settings, including certain Riemannian manifolds. Joint work with D. Mikulincer and Y. Shenfeld.
The existence of steady vortex rings for the two-phase Euler equations with surface tension is studied, describing the evolution of a perfect bubble air ring in water. Such objects are created in nature by cetaceans such as dolphins or beluga whales, and they appear to be surprisingly stable configurations. The mathematical model features a vortex sheet on the surface of the air bubble. We construct such vortex rings with small cross sections with the help of an implicit function theorem and derive the asymptotics of various quantities for small cross sections. Joint work with David Meyer (Münster)
We consider effective properties of suspensions of inertialess, rigid, anisotropic, Brownian particles in Stokes flows. Recent years have seen tremendous progress regarding the rigorous justification of effective fluid equations for non-Brownian suspensions, where the complex fluid can be described in terms of an effective viscosity. In contrast to this (quasi-)Newtonian behavior, anisotropic Brownian particles cause an additional elastic stress on the fluid which leads to a variety of fascinating phenomena. A rigorous derivation of viscoelastic fluid systems starting from particle models is completely missing so far. In this talk I will present first results in this direction starting from a simplified microscopic model where the particles evolve only due to rotational Brownian motion and cause a Brownian torque on the fluid. In the limit of infinitely many small particles with very small particle volume fraction, we rigorously obtain an elastic stress on the fluid in terms of the particle density that is given as the solution to an (in-)stationary Fokker-Planck equation.
Joint work with Marta Leocata (Scuola Normale Superiore Pisa) and Amina Mecherbet (Université Paris Cité)
After having exhibited some lack of pertinence of standard Hilbert-Schmidt or trace class (or more general $L^p$-Schatten class) topologies usually used for linear PDEs, I will present a quantum notion of the Wasserstein-Monge-Kantorovich distance of order two canonically obtained through a simple dictionary between classical and quantum mathematical paradigms. This will lead to a quantum definition of optimal transport, actually shown to be ¨cheaper¨ than the classical one e.g. for the bi-partite problem and to make sense in situations where the standard classical (Brenier's) one fails to be true. As a bi-product I will show how quantization can be seen as a kind of Wasserstein geodesic path between a classical function and a quantum operator, thanks to a ¨semiquantum¨ Legendre transform. No quantum mechanics prerequisites will be necessary for following the lecture.
In this talk we will explore when the invariant Gibbs measure of an N -particle system of weakly interacting diffusion is well approximated by the unique minimiser of the mean field energy. We will compute the exact limit of the associated partition function, by re-interpreting the associated integral as the fluctuation of sampling from the mean field limit measure. With this technique we can obtain sharp estimates all the way up to the phase transition, colloquially defined as the loss of analyticity of the partition function with respect to the inverse temperature.
We consider a stochastic partial differential equation close to bifurcation of pitchfork type, where a one-dimensional space changes its stability.
For finite-time Lyapunov exponents we characterize regions depending on the distance from bifurcation and the noise strength where finite-time Lyapunov exponents are positive and thus detect changes in stability. One technical tool is the reduction of the essential dynamics of the infinite dimensional stochastic system to a simple ordinary stochastic differential equation, which is valid close to the bifurcation. This talk is based on joint works with Alex Blumenthal, Maximilian Engel and Dirk Blömker.
Strongly correlated quantum systems include some of the most challenging problems in science. I will present the first numerical analysis for the coupled cluster method tailored by matrix product states, which is a promising method for handling strongly correlated systems. I will then discuss recent applications of the coupled cluster method and matrix product states for solving the magic angle twisted bilayer graphene system at the level of interacting electrons.
We will investigate an old problem posed by Isaac Newton on shapes that do not produce gravity in the cavity. We will see that it can be equivalently formulated as a question on the rigidity of global solutions of the obstacle problem, a question that recently has been answered in full generality. This is joint work with A. Figalli, H. Shahgholian and G. S. Weiss.
When studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, a ''Loewner energy", and "Loewner potential'', that describe the rate function for the LDP, were recently introduced. While these objects were originally derived from SLE theory, they turned out to have several intrinsic, and perhaps surprising, connections to various fields. I will discuss some of these connections and interpretations towards Brownian loops, semiclassical limits of certain correlation functions in conformal field theory, and rational functions with real critical points (Shapiro-Shapiro conjecture in real enumerative geometry).
(Based on joint work with Yilin Wang - IHES, France.)
We establish existence of infinitely many stationary solutions as well as ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. The result is based on a stochastic variant of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces.
The concept of paired calibrations due to Lawlor and Morgan (Pacific J.\ Math., 166, 1994) provides a particularly elegant tool for proving global area minimization of a network of branched interfaces as appearing in multiphase materials. In this talk, I will present a result on local interface area minimization for a class of stationary points of the interface energy which, due to Kinderlehrer and Liu (Math.\ Models Methods Appl.\ Sci., 11, 2001), occur as long-time asymptotic limits of multiphase mean curvature flow. This class contains continuous one-parameter families of stationary points which are not global minimizers. In particular, paired calibrations do not exist for those and consequently, minimality properties for such networks remained an open problem. Our result is based on a new concept of paired local calibrations allowing to overcome the difficulties associated with the above mentioned degeneracy of the energy landscape. This is joint work with Julian Fischer, Tim Laux and Theresa Simon (arXiv:2212.11840).
The study of stochastic PDEs has known tremendous advances in recent years and, thanks to Hairer's theory of regularity structures and Gubinelli and Perkowski's paracontrolled approach, (local) existence and uniqueness of solutions of subcritical SPDEs is by now well-understood. The goal of this talk is to move beyond the aforementioned theories and present novel tools to derive the scaling limit (in the so-called weak coupling scaling) for some stationary SPDEs at the critical dimension. Our techniques are inspired by the resolvent method developed by Landim, Olla, Yau, Varadhan, and many others, in the context of particle systems in the supercritical dimension and might be well-suited to study a much wider class of statistical mechanics models at criticality.
I will report on mathematical progress on certain (non-linear, singular) stochastic PDEs that model the mesoscopic behaviour of some out-of-equilibrium physical systems, most notably stochastic interface growth and driven diffusive systems. Namely, the d-dimensional Stochastic Burgers equation and the KPZ equation. Scaling and Renormalization Group arguments suggest that, above the critical dimension d=2, the large-scale behaviour should be Gaussian. Our results (joint works with Cannizzaro, Erhard, Gubinelli) imply, indeed, Gaussian scaling limits in dimension d\ge 3 (for the stochastic Burgers equation) and also, at least in the regime of weak non-linearity, in the critical dimension d=2 (both for stochastic Burgers and for the Anisotropic KPZ equation). The weak non-linearity limit will be discussed in much more detail in Giuseppe's talk
To understand mechanical origin of probability in statistical and continuum mechanics, it is useful to study hydrodynamic limit for interacting particles following deterministic Hamiltonian dynamics. Traditional approach on such a program faces many difficulties. One of them is about rigorous justification of canonical type ensembles. This is because that relevant deterministic ergodic theory is still largely out of reach. Another huge barrier is on making rigorous sense out of hyperbolic conservation laws. Such PDEs are used to express F=ma and thermodynamic relations in the continuum.
We examine a new line of thoughts by formulating the hydrodynamic limit program as a multi-scale abstract Hamilton-Jacobi theory in space of probability measures.
This talk will focus on derivation of an isentropic model. Through mass transport calculus, we develop tools to reduce the hydrodynamic problem to known results on finite dimensional weak KAM (Kolmogorov-Arnold-Moser) theory, showing sufficiency of using a weak version of ergodic results on micro-canonical type ensembles, instead of the canonical ones. We will also reply on recent progress of viscosity solution theory for abstract Hamilton-Jacobi equation in space of probability measures (an example of Alexandrov space). Such approach gives a weak and indirect characterization on evolution of the limiting continuum model using generating-function formalism at the level of canonical transformation in calculus of variations. It avoids the use of hyperbolic systems of PDEs, which operates at the level of abstract Euler-Lagrange equations from the action functionals.
All together, these techniques enable us to realize a weaker but rigorous version of the hydrodynamic limit program for some nontrivial cases.
This is a joint work with Toshio Mikami from Tsuda University, Tokyo, Japan.
Entropy solutions of the 2D eikonal equation arise as limits of sequences of bounded Aviles-Giga energy. They might be much less regular than viscosity solutions, which have BV gradient, nevertheless their gradients share several structural properties with BV maps, as discovered by De Lellis and Otto. Fine estimates on their Lebesgue points are however still open. In a joint work with Elio Marconi, we obtain a bound on local oscillations in terms of the entropy production, implying in particular that non-Lebesgue points have codimension 1 (as they would for BV maps).
This talk has two parts. First we present a possible extension of the Gromov-Wasserstein problem to the setting of metric measures spaces, whose total mass is not necessarily equal to 1. We propose a true distance and a lower bound which is more friendly for computations. Second, we study the existence of Monge maps as optimizer of the standard Gromov-Wasserstein problem for two different costs in euclidean spaces. The first cost for which we show existence of Monge maps is the scalar product, the second cost is the quadratic cost between the squared distances for which we show the structure of a bi-map. We present numerical evidence that the last result is sharp.
In the Bernoulli model of an elastic rod described by a curve, the elastic energy is given by integral of the curvature squared with respect to arc-length. We study the minimization of this energy on curves given by the graph of a sufficiently smooth function satisfying Dirichlet boundary conditions. Using invariances of the problem, we are able to integrate the Euler-Lagrange equation once in two different ways. To illustrate the idea and the power of the method, we give also another application to unstable Willmore surfaces of revolution.
We will discuss several interacting particle systems for sampling, or more precisely quantization of target measures. Namely instead of seeking an i.i.d. sample of the target measure we look to approximate the target probability distribution by a family of particles.
This can be cast as an optimization problem where the objective functional measures the dissimilarity to the target. This optimization can be addressed by approximating Wasserstein and related gradient flows. We will compare and contrast the Stein Variational Gradient Descent, projected gradient flows and gradient flows of Maximum Mean and Kernel Stein Discrepancy.
In practice, these are simulated by interacting particle systems, whose stationary states define an empirical measure approximating the target distribution. We investigate, theoretically and numerically, quantization properties of these approaches, i.e. how well is the target approximated by the empirical measure. In particular, we will discuss upper bounds on the quantization error of MMD and KSD with various kernels. The talk is based on joint work with Anna Korba and Lantian Xu.
The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature in type II superconductors is the occurrence of vortex lines, which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this talk I will present a sharp estimate of this value and report on a joint work with Etienne Sandier and Sylvia Serfaty in which we study the onset of vortex lines and derive an interaction energy for them. In particular, we will show that this onset of vorticity is directly related to an "isoflux problem" on curves (finding a curve that maximizes the ratio of a magnetic flux by its length).
This talk focuses on a variational approach to explain pattern formation in helimagnetic compounds. Such materials are often modelled in terms of (discrete) frustrated spin systems. We derive (in the sense of Gamma-convergence) a limiting continuum model at the helimagnetic/ferromagnetic transition point and discuss relations to models from the literature for other pattern forming systems. We focus on the case of incompatible boundary conditions for the spin field, and discuss in particular the scaling laws for the minimal energy. The latter indicate that the formation of complex spin patterns is expected in certain parameter regimes. This is based on joint work with Janusz Ginster and Melanie Koser (both Humboldt-Universität zu Berlin).
Motivated by problems from Industrial Mathematics we further developed the concepts of hypocoercivity. The original concepts needed Poincaré inequalities and were applied to equations in linear finite dimensional spaces. Meanwhile we can treat equations in manifolds or even infinite dimensional spaces. The condition giving micro- and macroscopic coercivity we could relax from Poincaré to weak Poincaré inequalities. In this talk an overview and many examples are given.
Dislocations are the physical defects whose motion and interaction are responsible for the plasticity of crystalline solids. The physics can be characterized by a system of nonlinear PDE which does not naturally arise from a variational principle. With a view towards an eventual path integral formulation to understand statistical properties, we describe a first step related to the development of a family of dual variational principles for this primal system with the property that the Euler-Lagrange equations of each of its members is the primal system in a well-defined sense. We illustrate the main idea of the scheme and its viability by applying it to compute approximate solutions to the linear heat, and first-order, scalar wave equations, and 1-d, nonconvex elastostatics. This is joint work with Udit Kouskiya and Siddharth Singh.
We review a series of works that address homogenization for partial differential equations with highly oscillatory coefficients. A prototypical setting is that of periodic coefficients that are locally, or more globally perturbed. We investigate the homogenization limits obtained, first for linear elliptic equations, both in conservative and non conservative forms, and next for nonlinear equations such as Hamilton-Jacobi type equations. The connection between the above theoretical endeavour and strategies for modeling actual materials and simulating them using computational mutiscale approaches will also be addressed.
The works presented have been completed in collaboration with a number of colleagues, in particular with Y. Achdou, X. Blanc, P. Cardaliaguet, P.-L. Lions, P. Souganidis, and R. Goudey.
Understanding the contributions of boundary layers is important in many quantitative homogenization problems: In stochastic homogenization, boundary layer effects impact the accuracy of representative volume approximations for effective material properties. In both periodic and stochastic homogenization, the derivation of higher-order homogenized approximations on bounded domains necessitates an understanding of the homogenization problem for fluctuating boundary data. We give an overview of recent progress on the analysis of boundary layers for elliptic homogenization problems, including in particular decay estimates for boundary layers for random elliptic operators. based on joint works with Peter Bella, Marc Josien, Claudia Raithel
The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition by another class of methods known as threshold dynamics. Based on this connection, we revisit median filters in light of recent work on the threshold dynamics method.
We prove homogenization of Hamilton-Jacobi-Bellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is non-elliptic and its law is non-stationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finite-range dependence (i.i.d.) on the environment. In this set up, we develop an approach based on the coercivity property of the Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB) and make use of the random geometry of continuum percolation. Joint work with Rodrigo Bazaes (Münster) and Alexander Milke (Berlin).
We introduce a probabilistic formulation for the Nash embedding theorems. Our approach inverts the usual relation between mathematics and physics. We use rigorous mathematical results, including Nash’s work, results of De Lellis and Szekelyhidi, and work of the speaker and Rezakhanlou, to guide the design of algorithms and evolution equations. We use relaxation as in Nash’s work, but replace his iteration (in low codimension) or continuous flow (in high codimension) with a stochastic flow. The main issue in the derivation of our flow is a principled resolution of a semidefinite program. The same fundamental structure applies to several hard constraint systems and nonlinear PDE.
Optimal transport asks the question: what is the optimal way to transport a distribution of mass from one configuration to another. One of its variant, regularized optimal transport, is closely connect to large deviation principle and entropy minimization with respect to the law of the Brownian motion (a.k.a. the Schrödinger problem). In short: regularized optimal transport has a neat and fruitful probabilistic interpretation. I will explain what happens when we replace Brownian motion by branching Brownian motion (that is, when particles may also split or die at random instants): the optimal transport counterpart becomes regularized unbalanced optimal transport, enabling to match distributions of unequal mass. This is joint work with Aymeric Baradat, see Arxiv preprint 2111.01666.
The Sineβ point process is a stationary point process that appears as the limit of a certain system of interacting particles with a logarithmic repulsion, the so called one-dimensional log-gas. In this talk, we give a characterisation of the Sineβ process as the unique minimizer of a free energy functional. Our argument is based on the combination of optimal transport ideas for point processes and approximation techniques for log-gases (screening). (Joint work with Matthias Erbar and Thomas Leblé.)
In the limit of vanishing but moderate external magnetic field, we derived a few years ago together with S. Conti, F. Otto and S. Serfaty a branched transport problem from the full Ginzburg-Landau model. In this regime, the irrigated measure is the Lebesgue measure and, at least in a simplified 2d setting, it is possible to prove that the minimizer is a self-similar branching tree. In the regime of even smaller magnetic fields, a similar limit problem is expected but this time the irrigation of the Lebesgue measure is not imposed as a hard constraint but rather as a penalization. While an explicit computation of the minimizers seems here out of reach, I will present some ongoing project with G. De Philippis and B. Ruffini relating local energy bounds to dimensional estimates for the irrigated measure.
The talk is based on joint works with Siva Athreya, Leonid Mytnik and Khoa Le. It is well-known that an SDE $$dX_t = b(X_t) dt +dW_t$$might have a unique solution even if the corresponding noiseless ODE$$dX_t =b(X_t)dt $$has no or infinitely many solutions. This is called regularization-by-noise. While this phenomenon is quite well understood in the case of Brownian forcing, much less is known if the forcing is non-Markovian (for example, fractional Brownian) or infinite-dimensional. This happens not because regularization-by-noise is specific to the Brownian case, but rather because there are (almost) no appropriate tools to study this problem in other setups. We will talk about new technique, stochastic sewing, and its latest modifications, which is surprisingly effective in tackling this problem in the non-Brownian setting.
We consider models in the Kardar-Parisi-Zhang universality class of stochastic growth models in one spatial dimension. We study the correlations in space and time of the height function. In particular we present results on the decay of correlations of the spatial limit processes and on the universality of the first order of the covariance at macroscopically close times.
The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic partial differential equation introduced in physics in 1986. In one spatial dimension, for the KPZ equation on the line, it has been known for a long time that the Brownian measure is stationary. For the equation on an interval or a half-line, however, stationary measures are more complex and their explicit description have been obtained very recently. The method involves the detailed study of integrable probabilistic models that can be viewed as discretizations of the KPZ equation. The talk is based on joint works with Pierre Le Doussal and Ivan Corwin.
Scaling limits of stochastic PDEs which result in a deterministic PDE have many feature in common with hydrodynamic limits of interacting particle systems and singular limits of deterministic PDEs, yet require different techniques. We will explain how to use methods from nonlinear stochastic homogenization. This is based on joint work with Pierre Cardaliaguet and P.E. Souganidis
Martin Hairer’s regularity structures are a recent discovery and, despite their spectacular applications, they are still relatively little known in the mathematical community. Recently, in joint work with F. Caravenna and L. Broux/F. Caravenna we have revisited the main analytical tools of this theory with the aim of making these notions both more general and simpler to understand. The aim of this talk is to present the main ideas of this construction.
We present two frequency separation approaches in the study of Navier-Stokes equation. One is designed by accounting the competition between the nonlinear term and the linear dissipation term. It leads to improved regularity criterion which only poses condition on the low modes part of the solution. The other one arises from the parabolic structure of the linear part of the Navier-Stokes equation, which can be used to study the long time behavior of the solution.
Loop-erased random walk (LERW) is a model for a random simple path, which is created by running a simple random walk and, whenever the random walk hits its path, removing the resulting loop and continuing. LERW was originally introduced by Greg Lawler in 1980. Since then, it has been studied extensively both in mathematics and physics literature. Indeed, LERW has a strong connection with other models in statistical physics, especially the uniform spanning tree which arises in statistical physics in conjunction with the Potts model. In this talk, I will talk about some recent progress on LERW while focusing on the three-dimensional case. This is joint work with Xinyi Li.
We present a novel method developed for the 1-d Cahn Hilliard equation in Otto, Scholtes, and W. (2019) and extended in Biesenbach, Schubert, and W. (2021). We also mention work in progress with Otto and Schubert in which this method is applied to the Mullins Sekerka evolution in dimensions 2 and 3.
The discrete Yang Mills measure was introduced by Wilson (1974), who made several interesting conjectures about the observables of the theory, now known as Wilson loops. Shortly after, `t Hooft (1974) discovered the 1/N expansion of the theory with structure group U(N), inspiring a large body of physics literature known as 'large N problems'. More recently, Chatterjee (2015) studied the large N limit rigorously in the strong coupling regime, and a key tool is the finite N master loop equation, a recursive formula satisfied by Wilson loops. In this talk, I will discuss the Langevin dynamic associated to the discrete Yang Mills measure and show that Chatterjee's loop equation can be derived as a simple consequence of Ito's formula. This is joint work with Hao Shen and Rongchan Zhu.
This talk aims to give an overview on various notions of Lyapunov exponents (LEs) in random dynamical systems, depending on the timescale one wants to study: from finite-time LEs to classical asymptotic LEs and corresponding spectra up to LEs for processes conditioned on staying in bounded domains. We demonstrate how these notions become relevant in the context of stochastic bifurcations, in finite and infinite dimensions.
We study speeds of fronts in bistable, spatially inhomogeneous media at parameter regimes where speeds approach zero. We provide a set of conceptual assumptions under which we can prove power-law asymptotics for the speed, with exponent depending a local dimension of the ergodic measure near extremal values. We also show that our conceptual assumptions are satisfied in a context of weak inhomogeneity of the medium and almost balanced kinetics. The talk is based on a joint work with A. Scheel
Nonconvex and nonlocal variational problems are pervasive in energy-driven pattern formation. Two central issues are: (Q1) can one conjecture and prove asymptotic statements on the (geometric) nature of global minimizers. (Q2) can one develop systematic, hybrid numerical algorithms to navigate (or probe) the energy landscape and access low energy states whose basin of attraction might be "tiny". In this talk, we will mostly explore (Q1) in the context of the simple, yet rich, paradigm of optimal quantization and optimal centroidal Voronoi tessellations (CVT) on a 3D Euclidean domain and the 2-sphere. Gersho's conjecture, which may be viewed as a crystallization conjecture, asserts the periodic structure, as the number of generators tends to infinity, of the optimal CVT. In joint work with Xin Yang Lu (Lakehead University) we present certain bounds which, combined with a 2D approach of P. Gruber (following L. Fejers Toth), reduce the resolution of the 3D Gersho's conjecture to a finite (albeit very large) computation of an explicit convex problem in finitely many variables. We then discuss the analogous problem on the 2-sphere. We further address and support some interesting numerical observations about defect structures vs. lattice CVTs for the optimal CVT with a "small" number of generators on the 2D square torus. This is joint work with I. Gonzalez, C, Mantos, J. Tisdell and JC Nave at McGill.Finally (time permitting), we will mention work (with I. Gonzalez and JC Nave) pertaining to issue (Q2) by presenting a new hybrid algorithm which alternates gradient descent with movement away from the closest generator.
The Einstein equations describe the dynamics of space-time in general relativity. It is well-known that — analogous to the case of Maxwell’s equations — initial data for the Einstein equations needs to satisfy constraint equations. One approach to study the rigidity and flexibility of the Einstein equations is by considering gluing problems for initial data. The so-called spacelike gluing problem for initial data on slices of constant time has been intensively studied by Riemannian geometers. In this talk I will present recent work with S. Aretakis and I. Rodnianski, where we introduce the so-called characteristic gluing problem for initial data along light cones (i.e. characteristic hypersurfaces for the Einstein equations). The characteristic gluing problem is fundamentally different from the spacelike problem, and displays novel rigidity and flexibility features. We moreover show how to apply our characteristic gluing to prove gluing constructions for spacelike initial data. Towards the end of the talk, I will discuss future directions.
We consider Potential Mean Field Games and are focusing on the (deterministic/stochastic) HJB. In order to compute a semi-global solution, we consider first a Lagrangian perspective which is related to dynamical programing. We consider control affine dynamical systems and quadratic cost for the control. For many high dimensional PDEs of practical interest, e.g. Backward Kolmogorov equations, HJB etc., the PDE operator cannot be easily expanded in tensor form. In this case, we propose a machine learning approach confined to the manifold of tree based tensors with fixed multi-rank. We compare a Lagrangian approach with an Eulerian method. In the Lagrangian picture we apply policy iteration and solve the linearized HJB by integrating along trajectories, defined by the corresponding dynamical system (characteristics) for samples of initial values. From the computed point values we infer the sought value function. In the stochastic case we have many paths instead of a single trajectory. There the HJB can be reformulated by an (uncoupled) Forward Backward SDE system. The forward dynamics can be computed easily by standard Euler-Mayurana scheme. For the backward equation for the value function, we use variational interpolation (Bender et al.) by solving a regression problem in each time step. For this purpose we use e.g. tree based tensor networks, in particular MPS/TT, and/or Neural Networks. The forward backward SDE is linked with (parabolic) PDEs by a non-linear Feynman-Kac theorem. Solving regression problems by means of HT/TT tensors with good approximation of the gradients requires additional attention, and has been the technical key for a successful treatment. Joint work with M. Oster and L. Sallandt.
In this talk, we will discuss several algebraic structures relevant to the concise description of relations between the R- and S-transforms in non-commutative probability theory. This includes shuffle, pre- and post-Lie algebras. We will focus on Dykema’s so-called twisted factorisation formula for Voiculescu's S-transform, which can be understood in an operadic framework.
Based on joint work F. Patras (CNRS, Nice, FR) and N. Gilliers (IMT, Toulouse, FR).
In 1905, Albert Einstein published two papers that led to an experiment to measure the Avogadro number. The first paper studies the Brownian motion and establishes the Einstein relation in kinetic theory. The second paper studies a Stokes fluid with a suspension of rigid particles, and establishes the Einstein formula for the effective viscosity of this "complex fluid". The Avogadro number, which appears in both relations, can then be deduced from two measurements of fluid mechanics. Performed by Jean Perrin in 1907-1909, these experiments confirmed the atomistic nature of matter (for which he was awarded a Nobel prize in 1926) using classical fluid mechanics! The aim of the talk is to give an elementary (quantitative and robust) proof of Einstein's effective viscosity formula. This is a joint work with Mitia Duerinckx (ULB).
In this talk, we will present the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant properties. We describe in particular the renormalised equation for a very large class of spacetime dependent renormalization schemes. Our approach is based on local renormalisation maps that act directly at the level of the equation. This is joint work with Ismael Bailleul.
Community detection in complex networks has been very successful in understanding several systems where interactions are inherent. First, I introduce a novel trajectory-based method for identifying and predicting subtypes in heterogeneous and longitudinal data, i.e., that are characterized by time-varying interactions between various factors. The conventional Laplacian encodes many dynamical properties of a network, including community structure and flow of information along the network. Through spectral community detection in the graphical domain, I perform community detection on trajectory-based networks, for the application of identifying and predicting subtypes of diseases several years in advance.
While networks are a useful tool to represent data in the graphical domain, most systems naturally evolve to contain simultaneous interactions between more than two entities, represented as simplices - triangles, tetrahedra etc. Here I present the first work on revealing the relationship between a higher order equivalent of the Laplacian (Hodge Laplacian) and higher-order simplicial communities, demonstrating our results on both synthetic networks, as well as social and language networks.
I discuss the implications of Hodge decomposition on simplicial communities, and their relationship with clique communities. Lastly, I present a method to infer higher-order simplicial complexes from the network backbone, a question of some importance as simplicial datasets are rare.
Mean curvature flow is one of the most fundamental geometric evolution equations and appears in many surface-tension driven problems. Although the equation has an instantaneous smoothing effect, generically, singularities appear in finite time. One is led to consider weak solutions which persist through these singular events. Folklore says that mean curvature flow is a gradient flow with the caveat that the underlying metric is completely degenerate. In this talk, after discussing known weak notions of solution, I will present a new concept which has its roots in the theory of gradient flows and relies on basic geometric measure theory. I will show that these solutions arise naturally in the sharp-interface limit of the Allen-Cahn equation and in addition satisfy a weak-strong uniqueness principle. The latter property is a fundamental difference to well-known Brakke solutions, which a priori may disappear at any given time and are therefore fatally non-unique.
This is joint work with Sebastian Hensel (U Bonn).
Complex microstructures often involve multi-scale heterogeneous textures, that can be modelled by random closed sets derived from Mathematical Morphology [1]. Starting from 2D or 3D images, a complete morphological characterization is performed, and used for the identification of a probabilistic model of random structure.
This presentation briefly reviews some random models and their probabilistic properties, illustrated by examples of application and by simulations. Extensions of the Boolean random closed sets model provide multi-scale models:
Cox Boolean models, long range random sets generated by Boolean varieties, iteration of Poisson varieties, sequential Cox Boolean models.
Simulations of realistic microstructures generated by these models can be introduced in a numerical solver to compute appropriate fields (electric, elastic, velocity,...) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields.
[1] Jeulin D. (2021) Morphological Models of Random Structures, Springer.
Turbulent thermal convection which occurs due to the temperature differences imposed at the boundaries of the domain is omnipresent in nature in technology. In this talk, we will focus on Rayleigh-Benard convection but also consider some other flow configurations: vertical and horizontal convection. For these systems, we will discuss the problem of scaling relations of the global heat and momentum transport, the proper boundary layer equations, and the large-scale flow organization. We will show that in some cases, the scaling of the global heat transport and the temperature profiles can be predicted by solving the proper boundary-layer equations that include turbulent fluctuations and correct boundary conditions. For Rayleigh-Benard convection, we will discuss also the onset of convection in slender containers, the proper length scales, and the formation of global flow structures in turbulent regimes.
Kerr solutions to Einstein's equations of general relativity describe stationary, rotating black holes in vacuum. When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes can be thought of as critical solutions, featuring dynamics that are rich with interesting phenomena. I will introduce upcoming work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss the connection of instabilities with the precise shape of late-time power law tails in the emitted radiation.
This talk is about vortex-like structures in planar ferromagnetic materials which are stabilised by the so-called Dzyaloshinski-Moryia (DM) interaction. The vortex structures, called magnetic skyrmions in this context, are widely studied in physics because of their potential role in future magnetic information storage devices. The DM term is linear in the gradient of the magnetisation field and can be interpreted in terms of a non-abelian gauge field. The applicability of techniques from complex geometry and gauge theory is rather surprising in this model, and leads to an infinite family of exact solutions for special values of coupling constants. I will discuss the solutions and their relation to the variational equations at generic values of couplings, and also comment on the dynamics of magnetic skyrmions in response to an applied current.
We consider an optimal transport problem arising from stopping the Brownian motion from a given distribution to get a fixed or free target distribution; the fixed target case is often called the optimal Skorokhod embedding problem in the literature, a popular topic in math finance pioneered by many people. Our focus is on the case of general dimensions, which has not been well understood. We explain that under certain natural assumptions on the transportation cost, the optimal stopping time is given by the hitting time to a barrier, which is determined by the solution to the dual optimization problem. In the free target case, the problem is related to the Stefan problem, that is, a free boundary problem for the heat equation. We obtain analytical information on the optimal solutions, including certain BV estimates. The fixed target case is mainly from the joint work with Nassif Ghoussoub and Aaron Palmer at UBC, while the free target case is the recent joint work (in-progress) with Inwon Kim at UCLA.
We study the stability of a 1D shock for the p-system (the 1D isentropic Euler equation in the Lagrangian space). We show that such a shock is stable with respect to initial values perturbations in the energy space, in the class of inviscid limits of Navier-stokes equation.
The result is based on the theory of a-contraction with shifts for viscous shocks of Navier-Stokes equation. The method allows to show the uniform stability with respect to the viscosity. Stability results on the inviscid model are then inherited at the inviscid limit, thanks to the fact that large perturbations, independent of the viscosity, can be considered at the Navier-Stokes level. These stability results hold in the class of wild perturbations of inviscid limits, without any regularity restriction (non even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.
This is a joint work with Moon-Jin Kang.
In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a quantitative version of Hörmander’s hypoelliptic regularity theory in an L1 framework which estimates this Fisher information from below by a fractional Sobolev norm using the Kolmogorov equation. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and that this class includes the Lorenz 96 model in any dimension greater than or equal to 7, provided the additive stochastic driving is applied to any consecutive pair of unknowns. This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for stochastically driven Lorenz 96, despite the overwhelming numerical evidence (the deterministic case remains far out of reach). If time permits, I will discuss the application of the method to prove similar results for finite dimensional truncations of the classical shell models of hydrodynamic turbulence, GOY and SABRA.
The (time-dependent) continuum mechanics of dislocations at unrestricted geometric and material nonlinearity will be discussed. The theory will be illustrated with two results pertaining to dislocation pattern formation observed in experiments: i) dislocation cell formation in a time-dependent context describing collective mesoscale behavior ii) the phenomenon of polygonization involving dislocation walls, in a static context. Contact will be made with existing mathematical results.
This is joint work with Rajat Arora.
The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.
We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show - in a quantitative sense - that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general.
This is joint work with Max Engelstein and Luca Spolaor.
Variational problems are concerned with determining the state of minimal energy (broadly construed) of a given system within a competition class of admissible states the system can achieve. It is known that solutions to variational problems arising in Geometry and Physics may, in general, exhibit singularities. A fine analysis of the size and structure of such singular sets is of pivotal importance, both from the purely theoretical perspective, and in view of the applications, in particular as a confirmation of the suitability of the variational model towards a correct description of the observed phenomena.
In this talk, I will describe my work on a variety of aspects concerning the physical relevance, the analytic properties, and the evolution of the singular structures arising in the solutions to some geometric variational models pertaining to the description of physical systems governed by surface tension-type energies, with an emphasis on (measure-theoretic generalizations of) minimal surfaces and mean curvature flows.
I will discuss recent results on localization for the Anderson--Bernoulli model. This will include my work with Ding as well as work by Li--Zhang. Both develop new unique continuation results for the Laplacian on the integer lattice. One day before the seminar, an announcement with the link will be sent to the mailing list of the Oberseminar. If you are not on the mailing list, but still want to join the broadcast, please contact Jonas Sauer.
In this talk, we will focus on the standard Ginzburg-Landau functional for N-dimensional maps defined in the unit ball that are equal to the identity on the boundary. A special critical point is the so-called degree-one vortex map given by the identity map multiplied with a scalar radial profile. We will prove the minimality of this solution and also discuss about the uniqueness result. This is a joint work with L. Nguyen, V. Slastikov and A. Zarnescu.
One day before the seminar, an announcement with the link will be sent to the mailing list of the Oberseminar. If you are not on the mailing list, but still want to join the broadcast, please contact Jonas Sauer.
In the talk we discuss some recent results on self-improving properties for gradients of solutions for parabolic evolutions with fast or slow diffusion. The model case is the porous medium equation. We show how local higher integrability estimates can be derived via the celebrated Gehring lemma. The estimates relay on a Calderon Zygmund theory that is developed with respect to an intrinsic metric that depends on the solution; taking into account the local speed of the diffusion. The concept turns out to be flexible enough to show self-improving properties for large classes of diffusions depending on the solution and the gradient. One day before the seminar, an announcement with the link will be sent to the mailing list of the Oberseminar. If you are not on the mailing list, but still want to join the broadcast, please contact Jonas Sauer.
Imagine a particle flying through a dense gas, interacting with the particles of that gas. Due to the interaction the particle will experience dissipation and fluctuation. Both effects will typically increase as the density goes to infinity.
While this is true for a classical gas and also for Bose gases, the behaviour is very different for gases of Fermions: A charged particle moving through a Fermi sea of high density behaves almost like a free particle.
Here the Fermi pressure leads to a suppression of the fluctuations in the gas and eventually a suppression of fluctuation and dissipation.
While this is easy to prove in one dimension, the two dimensional case is highly non trivial. I will present recent results on this question.
The obstacle problem arises in several important physical models. We will present some recent work in collaboration with A. Figalli and X. Ros-Oton on the structure of the singular set for this problem.
We will start introducing some rather recent tools for the analysis of singularities in the obstacle problem, which are complementary to the classical theory of Caffarelli. These tools exploit a useful connection between singularities of the obstacle problem and solutions of the so-called thin obstacle problem.
With careful enough analysis, we are able to achieve a precise understanding of the behavior of solutions near "generic" singularities.
In particular we prove that the free boundary is generically smooth in dimensions 3 and 4, while in higher dimensions the singular set has, generically, co-dimension 3 inside the free boundary.
A while ago Nadirashvili proposed a beautiful idea how to attack problems on zero sets of Laplace eigenfunctions using quasiconformal mappings, aiming to estimate the length of nodal sets (zero sets of eigenfunctions) on closed two-dimensional surfaces. The idea have not yet worked out as it was planned. However it appears to be useful for Landis' Conjecture. We will explain how to apply the combination of quasiconformal mappings and zero sets to quantitative properties of solutions to $\Delta u + V u =0 on the plane, where $V$ is a real, bounded function. The method reduces some questions about solutions to Shrodinger equation $\Delta u + V u =0$ on the plane to questions about harmonic functions. Based on a joint work with E.Malinnikova, N.Nadirashvili and F. Nazarov.
In this talk I will survey recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, which was originally due to Hatcher. Second, we show that the space of metrics with positive scalar curvature on every 3-manifold is either contractible or empty. This completes work initiated by Marques.
Our proof is based on a new uniqueness theorem for singular Ricci flows, which I have previously obtained with Kleiner. Singular Ricci flows were inspired by Perelman’s proof of the Poincaré and Geometrization Conjectures, which relied on a flow in which singularities were removed by a certain surgery construction. Since this surgery construction depended on various auxiliary parameters, the resulting flow was not uniquely determined by its initial data. Perelman therefore conjectured that there must be a canonical, weak Ricci flow that automatically "flows through its singularities" at an infinitesimal scale. Our work on the uniqueness of singular Ricci flows gives an affirmative answer to Perelman's conjecture and allows the study of continuous families of singular Ricci flows leading to the topological applications mentioned above.
One day before the seminar, an announcement with the link will be sent to the mailing list of the Oberseminar. If you are not on the mailing list, but still want to join the broadcast, please contact Jonas Sauer.
We derive a sufficient condition under which a version of Kolmogorov's 4/5 law can be rigorously proved for stationary solutions of the 3D stochastic Navier-Stokes equations. We name this condition 'weak anomalous dissipation condition'. A similar condition allows to prove flux scaling laws for the 2D stochastic Navier-Stokes equations, including a scaling law for the inverse cascade. We also derive necessary conditions which are needed for the same scaling laws to hold. One day before the seminar, an announcement with the link will be sent to the mailing list of the Oberseminar. If you are not on the mailing list, but still want to join the broadcast, please contact Jonas Sauer.
Understanding fluid instabilities on micro and nanoscale is relevant for a variety of reasons. From scientific point of view, modeling systems involving fluid-solid interfaces is challenging. From mathematical side, there are significant challenges involved in formulating well-defined models, particularly in the settings that are more complex than simple Newtonian films. Numerous applications of thin films provide further motivation for their studies, in particular regarding fluid film evolution and resulting instabilities whose understanding is crucial for making progress in the field of self- and directed- assembly on nanoscale.
This talk will focus on recently developed asymptotic models and computational techniques for thin films. The models to be considered include long-wave asymptotic approach as well as full Navier-Stokes based models. Both types of models have been augmented to explicitly include fluid/solid interaction forces via disjoining pressure approach. The simulation techniques include algorithms for GPU computing that allow for simulations of large domains and detailed analysis of various instability mechanisms within long-wave approach, as well as volume-of-fluid based simulations of Navier-Stokes equations. Two case studies will be discussed: (i) Liquid crystal films, for which the challenge is to include liquid-crystalline nature of the fluid in the model in a tractable manner, and (ii) Liquid metal films irradiated by laser pulses; in this case, one of the challenges is to include complex thermal effects into consideration and understand their influence on the film instability and resulting pattern formation.
Particular issues that will be considered include the influence of the initial geometry on the instability development, Marangoni effects, and the instabilities in the case of multi-fluid configurations.
The Gross--Pitaevskii equation is a nonlinear Schrödinger equation that models the behavior of a Bose-Einstein condensate. The quantum vortices of the condensate are defined by the zero set of the wave function at time $t$. In this talk we will present recent work about how these quantum vortices can break and reconnect in arbitrarily complicated ways. As observed in the physics literature, the distance between the vortices near the breakdown time, say t=0, scales like the square root of t: it is the so-called $t^{1/2}$ law. At the heart of the proof lies a remarkable global approximation property for the linear Schrödinger equation. The talk is based on joint work with Daniel Peralta-Salas.
In this talk I will present a quantitative version of Alexandrov theorem with an application to the the long-time asymptotics of the discrete volume preserving mean curvature flow.
The Logarithmic Laplacian Operator arises as formal derivative of fractional Laplacians at order s= 0. In this talk I will discuss properties of this operator and its relevance in the derivation of asymptotics of Dirichlet eigenvalues and eigenfunctions of fractional Laplacians in bounded domains as the order tends to zero. A further applications arises in the study of the monotonicity of solutions to fractional Poisson problems with respect to the fractional order s. As a byproduct of this study, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which seem to be new even for the case s=1, i.e., for the classical local Dirichlet problem −Δu=f in Ω, u≡0 on ∂Ω.
This is joint work with Huyuan Chen, Sven Jarohs and Alberto Saldana.
Inverse problems research concentrates on the mathematical theory and practical implementation of indirect measurements. Applications are found in numerous research fields involving scientific, medical or industrial imaging; familiar examples include X-ray computed tomography and ultrasound imaging. Inverse problems have a rich mathematical theory employing modern methods in partial differential equations (PDEs), harmonic analysis, and differential geometry.
In this talk we outline a recent approach to develop general theory for inverse problems for PDEs (real principal type equations in particular). The work presents a unified point of view to inverse boundary value problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.
This is joint work with Lauri Oksanen (UCL), Plamen Stefanov (Purdue) and Gunther Uhlmann (Washington / IAS HKUST).
Employing a two-dimensional model for ferromagnetic thin-films and using standard methods of $\Gamma$-convergence we derive a reduced local model for thin films with perpendicular anisotropy.
We consider elliptic equations in the double-divergence form and perform a two-fold analysis. First, we present results on the improved regularity of the solutions along zero level-sets. In this context, solutions become asymptotically Lipschitz; under further conditions on the data of the problem, we produce asymptotically-Lipschitz regularity for the gradient of the solutions. A second instance of analysis regards the geometric properties of nodal sets; our findings concern the (local) regularity and the Hausdorff dimension of those sets. We close the talk with a discussion on applications to solid mechanics, as well as with comments on further directions of research.
One of the most interesting features of Erdös-Rényi random graphs is the `percolation phase transition', where the global structure intuitively changes from only small components to a single giant component plus small ones. In this talk we discuss the percolation phase transition in the random d-process, which corresponds to a natural algorithmic model for generating random regular graphs that differs from the usual configuration model (starting with an empty graph on n vertices, the random d-process evolves by sequentially adding new random edges so that the maximum degree remains at most d). Our results on the phase transition solve a problem of Wormald from 1997, and verify a conjecture of Balinska and Quintas from 1990.
Based on joint work with Nick Wormald.
In this talk I will give an overview on the few results available on the conjecture of Almgren regarding the convexity of drops subject to the action of an external convex potential. In particular I will present recent progress in this direction obtained with G. De Philippis on their connectedness. Together with an older result of McCann, this answers positively the conjecture in dimension two. The proof is inspired by the two-point function technique introduced by B. Andrews and is reminiscent of the doubling of variables trick in the context of viscosity solutions.
The flow equations of the renormalisation group are a universal tool to rigorously analyse perturbative quantum field theories. We give a short overview and then present (partly astonishing) results of an analysis beyond perturbation theory, in the mean field limit.
The asymmetric simple exclusion process (ASEP) is the evolution of a collection of particles on the integer lattice; particles interact according to simple rules and can be of various colors (equivalently, classes). In 2008 Amir-Angel-Valko established an interesting property of such processes: the color-position symmetry. We will discuss a generalization of this result and its new applications to the asymptotic behavior of this class of models.
Recently, D. Serre proposed the framework of divergence-free positive definite symmetric tensors (DPTs) to obtain better a priori estimates for various compressible fluid models. He proved, in particular, a Jensen-type inequality for the determinant to the power 1/(d-1), which suggests that a weak semicontinuity result could be obtained by virtue of by now standard methods of Fonseca-Müller. However, the positivity constraint excludes these methods and requires a novel approach. We elaborate such an approach within a very general framework. Joint work with Jack Skipper.
Evolution equations in spaces of measures describe a wide variety of natural phenomena. The theory for such evolutions has seen tremendous growth in the last decades, of which resulted in general metric space theories for analysing variational evolutions---evolutions driven by one or more energies/entropies. On the other hand, physics and large-deviation theory suggest the study of \emph{generalised} gradient flows---gradient flows with non-homogeneous dissipation potentials---which are not covered in metric space theories. In this talk, we introduce dynamical-variational transport costs (DVTs)---a large class of large-deviation inspired functionals that provide a variational generalisation of existing transport distances---to remedy this deficiency. The role in which these objects play in the theory of \emph{generalised} gradient flows will be illustrated with an example on Markov jump processes. Finally, open questions and challenges will be mentioned.
Modelling of global ocean dynamics is based on the Ocean Primitive Equation. These equations describe the ocean as an incompressible two-component fluid under the Boussinesq and the hydrostatic approximation. We describe a structure-preserving discretization of the Ocean Primitive Equations. This discretizations forms the foundation of the ocean general circulation model ICON-O. ICON-O is the ocean component of Max-Planck Institute for Meteorology's newly developed Earth System Model ICON-ESM and the ocean model of the ICON modelling system. The novel numerical approach of ICON-O rests on a discrete weak form that allows to control spurious modes in a way that is compatible with discrete conservations laws. We present a numerical analysis of the discrete algorithm and an experimental evaluation. Global ocean simulation are carried out and compared to observations in order to to demonstrate the physical soundness of the model.
In this joint work with Massimilano Gubinelli and Hiro Oh we prove local wellposedness for a renormalized wave equation with additive white noise. Key ingredients are an ansatz going back to Da Prato and Debussche, a paraproduct decomposition, and an analysis of various stochastic fields and of a stochastic operator.
We review an abstract and simple uncertainty principle from 'A. Boutet de Monvel, D. Lenz and P. Stollmann: An uncertainty principle, Wegner estimates and localization near fluctuation boundaries. Math. Z. 2010' and its application to graphs and divergence form operators with rough coefficients based on
D. Lenz, P. Stollmann and Gunter Stolz: An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs, arXiv: 1606.07476P. Stollmann and Gunter Stolz: Lower bounds for Dirichlet Laplacians and uncertainty principles, arXiv:1808.04202
We will review the main ideas behind our recent proof of the central limit theorem for the capacity of the range in dimension 5, and try to emphasize the specifities of this dimension. In particular estimates on probabilities of intersection for two independent ranges play a crucial role.
I will give a brief overview of very simple, hence maybe less investigated structures in interacting particle systems: reversible product blocking measures. These turn out to be more general than most people would think, in particular asymmetric simple exclusion and nearest-neighbour asymmetric zero range processes both enjoy them. But a careful look reveals that these two are really the same process. Exploitation of this fact gives rise to the Jacobi triple product formula - an identity previously known from algebra, combinatorics and number theory. I will show you the main steps of deriving it from pure probability this time, and I hope to surprise my audience as much as we got surprised when this identity first popped up in our notebooks.
We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is resampled at every unit of time; (II) the random environment is sampled at time zero only (i.e. the well-known RWRE model where the environment is static/frozen). In our model the random environment is resampled along a determinstic sequence of refreshing times.
Depending on the choice of these refreshing times the resulting long-term behavior can be close or not to the two interpolated models. We will make this clear as far as recurrence, asymptotic speed, fluctuations and large deviations are concerned. For those questions we show the emergence of a richer palette of behaviors with respect to the two interpolated models. In particular we will identify scenarios where either homogenization or localization occur, giving rise to qualitative different asymptotics, e.g. classical diffusive Gaussian regimes versus sub-diffusive regimes with mixed limiting laws.
The general philosophy is to explore how a well-understood stochastic process with a rich correlation structure gets affected by adding noise through local independence.
Joint work with Y. Chino, C. da Costa and F. den Hollander
Although numerical approximation, statistical inference and learning are traditionally seen as entirely separate subjects, they are intimately connected through the common purpose of making estimations with partial information. This talk is an invitation to explore these connections from the consolidating perspective of game/decision theory and it is motivated by the suggestion that these confluences might not just be objects of curiosity but can constitute a pathway to simple solutions to fundamental problems in all three areas. We will illustrate this point through problems related to numerical homogenization, operator adapted wavelets, computation with dense kernel matrices and to the kernel selection/design problem in Machine Learning. In these interplays, accurate reduced/multiscale models (for PDEs) can be identified as optimal bets for adversarial games describing the process of computing with partial information. Moreover, efficient kernels (for ML) can be selected by using relative energy content at fine scales (with a notion of scale corresponding to the number of data points) as an ordering criterion leading to the identification of (data driven) flows in kernel spaces (Kernel Flows), that (1) enable the design of bottomless networks amenable to some degree of analysis (2) appear to converge towards kernels with good generalization properties.
This talk will cover joint work with F. Schäfer, C. Scovel, T. Sullivan, G. R. Yoo and L. Zhang.
In nonlinear periodic media of arbitrary dimension $d$ we consider the small amplitude asymptotics of wavepackets. The wavepackets have $N \in \mathbb{N}$ carrier Bloch waves of equal frequency. We use the cubic Gross-Pitaevskii equation (GP) as a prototype of the governing equation. We discuss two classical asymptotic scalings, one (for $N=1$) leading to the nonlinear Schroedinger equation and one (for $N>1$) leading to first order coupled mode equations (CMEs) as effective amplitude equations. Both of these models can support solitary waves - thus predicting nearly solitary waves of the GP. In particular, the CMEs for $d=1$ for the case of the coupling of two counter-propagating Bloch waves support a family of solitary waves parametrized by the velocity $v\in (-1,1)$. Can this be generalized to $d$ dimensions such that in the CMEs a solitary wave family parametrized by $\vec{v}\in (-1,1)^d$ exists? Solitary waves are typically found in spectral gaps. For $d\geq 2$ at least four ($N=4$) carrier waves are needed to produce CMEs with a spectral gap that supports solitary waves. However, only standing solitary waves have been found so far. We also provide a validity result of the $d-$dimensional NLS-asymptotics as well as the CME-asymptotics over asymptotically large time intervals.
I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. I will discuss some recent results on existence/non-existence of solutions to hyperbolic Monge-Ampere equations with various degrees of regularity, with an emphasis on numerical methods for constructing "rough" solutions. I will then discuss some applications of our results to (i) the occurrence of "geometric" defects that are invisible to the energy, but play a crucial role in determining the global morphology, (ii) a generalization of the Sine-Gordon equation to describe "rough" hyperbolic surfaces with constant negative curvature, and (iii) the important role of regularity in quantitative versions of the Hilbert-Efimov theorem on the nonexistence of C^2 isometric immersions of the Hyperbolic plane into R^3, and (iv) studying the mechanics of leaves, flowers, and sea-slugs.This is joint work with Toby Shearman and Ken Yamamoto.
We study a system of cross-diffusion equations that results, as a formal limit, from an interacting particle system with multiple species. In the first part of the talk we exploit the (formal) gradient flow structure of the system to prove the existence of weak solutions. This is based on a priori bounds obtained from the dissipation of the corresponding entropy and the use of dual variables. In the second part, we discuss strong solutions and a weak-strong stability result under certain conditions on the diffusion constants.
The contact process is a model for the spread of an infection in a population. At any given point in time, vertices of a graph (interpreted as individuals) can be either healthy or infected; infected individuals recover at rate 1 and transmit the infection to neighbours at rate lambda. On finite graphs, the infection eventually disappears with probability one. In many cases, the time it takes for this to occur depends sensitively on the value of the parameter lambda, and this finite-volume phase transition can be linked to a phase transition of the contact process on a related infinite graph. We will survey recent works in this direction, including some general results which hold on large classes of graphs, as well as results on specific graphs, such as finite d-ary trees and the random graph known as the configuration model.
This talk includes joint work with M. Cranston, T. Mountford, J.C. Mourrat, Bruno Schapira and Q. Yao.
The three-dimensional Navier-Stokes equations describe the motion of incompressible viscous fluids. They date back to the 19th century. A breakthrough in their mathematical analysis came from the pioneering work of Leray in 1934. As far as we know, many questions about the behavior of solutions remain open, notably the uniqueness of weak solutions and their regularity or finite time blow-up.
In this talk, we will survey some aspects of the regularity theory for the Navier-Stokes equations. The other side of the coin is finding necessary conditions for solutions developing finite time singularities. In a recent work with Y. Maekawa (Kyoto University) and H. Miura (Tokyo Tech) we found a concentration phenomenon for blowing-up solutions. We will explain this result and a strengthened version, which is work in progress with T. Barker (ENS Paris).
On a non elementary, Gromov hyperbolic group of conformal dimension less than two - say a surface group - we consider a symmetric probability measure whose support generates the whole group and with a finite second moment. To such a probability mesaure, we associate a Besov space on the boundary of the group. All the Besov spaces constructed that way turn out to be isomorphic. They are also isomorphic to the Besov spaces already constructed by M. Bourdon and H. Pajot using the cohomology of the group. In our work, cohomology spaces are replaced by martingales.
We also provide a probabilistic interpretation of the Besov spaces as the Dirichlet spaces of the trace on the boundary of the random walks reflected on the boundary. Along with the definition of Besov spaces, come notions of sets of zero capacity, smooth measures and measures of finite energy on the boundary. Using heat kernel estimates, we obtain an integral criterion for a measure on the boundary to have finite energy.
We apply this criterion and deviation inequalities to show that harmonic measures of random walks with a driving measure with a finite first moment are smooth measures in our sense, thus obtaining a quite general regularity property of harmonic measures.
P. Mathieu (Aix-Marseille) and Tokushige Y. (Kyoto)
We study the impact of small additive space-time white noise on nonlinear stochastic partial differential equations on unbounded domains close to a bifurcation, where an infinite band of eigenvalues changes stability due to the unboundedness of the underlying domain. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, and we rely on the approximation via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains.
We study the stochastic one-dimensional Swift-Hohenberg equation on the whole real line, where the dominant mode is well approximated by a stochastic Ginzburg-Landau equation. In this setting, because of the weak regularity of solutions, the standard methods for deterministic modulation equations fail, and we need to develop new tools to treat the approximation.
One main technical problem for establishing error estimates comes from the spatially translation invariant nature of the noise, which causes the error to be always very large somewhere far out in space. Thus we need to work in weighted spaces that allow for growth at infinity. Using energy estimates we are only able to show that solutions of the stochastic Ginzburg-Landau equation are Hölder-continuous in spaces with a very weak weight, which provides just enough regularity to proceed with the error estimates.
The Bakry-Emery theorem states that if a probability measure is in some sense more log-concave than the standard Gaussian measure, then certain functional inequalities (such as the Poincare inequality and the logarithmic Sobolev inequality) hold, with better constants than for the associated Gaussian inequalities. I will show how we can combine Stein's method and simple variational arguments to show that if the Bakry-Emery bound is almost sharp for a given measure, then that measure must almost split off a Gaussian factor, with explicit quantitative bounds. Joint work with Thomas Courtade.
In this joint work with Amandine Aftalion we study an energy functional in two-dimensions describing a rotating two-component Bose-Einstein condensate. The mathematical difficulty in this model is that it exhibits defects which are both 1-dimensional (curves) and 0-dimensional (vortices).
The inverse conductivity problem, posed by A.P. Calderon in 1980, consists in determining the coefficient $A$ in the elliptic PDE $div(A \nabla u) = 0$ from the Cauchy data of its solutions. This problem is the mathematical model for Electrical Impedance Tomography. Various harmonic analysis, PDE and geometric techniques come into play in its study, and the Calderon problem remains a central question in the theory of inverse problems. We will survey known results and open questions, focusing on issues with low regularity, partial data and matrix coefficients.
One of the most important areas of applied analysis is in the development of robust bounds for physically motivated evolution equations. When the equations feature prominent nonlinear/nonlocal effects (which are notoriously difficult to handle), such bounds can nevertheless recover certain asymptotic properties that simplify the problem or even the equations themselves.
The focus of this lecture will be on recent results for three physical models: homogenization and asymptotics for nonlocal reaction-diffusion equations, a priori bounds for hydrodynamic equations with thermal effects, and the local well-posedness for the Landau equation. Each problem presents unique challenges arising from the nonlinearity and/or nonlocality of the equation(s), and the emphasis will be on the different methods and techniques used to treat these difficulties. The talk will touch on novelties in viscosity theory and precision in nonlocal front propagation for reaction-diffusion equations, as well as the emergence of dynamic self-regularization in the thermal hydrodynamic and Landau equations.
Molecular quantum dynamics are highly oscillatory and move in high dimensional configuration spaces. The talk reviews several numerical methods based on semiclassical approximations and presents part of their convergence theory.
We consider a divergence-form elliptic difference operator on the lattice $\mathbb Z^d$, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which improves the key decay rate from $−2d+\epsilon$ to $−3d+\epsilon$. (The optimal decay rate is conjectured to be $−3d$.) As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works.
I will describe some diffusion processes on periodic comb-like structures and their scaling limit, which is a Brownian motion subordinated to an independent sticky Brownian motion. This limit process may exhibit anomalous diffusion. This work is motivated in part by studies of diffusion and homogenization in media which are not uniformly elliptic. I will describe the scaling limit from the point of view of SDEs and from the point of view of Ito excursion theory. The latter approach also makes connections with recent work on trapped random walks. This is joint work with Sam Cohn, Gautam Iyer, and Bob Pego from Carnegie Mellon University.
The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers for the sharp constant are not radial. In this talk I indicate a proof that, in the remaining parameter region, the optimizers are in fact radial.
The proof will proceed via a well chosen flow that decreases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.
This talk proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales.
The method compresses the random partial differential operator to an effective quasilocal deterministic operator that represents the expected solution on a coarse scale of interest. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.
Martensitic transformation is a shear-dominant, lattice distortive and diffusionless solid-solid transformation occurring by nucleation and growth. Shape memory alloy exhibits a martensite microstructure, which is a complex pattern of martensitic domains. In this study, the character of the interfaces between the martensite domains, dynamics of the formation of the microstructure and the emergence of power-law in the domain size distribution are investigated by various recent microscopy techniques in shape memory alloys. The experimental results are analyzed in the framework of the nonlinear elasticity theory of the microstructure which was founded by Ball and James, to bridge the theory and experiment and to elucidate underlying problems to be solved.
Graphene is a two-dimensional material made up of a single atomic layer of carbon atoms arranged in honeycomb pattern. Many of its remarkable electronic properties, e.g. quasi-particles (wave-packets) that propagate as massless relativistic particles and topologically protected edge states, are closely related to the spectral properties of the underlying single-electron Hamiltonian: -Laplacian + V(x), where V(x) is a potential with the symmetries of a hexagonal tiling of the plane. Taking inspiration from graphene, there has been a great deal of activity in the fundamental and applied physics communities related to the properties of waves (photonic, acoustic, elastic,…) in media whose material properties have honeycomb symmetry. In this talk l will review progress on the mathematical theory.
Using elliptic regularity results for sub-Markovian $C_0$-semigrous of contractions in $L^p$-spaces, we construct for every starting point weak solutions to SDEs in $d$-dimensional Euclidean space up to their explosion times under the following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient $A=(a_{ij})$ is supposed to be locally uniformly strictly elliptic with functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and for the drift coefficient $\mathbf{G}=(g_1,\dots, g_d)$, we assume $g_i\in L^p_{loc}(\mathbb{R}^d)$. Subsequently, we develop non-explosion criteria which allow for linear growth, singularities of the drift coefficient inside an arbitrarily large compact set, and an interplay between the drift and the diffusion coefficient.
Moreover, we show strict irreducibility of the solution, which by construction is a strong Markov process with continuous sample paths on the one-point compactification of $\mathbb{R}^d$. Constraining our conditions for existence further and respectively to the conditions of several well-known articles, as for instance Gyöngy and Martinez (CMJ 2001), X. Zhang (SPA 2005, EJP 2011), Krylov and Röckner (PTRF 2005) and Fang and T.-S. Zhang (PTRF 2005), where pathwise unique and strong solutions are constructed up to their explosion times, we must have that both solutions coincide. This leads as an application to new non-explosion criteria for the solutions constructed in the mentioned papers and thereby to new pathwise uniqueness results up to infinity for It\^o-SDEs with merely locally integrable drifts and Sobolev diffusion coefficients. This is joint work with Haesung Lee (Seoul National University).
The Parisi theory makes astounding predictions about the large volume behavior of mean field disordered systems, such as the paradigmatic SK-model. Some of these predictions have been meanwhile rigorously confirmed, yet many most of the deep conceptual puzzles raised by the Parisi theory remain to these days unanswered, in fact: un-addressed. Even a non-rigorous (!) yet hands-on, microcanonical treatment of spin glasses within the Boltzmann-Gibbs framework is to date lacking.
I will discuss these issues, and suggest through a case-study that the proper framework to address them rigorously is provided by the treatment pioneered by Thouless-Anderson-Palmer, and Plefka.
Joint work with David Belius (Zurich).
We discuss the long-time transport properties of linear wave equations in heterogeneous media that are small random perturbations of periodic media. Although periodic Bloch waves cannot be simply deformed into an exact diagonalisation of the perturbed operator, we construct a natural approximate diagonalisation that leads to a very precise description of the solution as long as the transport remains ballistic. In the case of a quasiperiodic perturbation, this approach establishes ballistic transport up to exponential time scales.
In this talk I will describe some results obtained in collaboration with Sergio Conti (Bonn), Gilles Francfort (Paris Nord), Flaviana Iurlano (Paris) and Vito Crismale (Palaiseau) on the existence and properties of minimizers of energies which arise in the variational approach to fracture. I will recall the basics of the theory, discuss some mathematical difficulties and describe some approaches to address them.
Uncertainty Quantification (UQ) is a rapidly developing field within applied mathematics and computational science concerned with identifying, modeling and quantifying the myriad uncertainties arising in computational models, specifically those involving differential equations. Besides discretization error and rounding error, which by now are well understood and can be managed, uncertainties in problem specifications such as coefficient functions, source and boundary data or domain geometry can often dominate the previously mentioned uncertainty and error sources.
In this talk we discuss the prevalent model problem in UQ, stationary diffusion with an uncertain diffusion coefficient which is modeled as a lognormal random field. We emphasize uncertainty modeling aspects in connection with a groundwater contamination problem leading to such a random diffusion equation and present numerical approaches for approximating its solution. In particular, the numerical representation of rough lognormal fields by Karhunen-Loève expansion raises interesting questions of stochastic homogenization. Continuing with the groundwater application, we also indicate how observational data can be incorporated to reduce uncertainty by large-scale Bayesian inference.
I will present some recent results on global solutions to singular SPDEs on $\mathbb{R}^d$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions $d=4,5$ and in the parabolic setting for $d=2,3$. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $\Phi^4_d$ Euclidean quantum field theory via Parisi--Wu stochastic quantization, while the elliptic equations are linked to the $\Phi^4_{d-2}$ Euclidean quantum field theory via the Parisi--Sourlas dimensional reduction mechanism. We prove existence for the elliptic equations and existence, uniqueness and coming down from infinity for the parabolic equations. Join work with Massimiliano Gubinelli.
In 1904, Prandtl introduced his famous boundary layer theory to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary in the inviscid limit. His Ansatz was that the solution of Navier Stokes can be described as a solution of Euler, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$. In this talk, I will present a recent joint work with E. Grenier (ENS Lyon), proving that, for a class of regular solutions of Navier Stokes equations, namely for shear profiles that are unstable to Rayleigh equations, this Prandtl's Ansatz is false. In addition, for shear profiles that are monotone and stable to Rayleigh equations, the Prandtl's asymptotic expansions are invalid.
Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp.~flat chain mod 2) such that $\partial S = \Sigma$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $\Sigma$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $ 6 \sqrt{\pi}\, \mathbf{M}[S] \leq (\mathbf{M}[\Sigma])^{3/2} $. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by $-\kappa < 0$ and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in $L^2$.
In the talk we discuss a model of motion of the non-smooth curve in the random media.The self-intersection local times will be used as the geometric characteristics of the curve. Different characteristics of the compact sets in Hilbert space, relationships between them and Kolmogorov entropy, and applications to the extension of Dynkin renormalization procedure on unbounded weight is obtained. This talk is based on joint work with O. L. Izyumtseva.
I will present some recent results obtained in collaboration with V. Banica and F. de la Hoz on the evolution of vortex filaments according to the so called Localized Induction Approximation (LIA). This approximation is given by a non-linear geometric partial differential equation, that is known under the name of the Vortex Filament Equation (VFE). The aim of the talk is threefold. First, I will recall the Talbot effect of linear optics. Secondly, I will give some explicit solutions of VFE where this Talbot effect is also present. Finally, I will consider some questions concerning the transfer of energy and momentum for these explicit solutions.
The continuum Gaussian free field (GFF) can be seen as a generalization of Brownian motion to higher dimensions and it is a canonical example of a random height function. I will discuss some recent progress in describing geometric and probabilistic properties of the 2D GFF : How do its level sets look like? What is the structure of its excursions off the level sets? Answering these questions will reveal connections to SLE processes, to Brownian loop soups and even to the probabilistic theory of 2D Liouville quantum gravity.
Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. In particular, PDEs are a fundamental tool in portfolio optimization problems and in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high dimensional as the dimensionality of the PDE corresponds to the number of financial asserts in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and developing efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called "curse of dimensionality" in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE and there is only a very limited number of cases where a practical PDE approximation algorithm with a computational effort which grows at most polynomially in the PDE dimension has been developed. In the case of linear parabolic PDEs the curse of dimensionality can be overcome by means of stochastic approximation algorithms and the Feynman-Kac formula. We first review some results for stochastic approximation algorithms for linear PDEs and, thereafter, we present a stochastic approximation algorithm for high dimensional nonlinear PDEs whose key ingredients are deep artificial neural networks, which are widely used in data science applications. Numerical simulations and first mathematical results sketch the efficiency and the accuracy of the proposed stochastic approximation algorithm in the cases of several high dimensional PDEs from finance and physics.
The most widely employed method for determining the effective large-scale properties of random materials is the representative volume element (RVE) method: It basically proceeds by choosing a sample of the random medium - the representative volume element - and computing its effective properties. To obtain an accurate approximation for the effective material properties, the RVE should reflect the statistical properties of the material well. Hence, it is desirable to choose a large sample of the random medium as an RVE. However, an increased size of the RVE comes with an increased computation cost. For this reason, there have been attempts in material science and mechanics towards capturing the statistical properties of the material in a better way in an RVE of a fixed size. We provide an analysis of an approach by Le Bris, Legoll, and Minvielle, which has been capable of improving the computational efficiency by a factor of 10-50 in some numerical examples by such an ansatz.
The aim is to study the symmetry of transition layers in Ginzburg-Landau type functionals for divergence-free maps in R^N. Namely, we determine a class of nonlinear potentials such that the minimal transition layers are one-dimensional.
In particular, this class includes in dimension N=2 the nonlinearities w^2 with w being an harmonic function or a solution to the wave equation, while in dimension N>2, this class contains a perturbation of the standard Ginzburg-Landau potential as well as potentials having N+1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations for divergence-free maps in R^N(similar to the theory of entropies for the Aviles-Giga model when N=2).
This is a joint work with Antonin Monteil (Louvain, Belgium).
Stationary solutions of interacting system usually play a significant role in physics as (potential) attractors of the evolution, i.e. equilibrium states. In the case of the Einstein field equations, which in the simplest case describe the interaction of the gravitational field with itself "in empty space", finding the stationary solutions is a rather non-trivial mathematical problem. In spacetime dimension four, the celebrated uniqueness theorem (a culmination of separate arguments due to Carter, Hawking, Robinson, Bunting, and others) states that the so called Kerr solution is the only such solution describing a black hole in empty space. Here I review some of the arguments leading to this conclusion and present a generalisation to higher dimensional spacetimes.
In their celebrated theory of renormalized solutions, DiPerna and Lions (1989) establish well-posedness and stability properties for transport equations with Sobolev vector fields. In this talk, I present a new approach to well-posedness that is based on stability estimates for certain logarithmic Kantorovich--Rubinstein distances. The new approach recovers some of DiPerna's and Lions's old results. In addition, it allows for two new major applications that were very inaccessible before: 1) We extend the theory to vector fields with $L^1$ vorticities and present applications to the 2D Euler equation (joint work with G. Crippa, C. Nobili and S. Spirito). 2) We derive optimal estimates on the error of the numerical upwind scheme (joint work with A. Schlichting).
We consider level set percolation for the Gaussian free field on the Euclidean lattice in dimensions larger than or equal to three. It had previously been shown by Bricmont, Lebowitz, and Maes that the critical level is non-negative in any dimension and finite in dimension three. Rodriguez and Sznitman have extended this result by proving that it is finite for all dimensions, and positive for all large enough dimensions. We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field. This talk is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).
I will introduce the model of spatial random permutations and talk about its connection to quantum physics, in particular to systems of interacting bosons. I will present the main open questions in the theory, and report on the (small) progress that has been made towards understanding some aspects of the model.
Formal considerations in mathematical physics often lead to so called “singular stochastic partial differential equations” (singular SPDEs), which have been mathematically ill posed for a long time. The problem is the interplay of very singular noise with nonlinearities, which creates small scale resonances that have be removed through a renormalization procedure. In the past five years we have seen a number of mathematical breakthroughs that allow us to finally solve and study such singular SPDEs. Now that solution theories are available we can rigorously study the predictions by mathematical physicists by proving the so called “weak universality” of these singular SPDEs.
The derivation of effective single-particle dynamics from interacting many-particle systems has a long history in the context of kinetic theory, and can pose challenging mathematical problems with the Boltzmann equation as a classical example. While effective dynamics are often used as a starting point to study stochastic particle systems in the theoretical physics literature, their rigorous derivation in this context has attracted attention only recently. We focus on the dynamics of cluster aggregation driven by the exchange of single particles, for which we derive effective rate equations in a mean-field scaling limit from an underlying particle system on a complete graph. We establish the propagation of chaos under generic growth conditions on particle jump rates, and the limit equation provides a Master equation for the single-site dynamics of the particle system, which is a non-linear birth-death chain. Conservation of mass in the particle system leads to conservation of the first moment for the limit dynamics, and to non-uniqueness of stationary measures. Our findings are consistent with recent results on exchange driven growth, and provide an interesting connection between well studied phenomena of gelation and condensation. This is joint work with Watthanan Jatuviriyapornchai.
The Monge-Ampere equation det D^2u = 1 arises in several applications. Examples of Pogorelov show that interior regularity is not expected in general. We will discuss optimal estimates on the Hausdorff dimension of the singular set, and sharp integrability conditions for the second derivatives that rule out singularities. Some of this is joint work with T. Collins.
We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born’s conjecture about the optimality of the rock-salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results holds for a class of completely monotone interaction potentials which includes Coulomb type interactions. In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.
We consider mathematical PDE models of motility of eukaryotic cells on a substrate and discuss them in a broader context of active materials. Our goal is to capture mathematically the key biological phenomena such as steady motion with no external stimuli and spontaneous breaking of symmetry.
We first describe the hierarchy of PDE models of cell motility and then focus on two specific models: the phase-field model and the free boundary problem model.
The phase-field model consists of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The key mathematical properties of this system are (i) the presence of gradients in the coupling terms and (ii) the mass (volume) preservation constraints. These properties lead to mathematical challenges that are specific to active (out of equilibrium) systems, e.g., the fact that variational principles do not apply. Therefore, standard techniques based on maximum principle and Gamma-convergence cannot be used, and one has to develop alternative asymptotic techniques.
The free boundary problem model consists of an elliptic equation describing the flow of the cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. This PDE system is of Keller-Segel type but in a free boundary setting with nonlocal condition that involves boundary curvature. Analysis of this system allows for a reduction to a Liouville type equation which arises in various applications ranging from geometry to chemotaxis. This equation contains an additional term that presents an additional challenge in analysis.
In the analysis of the above models our focus is on establishing the traveling wave solutions that are the signature of the cell motility. We also study breaking of symmetry by proving existence of non-radial steady states. Bifurcation of traveling waves from steady states is established via the Schauder's fixed point theorem for the phase field model and the Leray-Schauder degree theory for the free boundary problem model.
These results are obtained in collaboration with J. Fuhrmann, M. Potomkin, and V. Rybalko.
The class of entropy solutions to the eikonal equation arises in connection with the asymptotics of the Aviles-Giga energy. In a joint work with Francesco Ghiraldin we prove, using a new simple form of the kinetic formulation, that this class coincides with the class of solutions which enjoy a certain Besov regularity.
The $1$-harmonic flow is the formal gradient flow --with respect to the $L^2$-distance-- of the total variation of a manifold-valued unknown function. The problem originates from image processing and has an intrinsic analytical interest as prototype of constrained and vector-valued evolution equations in $BV$-spaces. I will introduce a notion of solution and I will present existence (and, in some cases, uniqueness) results when the target manifold is either the hyper-octant of a sphere or a connected subarc of a regular Jordan curve. I will also discuss a work in progress concerning local/global-in-time well-posedness in Lipschitz spaces. As all of the above is just a first, tentative step into a rather uncharted territory, I will conclude by highlighting a few basic open questions.
I will talk about a liquid drop sitting on a rough surface at equilibrium, which is formulated as a volume-constrained minimizer of the total interfacial energy.
The large-scale shape of such a drop strongly depends on the micro-structure of the solid surface. Surface roughness enhances wetting and de-wetting properties of the surface, altering the equilibrium contact angle between the drop and the surface. Our goal is to understand the shape of the drop with fixed small scale roughness. We will discuss existing literature, our result, and some open questions.
Recent theoretical and numerical results have shown that invisvid models in gas dynamics, such as the (in)compressible Euler equations, are unstable with respect to initial data or even ill-posed. Going back to the roots of turbulence theory, we interpret instead these hyperbolic conservation laws in a probabilistic manner. In this talk I will survey some recent developments in so-called statistical solutions, both theoretical and numerical. These include well-posedness for scalar conservation laws; energy conservation for regular solutions of the incompressible Euler equations; and numerical evidence for the convergence of the mean flow, structure functions etc. for compressible Euler equations.
Since the late 70’s it is well-known that the addition of a random force in an ill-posed ODE may bring back the system well-posed. This kind of phenomenon is known as "stochastic regularization effect" or "regularization by noise". A breakthrough in this domain is contained in the paradigm known under the name of "The Itô-Tanaka trick" which links the time average of a non-smooth map $f$ along the solution of an SDE with the solution of a Fokker-Planck PDE. However, this approach is restricted to deterministic mappings $f$. The aim of this talk is to go beyond this limitation. More precisely, we propose an extended version of the Itô-Tanaka trick to random mappings $f$. This talk is based on a joint work with Romain Duboscq.
The plane, periodic Couette flow has served as a canonical problem in the field of hydrodynamic stability since the late 19th century. Obtaining a precise understanding of the stability and instability of this flow in the nonlinear equations has however remained elusive in both 2D and 3D. In this talk I will discuss some of the recent progress we have made in this direction in both 2D (joint with Nader Masmoudi, Vlad Vicol and Fei Wang) and 3D (joint with collaborators Pierre Germain and Nader Masmoudi). The dynamics of solutions have been determined in a variety of settings, and are governed by several important effects: namely inviscid damping and mixing-enhanced dissipation. Comparisons and contrasts will be made with recent results on Landau damping in kinetic theory.
The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 R. Finn and D. R. Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier-Stokes equations in a two-dimensional exterior domain that describe this type of flows when the Reynolds number is sufficiently small. The asymptotic behavior of their solutions at spatial infinity has been studied in details and well understood by now, while its stability has been open due to the difficulty specific to the two-dimensionality. In this talk we show that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.
Rayleigh-Bénard convection (RBC) in a fluid layer between two parallel walls, uniformly heated from below and cooled from above, is one of the fundamental flows with numerous applications in nature and technology. The three-dimensional Boussinesq equations which couple the velocity and temperature fields are studied by direct numerical simulations applying a spectral element method. We investigate two configurations of convection: (i) RBC in closed cylindrical cells at high Rayleigh numbers, the typical laboratory experiment configuration; (ii) RBC in horizontally extended domains at moderate Rayleigh numbers. We will discuss the global transport of heat and momentum and its close connection to the dynamics in the boundary layers of temperature and velocity fields at the walls for the first case. We will also present recent analysis results on the structure formation at large-scales for the second case. This includes the detection of defects in the time-averaged temperature and velocity patterns as well as Lagrangian trajectory-based methods that reveal how the heat is carried across the fluid layer.
We introduce and analyze a continuous time martingale optimal transport problem (MOT) which can be seen as the "Benamou-Brenier" formulation of MOT. It is naturally linked to the discrete MOT problem via a weak length relaxation. We present two different solutions to this problem. The first solution is based on a convex duality result and allows to derive a "geodesic equation" for the optimizer for a wide class of cost functions. The second is an explicit probabilistic representation in the case of a specific cost function. We will show that this solution has several applications as well as a remarkable additional optimality property.
(based on joint work with Julio Backhoff, Mathias Beiglböck, Sigrid Källblad, and Dario Trevisan)
In epitaxial thin film growth, elasticity effects often lead to self-organizing pattern formation which can be important in the fabrication of nanostructures. We discuss an elasticity model that takes into account of the lattice misfit between the substrate and the film, and the broken-bond effect due to surface steps. The former is an attractive while the latter is a repulsive interaction. It is found that uniform step train is unstable and will evolve into structures consisting of macroscopic step bunches. For the case of vicinal surface which consists of a sequence of monotonically decreasing steps, using a variational formulation, we analyze the properties of these bunches, notably their energy scaling and bunch width. We emphasize on a discrete model but continuum description will also be discussed. This is joint work with Tao Luo and Yang Xiang.
The problem of quantization of a d-dimension probability distribution by discrete probabilities with a given number of points can be stated as follows: given a probability density $\rho$, approximate it in the Wasserstein metric by a convex combination of a finite number N of Dirac masses.In collaboration with E. Caglioti and F. Golse we studied a gradient flow approach to this problem in one and two dimensions. By embedding the problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.
From a physical perspective, the projected anthropogenic climate change is insufficiently quantified. The largest uncertainty is due to the response of clouds to the anthropogenic perturbation of the composition of the atmosphere. The presentation will discuss the hypotheses how clouds respond to increases in atmospheric carbon dioxide concentration increases and atmospheric particle (aerosol) concentration increases. It presents approaches to assess the resulting effects on the energy budget of the Earth system from observations and numerical modelling.
In this talk we consider a Fokker-Planck equation modelling nucleation of clusters very similar to the classical Becker-Döring equation. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. In this way the equation has formally the structure of a McKean-Vlasov equation, but with a non-local boundary condition.
We briefly discuss the well-posedness and regularity properties of the Cauchy-Problem. Here the main difficulty is to improve basic a priori regularity properties of the non-local order parameter. The main part of the talk focuses on the longtime behaviour of the system. The system posses a free energy, which strictly decreases along the evolution and leads to a gradient flow structure involving boundary conditions.
We generalize the entropy method based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates. In this way, we obtain an explicit characterization of the convergence to equilibrium with algebraic or even exponential rates depending on the particular assumptions on the vector fields, diffusivity and initial data.
(joint work with J. Canizo, J. Conlon)
The Erdös-Rényi random graph was defined in a seminal 1973 paper by Erdös and Rényi, an English translation of which appeared in 1976. The model is of percolation type, and exhibits a phase transition as the number of particles (vertices, points) diverges, and the connective probability (of each pair of vertices) is about its reciprocal. Since 1970's random graphs have been widely studied, initially mostly by combinatorists. In 1997 the probabilistic approach began with a famous work by Aldous on random graphs and the standard (i.e. Aldous') multiplicative coalescent. The talk is intended to serve as an introduction to this research direction. It will try to explain the core of the 20 year old results, as well as certain recent improvements and developments. In anticipation of a mixed audience (analysits+probabilists+possibly others?) the talk will resemble more to a colloqium than to a probability working seminar.
In this talk I would like to present a joint work with B. Merlet and V. Millot about a variational model for defects in the ripple phase of lipid bilayers. The model is of Ginzburg-Landau type with an additional jump term. This adds a Mumford-Shah flavor to the problem. We will show that the limiting problem contains both points and line singularities. The limiting energy sees competition between the classical renormalized energy of the vortex points and a term reminiscent of the Steiner problem connecting these points. If time permits, I will also show how the regularity results obtained for the limiting problem can be lifted to the minimizers of the original one.
In this talk, a framework to solve elliptic interface problems on geometries involving spherical shapes is presented. Applications range from implicit solvation models in computational chemistry to electrostatic interaction of charged dielectric spheres to stochastic homogenisation with spherical inclusions. One of the main issues of elliptic problems is that the Green’s function is decaying very slowly. Therefore, the strategy is in all cases the same: we first transform the problem into an integral equation allowing for capturing the right asymptotic behaviour of the solution towards infinity. The integral equation is subsequently discretised by means of spherical harmonics on each sphere. In particular in the case of stochastic homogenisation, this approach allows for a new strategy of the corrector problem as one can solve problems on the unbounded domain where the stochastic medium is embedded in an infinite continuous medium. Numerical illustrations will underline the methodological developments.
This talk gives a short review of recent work of A. Acharya (CMU), G.-Q. Chen (Oxford), M. Slemrod (Univ. of Wisconsin-Madison), D. Wang (Pittsburgh) on the direct connection between the 2d Euler equations of incompressible fluid flow and motion of a 2d surface in 3-dimensional Euclidean space. In particular we note how this link provides a direct application of the Nash-Kuiper non-smooth isometric embedding result which in turn provides information on ill-posedness for the initial value problem for weak solutions of the Euler equations as well as neo-Hookian elasticity. Furthermore it suggests that the Nash-Kuiper solutions may represent fluid turbulence. The work was strongly motivated by the sequence of papers of C. De Lellis and L. Szekelyhidi, Jr.
A main goal in the field of enumerative combinatorics is to derive asymptotic formulas for combinatorial sequences. It has been shown in the past that many such sequences obey a central limit theorem, from which asymptotic estimates can be obtained. In particular, for a large class of combinatorial sequences (i.e., assemblies, multisets, and selections), there is a general principle that allows one to derive an asymptotic formula via a local central limit theorem (LCLT) for integer-valued random variables. We present a novel and robust method to derive quantitative LCLTs under conditions which are easy to verify, which then yields quantitative error estimates for the associated asymptotic formulas. In certain cases we are able to provide new quantitative error estimates unknown before. Joint work with Stephen De Salvo.
We consider a singularly perturbed variational problem of minimizing the sum of the surface and elastic energies of the order parameter $u$ in a two-dimensional rectangular domain. This model, originally suggested by Kohn and Muller, comes from martensitic phase transitions, in which two distinct phases of the martensite correspond to $u_y(x,y)=1$ and $u_y(x,y)=-1$. It was observed that minimizers develop self-similar branched microstructures in the case when the boundary condition is not compatible with either of the phases, and may have zigzag microstructures otherwise. In my talk, I will describe several patterns of the behavior of minimizers depending on the choice of boundary conditions, derive sharp global and local energy bounds, and discuss the applications to 2D and 3D linear elasticity models.
In 1989, Di Perna and Lions showed that Sobolev regularity for vector fields in R^d, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption on the vector field which prevents the trajectories from blowing up in finite time; in particular, it does not apply to fast-growing, smooth vector fields. In this seminar we present a notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show existence and uniqueness under only local assumptions on the vector field and we apply the result to a kinetic equation, the Vlasov-Poisson system, where we describe the solutions as transported by a suitable flow in the phase space. This allows, in turn, to prove existence of weak solutions for general initial data.
We will introduce some recent results in collaboration with L. Caffarelli and X. Ros-Oton on the optimal regularity of the solutions and the regularity of the free boundaries (near regular points) for nonlocal obstacle problems. The main novelty is that we obtain results for different operators than the fractional Laplacian. Indeed, we can consider infinitesimal generators of non rotationally invariant stable Lévy processes.
The incompressible three-dimensional Euler equations are a basic model of fluid mechanics. Although these equations are more than 200 years old, many fundamental questions remain unanswered, most notably if smooth solutions can form singularities in finite time. In this talk, I discuss recent progress towards proving a finite time blowup for the Euler equations, inspired numerical work by T. Hou and G. Luo and analytical results by A. Kiselev and V. Sverak. My main focus lies on various model equations of fluid mechanics that isolate and capture possible mechanisms for singularity formation. An important theme is to achieve finite-time blowup in a controlled manner using the hyperbolic flow scenario in one and two space dimensions. This talk is based on joint work with T. Do, B. Orcan-Ekmecki, M. Radosz, X. Xu and H. Yang.
Widths of a Riemannian manifold can be informally described as critical points of the volume functional corresponding to distinct homology classes of the space of cycles. We prove that widths satisfy a Weyl's law that was proposed by Gromov.
This is a joint work with F.C. Marques and A. Neves.
Consider the eigenvalue problem \(\left. \begin{array}{cc} -y''+q(x)y=\lambda y\\ y'(0)=y'(1)=0 \end{array} \right\}.\) If $q=0$ then the eigenvalues are $\lambda_n=n^2\pi^2$ ($n\ge 0$). What is surprising is that the converse is also true; this is Ambarzumian's theorem. We discuss some similar statements on graphs and in a PDE setting.
The notions mentioned in the title are roughly related as follows. Isotropicity of a minimal surface is characterised by the vanishing of certain holomorphic differentials. Holomorphic curves in a complex torus with a flat metric are precisely the minimal surfaces which are maximally isotropic. And it is well known that a surface which is holomorphic in a Kähler manifold minimizes area in its homology class.
I will present various results in this area and mention some open problems. In particular, I will discuss the deformation of a holomorphic curve in a complex torus with a flat metric to a minimal surface which is isotropic to a sufficiently high order (but less than maximal!). The holomorphicity of stable minimal surfaces which are isotropic to the same degree will also be described. This is joint work with Elisabeta Nedita and it is related to (some old) work with Claudio Arezzo and Jon Wolfson.
During the 17th Century, Edme Mariotte observed that floating rigid bodies could attract or repel each other, and he attempted to develop a quantitative description of the phenomenon. He did not succeed, which in retrospect is not a surprise, as the behavior seems to depend on surface tension, for which there was no conceptual theory at that time. There are now elaborate surface tension theories, nevertheless very little quantitative information on the original question has appeared. I will describe some recent progress on a simplified model proposed by Laplace in 1806. This is nominally the second of two lectures, but was delayed due to unfortunate circumstances; I have reorganized it to be self-contained.
We give a constructive proof of smoothness (in ultradifferentiable classes) of Lagrangian trajectories for 3D incompressible Euler flows in an impermeable bounded domain whose boundary is ultradifferentiable, i.e. may be either analytic or have a regularity between indefinite differentiability and analyticity. Based on a little-known Cauchy Lagrangian formulation of the 3D incompressible Euler equations, we establish novel explicit recursion relations that include contributions from the boundary.
This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy-Lagrangian method to study numerically, among other, the open issue of finite time blow up of classical solutions.
There are several ways to quantify the "complexity" of a minimal subvariety: on the one hand we have analytic data (like the Morse index, the value of the p-th eigenvalue of the Jacobi operator etc...), on the other we have geometric invariants (like the Betti numbers, the sigma invariant etc...). But what is the relation between these pieces of information? Are these measures equivalent in some sense? I will give a broad overview of this class of problems and will then focus on recent joint work with L. Ambrozio and B. Sharp: Motivated by a well-known conjecture due to Schoen and recently presented in extended form by F. Marques and A. Neves in their ICM lectures, we study the relation between the Morse index and the first Betti number of minimal hypersurfaces inside positively curved closed Riemannian manifolds. We present a unified framework to address such conjecture and we settle it for a wide class of ambient spaces. More specifically, we give a curvature condition on the ambient manifold which ensures that the Morse index of any minimal hypersurface is bounded from below by a fixed linear function of its first Betti number. As a result, such conclusion is shown to hold true on all compact rank one symmetric spaces, on product of spheres and on suitably pinched submanifolds of the Euclidean space. Differently from earlier works of Ros and Savo, our methods are "flexible" meaning that they also apply to spaces that are neither rigid nor special in any reasonable sense.
It is well known that for real-valued functions on matrices polyconvexity implies quasiconvexity implies rank-one convexity. While quasiconvexity arises as a natural condition in the Calculus of Variations it is also the most difficult to verify and instead the only accessible path to many problems is to work with the more tractable notions of poly- and rank-one convexity. In this talk we will define the new concept of n-polyconvexity. For $f:ℝ^{d×D}→ℝ∪{+∞}$ n-polyconvexity unifies polyconvexity and rank-one convexity in the following sense: If $n=d∧D:=min{d,D}$ then $(d∧D)$-polyconvexity is equivalent to polyconvexity and if n=1 then 1-polyconvexity is equivalent to rank-one convexity. Additionally one gains the new convexities for $n=(d∧D-1)...2$ in weakening order. We will discuss some basic properties of n-polyconvexity, including non-locality, subdifferentiability and the formation of generalized versions of $T_k$-configurations. Furthermore the concept of n-polyconvexity may prove to be useful in relation to quasiconvexity the same way as poly- and rank-one convexity do and we will in particular discuss whether in $ℝ^{3×3}$ quasiconvexity implies or is implied by the previously unknown concept of 2-polyconvexity or not.
We present a variational model from micromagnetics involving a nonlocal Ginzburg-Landau type energy for S^1-valued vector fields. These vector fields form domain walls, called Neel walls, that correspond to one-dimensional transitions between two directions within the unit circle S^1. Due to the nonlocality of the energy, a Neel wall is a two length scale object, comprising a core and two logarithmically decaying tails. Our aim is to determine the energy differences leading to repulsion or attraction between Neel walls. In contrast to the usual Ginzburg-Landau vortices, we obtain a renormalised energy for Neel walls that shows both a tail-tail interaction and a core-tail interaction. This is a novel feature for Ginzburg-Landau type energies that entails attraction between Neel walls of the same sign and repulsion between Neel walls of opposite signs. This is a joint work with Roger Moser (University of Bath).
Many nonlinear systems admit families of striped solutions, which are periodic in one spatial variable. A prototypical system is the Swift-Hohenberg equation with cubic nonlinearity. I will discuss attempts to describe patterns that deviate from exact spatial periodicity due to the presence of boundary conditions, inhomogeneities, or ``self-organized'' defects.
A classical problem consists in optimizing the structure of a composite material, for instance to achieve high rigidity against a prescribed mechanical loading. In the simplest case, the material is a composite of void and the elastic base material. The problem then reduces to finding the optimal topology and geometry of the structure. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal material designs, but also represents a prototype problem to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that can hardly be resolved numerically. However, for certain geometries an energy scaling law can be proven that describes how the minimum of the objective functional scales with the model parameters and that provides near-optimal designs. The optimal design problem is strongly related to a few well-known pattern formation problems but has several distinct features.
Many nonlinear stochastic PDES arising in statistical mechanics are ill-posed in the sense that one cannot give a canonical meaning to the nonlinearity. Nevertheless, Martin Hairer’s theory of regularity structures provides us with a good notion of solution for a large class of such equations (KPZ equation,stochastic quantization,...). One of the simplest equations to which this theory can (and should) be applied is the generalized 2D Parabolic Anderson Model : $$(\partial_t - \Delta) u = f(u) \xi, \;\;\; u(0)=u_0,$$ where $\xi$ is a spatial white noise on the torus $\mathbb{T}^2$.In my talk, after a quick overview of the theory of regularity structures (in the particular case of this equation), I will explain how it can be combined with classical tools of Malliavin calculus, which allows in particular to obtain absolute continuity results for the marginal laws of the solutions. This is based on joint work with G. Cannizzaro and P. Friz (TU Berlin).
We aim to study the long time behaviour of the solution to a rough differential equation (in the sense of Lyons) driven by a random rough path. To do so, we use the theory of random dynamical systems. In a first step, we show that rough differential equations naturally induce random dynamical systems, provided the driving rough path has stationary increments. If the equation satisfies a strong form of stability, we can show that the solution admits an invariant measure.
This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).
We say that a regularization by noise phenomenon occurs for a possibly ill-posed differential equation if this equation becomes well-posed under addition of noise. In this talk we show such a regularization for stochastic linear transport-like equations, namely \begin{equation*} \partial_t v +b\cdot\nabla v +hv +\nabla v\circ \dot{W} =0, \end{equation*} where $b=b(t,x)$, $h=h(t,x)$ are given deterministic, possibly irregular vector fields, $W$ is a $d$-dimensional Brownian motion, $\circ$ denotes Stratonovich integration and $v=v(t,x,\omega)$ is the solution. We show, under a certain integrability assumption on $b$ and $h$ (the Ladyzhenskaya-Prodi-Serrin integrability condition), that this equation admits a unique distributional solution (in a suitable integrability class), which is also Sobolev regular for regular initial condition. The result is false in general without noise. The existence of a (Sobolev) regular solution is obtained by a priori estimates: using the renormalization property of the transport equation, we get a system of transport-like SPDEs for the powers of the derivative of the solution $v$; then we use parabolic estimates to bound the expectation of such powers. The uniqueness we get is of pathwise type (actually even stronger than pathwise) and is obtained through a duality method, using the regular solution to the dual SPDE. These results can be applied to get well-posedness for the associated SDE \begin{equation*} \rmd X = b(X)\rmd t +\rmd W, \end{equation*} again for $b$ non-smooth.
A function in $\Bbb R^n$ is called DC (d.c., delta-convex) if it is the difference of two continous functions $f_1$, $f_2$ (i.e., $f=f_1-f_2$). DC functions are a natural and important joint generalization of both $C^2$ smooth and convex functions. I will recall some properties of DC functions and will give some information on several (older and quite recent) their applications: a)\ to singularities of convex functions; b)\ to ``distance spheres'' in Riemann spaces and in convex surfaces; c)\ to the theory of curvature measures of non-regular sets. Several open questions will be presented.
Various concepts of weak solution have been suggested for the fundamental equations of inviscid fluid flow over the last few decades. A common problem is the vast degree of non-uniqueness that all these types of solution exhibit. Nevertheless, a conditional notion of uniqueness, the so-called weak-strong uniqueness, can be established in various situations. We present some recent results, both positive and negative, on weak-strong uniqueness in the realm of incompressible and compressible fluid dynamics.
In this talk we present a quenched invariance principle for the dynamic random conductance model, that is we consider a continuous time random walk on the integer lattice in an environment of time-dependent random conductances. We assume that the conductances are stationary ergodic with respect to space-time shifts and satisfy some moment condition. One key result in the proof is a maximal inequality for the corrector function, which is obtained by a Moser iteration. This is joint work in progress with A. Chiarini, J.-D. Deuschel and M. Slowik.
Suppose that C in n+1-dimensional Euclidean space is a closed countably n-rectifiable set whose complement consists of more than one connected component. Assume that the n-dimensional Hausdorff measure of C is finite or grows at most exponentially near infinity, but no further regularity is assumed. Examples of C are planar networks in 2D, soap bubble clusters in 3D, and co-dimension one simplicial complexes in general dimensions. Very recently, we have proved a global-in-time existence of nontrivial mean curvature flow in the sense of Brakke starting from C. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curvature flow. One may consider the open sets as grains and the flow as the mean curvature flow of grain boundaries. In this talk, mainly the background and results are discussed and if time permits, some key concepts will be discussed. This is a joint work with Lami Kim.
The Schrödinger problem is an entropy minimization problem on a set of path measures with prescribed initial and final marginals. It arises from a large deviation principle for the empirical measures of large particle systems. When the dynamics of the particles is slowed down while the prescribed marginals are unchanged, a second level large deviation phenomenon occurs. This leads to a sequence, indexed by the slowdown parameter, of Schrödinger problems which Gamma-converges to a dynamical optimal transport problem. We will illustrate these limits in the setting of L2 displacement interpolations on Rd and L1 displacement interpolations on graphs and Finsler manifolds.
In the first part of the talk we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler-Laplace operator on Finsler-Hadamard manifolds having finite reversibility constant.
The subject of the lecture is the numerical simulation of two-phase flow of immiscible fluids. Their motion is described by the incompressible Navier-Stokes equations with piecewise constant density and viscosity. The interface between the fluids is defined with the aid of the level-set method using a transport first-order hyperbolic equation. The Navier-Stokes system equipped with initial and boundary conditions and transmission conditions on the interface between the fluids is discretized by the Taylor-Hood P2/P1 conforming finite elements in space and the second-order BDF method in time. The transport level-set problem is solved with the aid of the space-time discontinuous Galerkin method (DGM). The second part of the lecture is devoted to the theoretical analysis of the DGM for the level-set problem. Numerical experiments demonstrate the applicability, accuracy and robustness of the developed method.
For mean values close to -1, it is easy to see that the constant state is a local energy minimizer of the Cahn-Hilliard energy. As already described in the seminal work of Cahn and Hilliard, stochastic fluctuations lead to nucleation of small, droplet-shaped regions of +1, which may then grow and coalesce. Moreover, whether the regions of +1 grow or shrink should depend on whether their mass is large enough to form a so-called critical nucleus. We describe recent (deterministic) work on the Cahn-Hilliard energy landscape in the regime of mean value close to -1 and large system size, which leads to quantitative bounds on the volume and approximate "droplet-shape" of a candidate for the critical nucleus. Our methods involve Gamma-limits, quantitative isoperimetric inequalities, and Steiner symmetrization.
In order to obtain uniqueness for solutions of scalar conservation laws with discontinuous flux, Kruzhkov's entropy conditions are not enough and additional dissipation conditions have to be imposed on the discontinuity set of the flux. Understanding these conditions requires to study the structure of solutions on the discontinuity set. I will show that under quite general assumptions on the flux, solutions admit traces on the discontinuity set of the ux. This allows to show that any pair of solutions satises a Kato type inequality with an explicit reminder term concentrated on the discontinuities of the flux. Applications to uniqueness is then discussed.
We deal with the computer simulation of the movement of pedestrian crowds. A macroscopic model describing the pedestrian flow consists of the continuity equation and momentum equation of fluid dynamics. Specifying the social and pressure forces in the momentum equation, we get the first order hyperbolic system of partial differential equations with a source term. The splitting technique is applied which leads to a combination of the finite volume method for the hyperbolic problem with the numerical solution of the system of ordinary differential equations. Additionally, the solution of the so-called eikonal equation plays an important role here. Such a solution determines the density dependent direction of pedestrian motion. The algorithm giving the time evolution of the density and velocity of pedestrians in the two-dimensional domain is described. The practical application of the algorithm for the evacuationof the 2D hall for various configurations of obstacles near to the exit is presented.
We consider particles on an infinite one-dimensional lattice whose evolution is governed by a discretization of a backward nonlinear diffusion equation. Such a system is one of the simplest examples of an infinite coarsening system with local interactions.
In this talk I will show how to prove global existence of solutions under a density assumption on the initial data, give examples of blow-up and non-uniqueness and discuss different coarsening scenarios.
We discuss several application of the concept of relative energy/entropy functional in th context of mathematical fluid dynamics. Applications to well-posedness problems, weak-strong uniqueness, stability of numerical schemes as well as stochastic PDE's modeling compressible fluid flows will be given. This talk is based on joint work with P.Bella, D.Breit, M.Hofmanova, M. Lewicka, and A.Novotny.
Non-dominated sorting is a fundamental problem in multi- objective optimization. We show that non-dominated sorting of a sequence of i.i.d. random variables in Euclidean space has a continuum limit that corresponds to solving a partial differential equation (specifically, a Hamilton-Jacobi equation) involving the probability density function of the random variables. We give an efficient numerical scheme for computing the viscosity solution of this Hamilton-Jacobi equation, and show how it can be used to design a fast algorithm for approximate non-dominated sorting. We demonstrate the advantages of this approach on a large scale real data application of anomaly detection in pedestrian trajectories captured from an overhead camera. Joint work with Jeff Calder and Alfred Hero (Ph.D. thesis work of Jeff Calder).
We prove that the Brouwer degree (a.k.a. mapping degree) for a Hölder continuous function is L^p integrable if the Hölder exponent is large enough in comparison to p times the box dimension of the boundary of the domain. This is supplemented by a theorem showing that convergence of a sequence of functions in an appropriate Hölder space implies convergence of the associated sequence of mapping degrees in L^p, again for a suitably chosen regime of the Hölder and Lebesgue exponents. In a different regime, we show the existence of counterexamples to the latter case.
A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrarily small positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the unit circle and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent τ of the degree distribution and the exponent δ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if τ < 2 + 1/δ , but fails to be robust if τ > 2 + 1/(δ−1) . This is the first instance of a scale-free network where robustness depends not only on its degree distribution but also on its clustering features. This is joint work with Emmanuel Jacob (ENS Lyon).
In this talk I will discuss the (asymptotic) shape of the minimizers of a Gross-Pitaievskii functional describing Bose-Einstein condensates with two components. We will be interested in the regime of strong segregation where the two components do not overlap. I will show that for different repulsion parameters, these minimizers sometime radially symmetric and sometime not.
This is based on joint works with B. Merlet and J. Royo-Letelier
We develop a method of proving limit theorems (CLT, and more generally diffusion approximation type theorems) for systems with random entries given by Markov processes which intrinsic memory. The method mainly exploits weak ergodic rates for the underlying Markov processes and is, to a great extent, insensitive w.r.t. to the structure of the process. The talk is based on a joint research with A.Veretennikov (Leeds, UK).
We consider the effective diffusivity (i.e. the covariance matrix in the central limit theorem) of a random walk among random conductances. It is interesting and non-trivial to describe this diffusivity in terms of the law of the conductances. The Einstein relation gives a different interpretation of the effective diffusivity as mobility. The mobility measures the response of the diffusing particle to a constant exterior force: Consider the perturbed process obtained by imposing a constant drift of strength $\lambda$ in some fixed direction. The perturbed motion satisfies (as one can show in many examples) a law of large numbers with effective velocity $v(\lambda)$. The mobility is the derivative of $v(\lambda)$ as $\lambda$ goes to 0. The Einstein relation says that the mobility and the diffusivity of a particle coincide. The Einstein relation is conjectured to hold for a variety of models, but it is proved insofar only for particular cases. We explain how it follows from an expansion of the invariant measures for the environment, seen from the particle.The talk is based on joint work with Jan Nagel and Xiaoqin Guo.
Mean curvature motion as well as surface diffusion constitute geometric interfacial motions which have received considerable attention. However in many applications a complex combination of coupled surfaces appear whose evolution may be described by coupling these two types of motion. In my lecture, a variety of physical problems will be described which may be reasonably modeled by such motions. While some these problems appear to require an anisotropic formulation, often an isotropic formulation is helpful to consider. A panoply of analytic and numerical results will be presented, in addition to some supporting experimental evidence.
Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. The grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the coarsening, or growth, process, an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, if the given energy density depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy. Why does such simplicity emerge from such complexity? Here we elaborate on this and attempt to explain the identification of the statistic in terms of mass transport. Other opportunities emerge and we discuss our hunt for ‘gradient flows in the wild’.
Derivation of the Vlasov-Poisson equation from Newtonian mechanics of interacting particles is challenging in the case of singular interactions.
In the talk I will present a new method which is based on probabilistic arguments and allows to consider Coulomb interactions with an N-dependent cut-off. In particular, I will show convergence of the empirical distribution to solutions of the Vlasov-Poisson equation for typical initial conditions.
We study the behavior of random walk on dynamical percolation, which is a random walk that moves in an environment (graph) that changes over time. In this model, the edges of a graph G are either open or closed, and refresh their status at rate $\mu$. At the same time a random walker moves on G at rate 1 but only along edges which are open. The regime of interest here is when $\mu$ goes to zero as the number of vertices of G goes to infinity, since this creates long-range dependencies on the model. When G is the d-dimensional torus of side length n, we prove that in the subcritical regime, the mixing times is of order $n^2\mu$. We also obtain results concerning mean squared displacement and hitting times. This is a joint work with Yuval Peres and Jeff Steif.
How much stuff can be transported by an incompressible flow flow containing a specified amount of kinetic energy (mean square velocity) or enstrophy (mean square vorticity)? We study this problem for steady 2D flows focusing on passive tracer transport between parallel impermeable walls, employing the calculus of variations to find divergence-free velocity fields with a given intensity budget that maximize transport between the walls. The maximizing velocity fields, i.e., the optimal flows, consist of arrays of (convection-like) cells of a particular form. Results are reported in terms of the Nusselt number Nu, the convective enhancement of transport normalized by the flow-free diffusive transport, and the Péclect number Pe, the dimensionless gauge of the strength of the flow. For each of the two flow intensity constraints, we also consider buoyancy-driven Rayleigh-Bénard flows with the same flow intensity constraints to see how parameter scalings for Nu (the heat transport in this context) reported in the literature compare with the bounds. We also consider the time-dependent optimal transport problems coming from Galerkin trunctions of a Rayleigh's original convection model, namely the Lorenz equations (a three Fourier-mode truncation) and another eight-mode model.
Much of this is joint work with Pedram Hassanzadeh (Berkeley/Harvard) and Gregory P. Chini (New Hampshire) published in /Journal of Fluid Mechanics /*751*, 627-662 (2014), and Andre Souza (Michigan) published in /Physics Letters A /*379*, 518-523 (2015).
In this talk we will review recent results for stochastic scalar conservation laws with random flux. In the first part we will focus on a well-posedness theory for the case of spatially inhomogeneous, random fluxes as they appear in mean field games. In the second part we will investigate the long time behavior and regularity of solutions to stochastic scalar conservation laws on the torus. In particular, we will observe a certain regularizing effect due to the noise.
This talk will be concerned with some classical results (CZ and Aronson theorems) in harmonic analysis and their application to homogenization of linear uniformly elliptic and parabolic PDE with random coefficients. The CZ theorem (1952) concerns the boundedness of singular integral operators on p integrable functions. The Aronson theorem (1967) gives bounds on the Green’s function for uniformly elliptic and parabolic PDE which depend only on the ellipticity constant. We will show how these theorems can be used to obtain rate of convergence results in homogenization of PDE with random coefficients, and in Euclidean field theories with uniformly convex Lagrangians.
Model reduction has become an ubiquitous tool in simulation and control for dynamical systems arising in various engineering disciplines. Often, models of physical processes contain parameters describing material properties and geometry variations, or arising from changing boundary conditions. These parameters may vary with time or be stochastic when modeling the uncertain knowledge about them. For purposes of design, optimization, and uncertainty quantification, it is often desirable to preserve these parameters as symbolic quantities in the reduced-order model (ROM). This allows the re-use of the ROM after changing the parameter so that the repeated computation of reduced-order models can be avoided. Significant savings in simulation times for full parameter sweeps, within optimization algorithms and Monte Carlo simulations can be achieved this way.
In this talk, we survey several approaches for computing ROMs for linear and nonlinear parametric systems. Parameter dependencies can be linear, polynomial, or nonlinear in general. The considered methods construct the ROMs in different way, using system theoretic concepts, rational interpolation, and/or greedy-type sampling approaches. We discuss the state-of-the-art of these methods and present recent advances. We illustrate the performance of the available methods using numerical examples, often in the context of industrially relevant applications.
The spatial Markov property of random planar maps is arguably one of the most useful properties of these random lattices. Roughly speaking, it says that after exploring a region of the map, the law of the remaining part only depends on the perimeter of the discovered region. It has been first heuristically used by Watabiki in the physics literature under the name of "peeling process" but was rigorously defined in 2003 by Angel in the case of the Uniform Infinite Planar Triangulation (UIPT). Since then, it has been used to derive information about the metric, (bernoulli and first passage) percolation, simple random walk and recently about the conformal structure of random planar maps. It is also at the core of the construction of "hyperbolic" random maps. In this talk, we will introduce smoothly the peeling process and present some of its main applications.
Understanding the coupling of physical models at different scales is important and quite challenging. In this talk, we focus on the issue of kinetic-fluid coupling, in particular, the half-space problems for kinetic equations coming from the boundary layer. We will present some recent progress in analysis and algorithm development for the linear half-space kinetic equations, and its application in coupling of neutron transport equations with diffusion equations. (joint work with Qin Li and Weiran Sun)
In the continuous setting, if the sources of Maxwell's equation satisfy the continuity equation and satisfy the divergence constraints at initial time, they satisfy those constraints for all time. This divergence constraint is in particular satisfied by sources computed from the Vlasov equation. However this is no more true at the discrete level.
Different methods have been developed to enable stable long time simulations of the Vlasov-Maxwell equations. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equation, for which additional boundary conditions are in some cases non trivial to find. The source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete Gauss’ law that needs to be defined in accordance with the discretization of the Maxwell propagation operator.
The source correction method can be applied quite naturally for conforming Finite Elements in the Framework of Finite Element Exterior Calculus. However, one needs to be careful when projecting the sources. A more difficult question is how this can be extended to Discontinuous Galerkin formulations. We shall see how the Finite Element Exterior Calculus can also be used in this setting along with appropriate projections and definition of the discrete divergence to get long time stability in this case.
Self similar fractals are often used in modeling porous media. Hence, defining a Laplacian and a Brownian motion on such sets describes transport through such materials. However, the assumption of strict self similarity could be too restricting. So, we present several models of random fractals which could be used instead. After recalling the classical approaches of random homogenous and recursive random fractals, we show how to interpolate between these two model classes with the help of so called V-variable fractals. This concept (developed by Barnsley, Hutchinson & Stenflo) allows the definition of new families of random fractals, hereby the parameter V describes the degree of `variability' of the realizations. We discuss how the degree of variability influences the geometric, analytic and stochastic properties of these sets. - These results have been obtained with Ben Hambly (University of Oxford) and John Hutchinson (ANU Canberra).
In this talk we discuss a class of stochastic PDEs which generalizes the famous KPZ equation. These equations must be renormalized in order to compensate several ill-defined products of random Schwartz distributions. We use the tools provided by the recent theory of regularity structures, with a particular focus on recursive procedures. (Joint project with Yvain Bruned and Martin Hairer)
Paracontrolled distributions combine the Fourier techniques of Bony's paradifferential calculus with ideas from the theory of controlled rough paths. This leads to a lightweight calculus for distributions that allows to handle nonlinear operations involving very irregular objects such as white noise, and which allows to give a meaning to and solve singular stochastic partial differential equations. Due to their good continuity properties, paracontrolled distributions are also a powerful tool for studying the convergence of discrete systems to continuum limits. I will present the basic ideas and techniques of paracontrolled distributions and how to apply them to solve the KPZ equation and to prove that it is the limit of the Sasamoto-Spohn discretization. This is joint work with Massimiliano Gubinelli and Peter Imkeller.
I will speak about the existence of weak solutions (and the existence and uniqueness of strong solutions) to the stochastic Landau-Lifshitz equations for multi(and one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire.
The talk is based on a joint works with B. Goldys and T. Jegaraj.
Self-assembly, a process in which a disordered system of preexisting components forms an organized structure or pattern, is both ubiquitous in nature and important for the synthesis of many designer materials. In this talk, we will address three variational models for self-assembly from the point of view of mathematical analysis and computation.
The first is a nonlocal perturbation of Coulombic-type to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. The functional has a rich and complex energy landscape with many metastable states. I will present a simple method for assessing whether or not a particular computed metastable state is a global minimizer. The method is based upon finding a ``suitable" global quadratic lower bound to the free energy.
The second model is a purely geometric and finite-dimensional paradigm for self-assembly which generalizes the notion of centroidal Voronoi tessellations from points to rigid bodies. Using a level set formulation, we a priori fix the geometry for the structures and consider self-assembly entirely dictated by distance functions. I will introduce a novel fast algorithm for simulations in two and three space dimensions.
Many mathematical models for hydrodynamic problems or for pattern forming systems are so complicated that a qualitative understanding of the full models being usable for practical applications does not seem within reach for the near future, neither analytically nor numerically.
Therefore, it is reasonable to approximate such models in various parameter regimes by appropriate reduced models whose qualitative properties are more easily accessible. To understand to which extent these reduced models yield correct predictions of the behavior of the original models it is important to justify the validity of these approximations by estimates of the approximation errors in the typical length and time scales.
In this talk, we present generic reduced model equations for slow modulations of typical solutions of hydrodynamic equations and reaction-diffusion systems and discuss mathematically rigorous justifications of these approximations. Special emphasis will be put on the approximations of the water wave equations by the Korteweg-de Vries equation and the Nonlinear Schrödinger equation.
Numerical methods for tasks like quadrature, optimization, linear algebra and the solution of differential equations estimate latent quantities from the observed result of tractable computations. In this sense, they are learning machines, and accessible to the framework of probabilistic inference.
What started as an entertaining observation has, over the past few years, turned up increasingly precise links between computation and inference, and begun to produce practically useful functionality. I will give a brief overview of recent developments, emphasizing a string of results identifying classic numerical methods -- Gaussian quadrature, conjugate gradients, BFGS , Runge-Kutta -- with maximum a posteriori estimators. I will also highlight several stirring potential applications that, coupled with a stack of fundamental questions, make probabilistic numerics an exciting area at the boundary between mathematics and computer science.
I will present a surprising connection between the inverse problem which consists of determining a potential from the scattering amplitude at fixed energy and Carlesson problem of convergence to initial data of the time dependent non elliptic Schrödinger equation. As a byproduct we show that potential with half derivative square integrable can be recovered from the scattering amplitude and that the method fails below that regularity. This is a joint work with Kari Astala from Helsinki and Keith M Rogers from Madrid.
In the original transport problem we are given two measures of equal mass and then look for an optimal map that takes one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed this problem by considering a measure whose marginals agree with given two measures instead of a bijection. This generalization linearizes the problem. Hence, allows for an easy existence result and enables one to identify its convex dual.
In robust hedging problems, we are also given two measures. Namely, the initial and the final distributions of a stock process.
We then construct an optimal connection. In general, however, the cost functional depends on the whole path of this connection and not simply on the final value. Hence, one needs to consider processes instead of simply the transport maps. The probability distribution of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy with the Kantorovich measure. But, financial considerations restrict the process to be a martingale. Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication) problem.
In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in the robust setting is given as the supremum of the expectations of the contingent claim over all martingale measures with a given marginal at the maturity.
This is joint work with Yan Dolinsky from Hebrew University.
I will present a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is based on Preiss' rectifiability result and on some elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David.
Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. In this talk we will show that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, including (i) Steiner formulae for closed convex cones, (ii) Stein's method and second order Poincaré inequality, and (ii) concentration of measure estimates. This work is joint with Laryy Goldstein (Southern California) and Giovanni Peccati (Luxembourg).
In this talk, we discuss several problems of the modern shadowing theory:- characterization of homeomorphisms having the Lipschitz shadowing property;- relations between shadowing and inverse shadowing;- C0-transversality and shadowing;- C1-interiors of sets of vector fields with shadowing properties.
Understanding intermittency in turbulent flows is still an outstanding open problem in classical physics. More than 15 years ago, for systems like passive advection and Burgers turbulence, the door for attacking this issue was opened with methods borrowed from field theory, most notably the instanton approach. Yet, conflicting measurements have let to a controversy around the usefulness and applicability of instantons in turbulent systems. In this talk, a novel method is presented to effectively compute minimizers of the Freidlin-Wentzell action functional that arises in the context of large deviation theory for stochastic partial differential equations. The presented method is then applied to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints in hydrodynamical equations to gain knowledge on statistics of rare events and the interplay between the nonlinear dynamics and the stochastic forcing. In particular it is applied to predict tail exponents of PDFs of turbulent flow quantities. This allows to solve the discrepancy between measurements and theory that was open for the last 15 years and reinstate instanton calculus in turbulence setups.
In recent years several engineering models, e.g., in the context of elasticity theory, were justified by a discrete to continuum analysis, i.e., by a passage from discrete/atomistic systems to continuum problems. For variational problems this involves the notion of Γ-convergence. I will talk about some analytical aspects of the so-called quasicontinuum method. This is a computational scheme that has been successfully applied to, e.g., continua with cracks. The idea there is to describe the material with an atomistic model close to the crack tips and with a continuum model away from the crack tips, i.e., away from singularities. This reduces the complexity of atomistic simulations. In joint work with M. Schäffner we analyze the validity of this quasicontinuum approximation. To this end we compare the limiting behavior of the quasicontinuum approximation with the limiting behavior of a fully atomistic system in the context of fracture mechanics by means of Γ-convergence techniques. This yields in particular that the discretization in the continuum region has an impact on the validity of the quasicontinuum approximation.
The non-backtracking matrix of a graph is a non-symmetric matrix on the oriented edge of a graph which has interesting algebraic properties and appears notably in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. It has also been used recently in the context of community detection. In this talk, we will study the largest eigenvalues of this matrix for the Erdos-Renyi graph G(n,c/n) and for simple inhomogeneous random graphs (stochastic block model). This is an ongoing joint work with Marc Lelarge and Laurent Massoulié.
The question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time is addressed. Analytical considerations and numerical simulations suggest high-symmetry flows being promising candidates for a blowup of the Euler equations in finite time [Pelz, 2001]. We present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations (effective resolution of up to 81923 mesh points) using Lagrangian tracer particles to verify the assumptions made by recent theorems by Deng et al. [2005,2006]. This approach connects vortex line geometry (curvature, spreading) and velocity increase with a non-blowup statement.We then ask the question, which role these singular structures play in turbulence statistics. More than 15 years ago, for certain systems like the problem of passive advection and Burgers turbulence the door for attacking this issue was opened by getting access to the probability density function to rare and strong fluctuations by the instanton approach. We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [2001]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement.
We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using Γ-convergence type arguments to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau-type functional and the minimizers of an exponentially weighted area-type functional. These minimizers yield the fastest moving traveling waves in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts. This is joint work with A. Cesaroni and M. Novaga.
We will discuss recent results regarding diffusion processes in random environment which are small, isotropic perturbations of Brownian motion in spacial dimension greater than two. Such processes were originally considered in the continuous setting by Sznitman and Zeitouni [4], where it was shown that, almost surely in the limit with respect to Brownian scaling, the diffusion converges in law to a Brownian motion with deterministic variance. This result was obtained through the construction of an inductive scheme which will be explained in the first portion of the talk.The remainder will be spent discussing more recent results regarding the existence of an invariant measure and proving a Liouville property for such environments. The invariant measure was constructed in Fehrman [2], and the method of proof relies upon aspects of the inductive scheme from [4]. The Liouville property for the environment was shown in Fehrman [3]. It relies upon the results of [2] and [4], and adapts a technique from the discrete setting of Benjamini, Duminil-Copin, Kozma and Yadin [1]. Finally, we will mention the applications of this work to the corresponding problems in random homogenization.[1] I. Benjamini, H. Duminil-Copin, G. Kozma, and A. Yadin. Disorder, entropy and harmonic functions. Arxiv:1111.4853, 2011.[2] B. Fehrman. On the existence of an invariant measure for isotropic diffusions in random environment. Arxiv:1404.5274, 2014.[3] B. Fehrman. A Liouville property for isotropic diffusions in random environment. Arxiv:1406.1549, 2014.[4] Alain-Sol Sznitman and Ofer Zeitouni. An invariance principle for isotropic diffusions in random environment. Invent. Math., 164(3):455–567, 2006.
For half a century, nuclear fusion has been a dream of humanity: clean, inexhaustible energy without the problems of fossil fuels, nuclear fission or conventional renewable energy sources. In this talk, I shall give a general introduction into fusion research, describe its current status (including the new Wendelstein 7-X experiment in Greifswald), but the emphasis of the talk will be on mathematics. In particular, I will describe a few concrete, unsolved mathematical problems whose solution would be greatly appreciated.
We show that an approximation of the hydrodynamical Euler equation describes the binormal mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively.
Defects play a crucial role in influencing the macroscopic properties of solids—examples include the role of dislocations in plastic deformation, dopants in semiconductor properties, and domain walls in ferroelectric properties. These defects are present in very small concentrations (few parts per million), yet, produce a significant macroscopic effect on the materials behavior through the long-ranged elastic and electrostatic fields they generate. The strength and nature of these fields, as well as other critical aspects of the defect-core are all determined by the electronic structure of the material at the quantum-mechanical length-scale. Hence, there is a wide range of interacting length-scales, from electronic structure to continuum, that need to be resolved to accurately describe defects in materials and their influence on the macroscopic properties of materials. This has remained a significant challenge in multi-scale modeling, and a solution to this problem holds the key for predictive modeling of complex materials systems.
In an attempt to address the aforementioned challenge, this talk presents the development of a seamless multi-scale scheme to perform electronic structure calculations at macroscopic scales. The key ideas involved in its development are (i) a real-space variational formulation of electronic structure theories, (ii) a nested finite-element discretization of the formulation, and (iii) a systematic means of adaptive coarse-graining retaining full resolution where necessary, and coarsening elsewhere with no patches, assumptions or structure. This multi-scale scheme has enabled, for the first time, calculations of the electronic structure of multi-million atom systems using orbital-free density-functional theory, thus, paving the way for an accurate electronic structure study of defects in materials. The accuracy of the method and the physical insights it offers into the behavior of defects in materials is highlighted through studies on vacancies and dislocations. Current efforts towards extending this multi-scale method to Kohn-Sham density functional theory will also be presented, which include: (i) the development of higher-order adaptive finite-element formulation for efficient real-space Kohn-Sham DFT calculations; (ii) the development of a linear-scaling approach that is applicable to both insulating and metallic systems.
In this seminar I will introduce a class of (possibly nonlocal) curvatures, and I will describe the level set approach to the corresponding geometric flows. Then, I will introduce a notion of generalized perimeter and describe the minimizing movements approach.
Such method is consistent with the level set approach whenever the curvature is the first variation of the perimeter in this generalized context. These results are in collaboration with A. Chambolle and M. Morini.
In the machine learning literature, one approach to "prediction" assumes that there are two more "experts"; the best prediction in this setting is the one that "minimizes regret", i.e. minimizes the shortfall relative to the best-performing expert. My talk is motivated by a model problem recently considered by Rina Panigraphy, Michael Kapralov, and Alexandr Andoni, involving the prediction of a binary sequence (loosely speaking: a stock whose price is restricted to a binomial tree). I'll discuss joint work with Kangping Zhu about a continuum limit, in which the optimal strategy is associated with a 2nd order PDE. While the PDE is very nonlinear, explicit solutions are available in many cases due to a surprising link to the linear heat equation. The continuum limit being considered here is related to my 2006 work with Sylvia Serfaty on a deterministic-game interpretation of motion by curvature.
We study periodic scalar first-order conservation laws with stochastic forcing. Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour in any space dimension. For sub-cubic fluxes, we show the existence of an invariant measure. Moreover for sub-quadratic fluxes we show uniqueness and ergodicity of the invariant measure.
We present a proof of existence of the density with respect to the Lebesgue measure for the finite dimensional marginal distributions of the law of solutions of the 3D Navier-Stokes equations forced by Gaussian noise, as well as of regularity of the density in Besov spaces.
Classical methods, such as the Malliavin calculus, do not work in this setting for reasons which are strongly related to the three dimensional case. Existence of a density is then ensured by an ad-hoc probabilistic method.
The same method provides also regularity in time of the densities, as well as absolute continuity of the laws of some quantities (the energy and dissipation) that depend on a infinite number of components.
When the random forcing has no full support in Fourier space, we prove existence of a density, although without any regularity, by using the backward local smoothness of trajectories, and weak--strong uniqueness.
The talk starts with a presentation of the dyadic model, a system of infinitely many differential equations that should capture the essential features of the evolution and interaction of energy in the motion of incompressible fluids. So far the model has provided heuristics insights on the original PDEs. Recent results [Tao2014] have shown that the connection is more robust and some insight can be made partially rigorous.
We present two results. In the first part we prove global existence of smooth solutions for a slightly supercritical dyadic model. This catches for the dyadic a conjecture that for Navier-Stokes equations was formulated in [Tao2009]. In the second part, we consider the model with non-linearity of strong intensity and driven by Gaussian noise. We prove the emergence of blow--up, that is loss of regularity after a random time. The contribution of the noise is essential to prove that blow-up occurs with full probability, by means of conditional recurrence.
Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a negligible set , showing in particular the validity of Young's law at almost every point (joint with Francesco Maggi).
A peacock is a continuous time stochastic process non-decreasing in convex ordering. A process is a peacock if and only if there is a martingale which has the same one-dimensional marginal distributions. Here we are interested in studying the ''extremal'' martingale associated to a given peacock. We will dicuss some results obtained by the martingale transport approach and the Skorokhod embedding approach.
I will discuss the fine structure of the branch set of multiple-valued solutions to the Laplace equation and minimal surface system. It was previously known that the branch set of a multiple-valued solution on a domain in $\mathbb{R}^n$ has Hausdorff dimension at most $n-2$. In joint work with Neshan Wickramasekera, we show that the branchset is countably $(n-2)$-rectifiable. This result follows from the asymptotic behavior of solutions near branch points, which we establish using a modification of the frequency function monotonicity formula due to F. J. Almgren and an adaptation to higher-multiplicity of a "blow-up" method due to L. Simon that was originally applied to "multiplicity one" classes of minimal submanifolds satisfying an integrability hypothesis. In ongoing independent work, I show that the branch set decomposes into locally real analytic submanifolds along which the frequency is constant and points where the frequency is not lower semicontinuous.
I will explain how ideas from the theory of non-linear waves, namely the paradifferential calculus of Bony, can be used to tackle problems in the theory of SPDEs and more generally for problems involving non-linear operation on distributions. These methods are suitable to solve problems going from differential equations driven by rough paths to the parabolic Anderson model in 2 dimensions, the Kardar-Parisi-Zhang equation, the equation of stochastic quantisation in 3d. If time permits we will may out also links with the recent theory of regularity structures of M. Hairer.
In discrete-time population models, the year to year development of the population is described by a map. Usually, the spectral radius of its derivative at the origin (the extinction state) acts as a threshold parameter that separates population extinction from population persistence. If it is taken into account that many populations have males and females that need to mate in order to reproduce, the first order approximation of the map at the origin is not longer additive but just homogeneous. A spectral radius can also be defined for homogeneous order-preserving maps, and it acts as a threshold parameter in a completely analogous way. Establishing population persistence relies on finding conditions for the spectral radius to be a lower eigenvalue with a positive lower eigenvector. As in the linear case, the spectral radius can be characterized and approximated by Collatz-Wielandt numbers and bounds, but different techniques are required. As motivation and illustration, a two-sex population is considered with very short reproductive seasons and diffusion of both males and females (impulsive PDE).
In this talk, we study a Thomas-Fermi-Dirac-von Weizsacker type density functional theory for the response of a single layer of graphene resting on a dielectric substrate to point charges some distance away from the layer. In particular, we obtain results on existence, regularity, and asymptotic decay behavior of the minimizer. This relates to questions in recent physics literature on out-of-plane charge screening in graphene. (Joint with Vitaly Moroz and Cyrill Muratov)
In three-dimensional turbulent flows, the flux of energy from large to small scales breaks time-reversal symmetry. I will discuss how this irreversibility can be observed by following the trajectories of tracers, moving with the fluid. For the problem of relative particle dispersion, irreversibility implies that particles separate slower forward in time than backward. This property can be understood in terms of an exact relation, established directly from the Navier-Stokes equations. A more surprising observation is that a tracer particle needs more time to build up large kinetic energy, than to dissipate it. This allows us to quantify the irreversibility of the turbulent fluid.
Enhancement of diffusion and reaction processes by advection is a classical subject that has been extensively studied by both physicists and mathematicians. In this talk I will consider reaction-diffusion on a bounded domain in the presence of a fast incompressible flow. I will also consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature.
In many applications structures can be described as (local) minimizer of suitable bending energies. The most prominent example is the variational characterization of shapes of biomembranes by the use of Helfrich- or Willmore-type functionals. Whereas the restriction to topological spheres is natural in many applications, it is sometimes more reasonable to consider the class of orientable connected surfaces of arbitrary genus instead. For example the inner membrane of mitochondria cells shields the inside matrix from the outside but shows - in contrast to old textbook illustrations - a lot of of handle-like junctions and therefore represents a higher genus surface. In this example another natural constraint comes into play, given by the confinement of the inner membrane to a 'container' that is given by the outer membrane of the mitochondria. This motivates to consider the following variational problem: Minimize Willmore's energy in the class of all compact, connected, orientable surfaces without boundary that are embedded in a bounded domain and have prescribed surface area. We consider a phase field approximation to this problem, i.e., we approximate the surface by a level set function $u$ admitting the value +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of $u$. Diffuse interface approximations for the area functional, as well as for the Willmore energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a nested minimization of two phase fields, the second one being used to identify connected components of the surface. We present a proof of Gamma-convergence of our model to the sharp interface limit. This is joint work with Matthias Röger (TU Dortmund) and Luca Mugnai (MPI Leipzig).
A thermodynamically consistent model for incompressible two-phase fluid flows with phase transitions driven by temperature is introduced and analyzed. Concentrating on the case of equal densities, we establish well-posedness and study the qualitative behavior of solutions. In particular, we characterize all equilibria and study their stability properties. The entropy turns out to be an important quantity in the stability analysis.
Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly, but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ''dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ''spine'' decomposition for the fragmentation, and Markov renewal theory.
This is joint work with Bénédicte Haas (Paris-Dauphine).
Transportation-cost inequalities can be used to characterize the concentration properties of probability measures. In this talk, we will investigate the law of diffusions driven by a multidimensional Gaussian process and we will show that Lyons' rough paths theory is a powerful tool to establish such inequalities on path spaces.
We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Zhan Shi (2003). Along the way, we investigate the law of the stable process evaluated at the first time its integral hits zero, extending classical formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion. This is joint work with Christophe Profeta (Evry-Val d’Essonne).
The self-organized emergence of spatio-temporal patterns is a ubiquitous phenomenon in nonlinear processes on large domains. In this talk the Landau-Lifshitz-Gilbert-Slonczewski equation is studied from this viewpoint, highlighting various aspects of the theory. The model describes magnetization dynamics in the presence of an applied field and a spin polarized current. Here we consider the case of axial symmetry and focus on one space dimension.
This is joint work with Christof Melcher (RWTH).
We consider a macroscopic limit for the Ohta-Kawasaki energy. This model has been used to describe phase separation for diblock-copolymers. We first investigate existence and shape of minimizers of the energy with prescribed volume (of the one phase) in the full space setting. We then consider situation of periodic configurations with prescribed density of the minority phase. We show that in a certain regime, the energy Gamma-converges to a homogenized problem.
This is joint work with C. Muratov and M. Novaga.
The Kac-Ising model is spin model on a grid in which every spin interacts with a large number of neighbours. It is a popular model in statistical physics as it captures some aspects of the "usual" Ising model, but it is often simpler to study. We study the Glauber dynamic associated to this Kac-Ising model on a finite sub-grid of $Z^2$ near its critical temperature. We show that in a suitable scaling the locally averaged spin field is well described by the formal stochastic PDE \[\partial_t \Phi = \Delta \Phi - (\Phi^3 - \infty \Phi) + \xi,\] where $\xi$ denotes space time white noise, and the "infinite constant" appears as the limit of a renormalisation procedure. This is joint work with J.C. Mourrat.
Sticky particle solutions to the one-dimensional pressureless gas dynamics equations can be constructed by a suitable metric projection onto the cone of monotone maps, as was shown in recent work by Natile and Savare. Their proof uses a discrete particle approximation and stability properties for first order differential inclusions. Here we give a more direct proof that relies on a result by Haraux on the differentiability of metric projections.
I'll describe the asymptotic behavior of solutions to a reaction-diffusion equation with KPP-type nonlinearity. One can interpret the solution to the PDE in terms of a branching Brownian motion. It is well-known that solutions to the Cauchy problem may behave asymptotically like a traveling wave. On the other hand, for certain initial data, M. Bramson proved that the solution to the Cauchy problem may lag behind the traveling wave by an amount that grows logarithmically in time. Using PDE arguments, we have extended this statement about the logarithmic delay to the case of periodic, spatially varying reaction rate. The PDE approach involves the study of the linearized equation with Dirichlet condition on a moving boundary. The method also gives refined asymptotics for the solution to the equation with time-varying coefficients. This is joint work with Francois Hamel, Jean-Michel Roquejoffre, and Lenya Ryzhik.
This talk addresses the issue of extending the PDE description of a dilute Bose gas beyond the regime of the Gross-Pitaevskii (cubic-nonlinear-Schroedinger equation) regime. In particular, I will discuss two interrelated aspects of this problem: (i) The modeling of atomic pair correlations (beyond the Gross-Pitaevskii limit). (ii) An effective, homogenized description of the Bose gas for a scattering length with periodic microstructure.
While classical stability analysis usually assumes that the base state is constant in time, many patten forming systems that arise in practice have non-constant base states. We suggest an approach that applies if the evolution is slow, and that takes into account the changing instability characteristics. We present results for several problems from continuum mechanics such as Rayleigh-B\'enard convection in evaporating liquid mixtures or receding liquid and solid layers.
We study the asymptotic behavior of boundary vortices and their interaction energy in a certain regime of thin-film micromagnetics. The key tool relies on the notion of “boundary Jacobian" which detects the topological defects at the boundary. The concentration of the energy around boundary vortices is proven via a Gamma-convergence result and we determine the renormalized energy that represents the interaction between the boundary vortices leading to their optimal position. The major difficulty consists in estimating the nonlocal part of the micromagnetic energy generated by the static Maxwell equation in order to detect the exact terms corresponding to the topological defects.
This is a joint work with Matthias Kurzke (Nottingham).
I will describe a new quantitative approach to stochastic homogenization for elliptic equations in divergence form. This gives the first quantitative results for nonlinear equations, but also new results for linear equations. The idea is to show that the energy of a minimizer spreads evenly over large scales, which can be thought of as a kind of quantitative compensated compactness.
I consider Ising models in two and three dimensions with nearest neighbor ferromagnetic interactions and long range, power law decaying, antiferromagnetic interactions. If the strength of the ferromagnetic coupling J is larger than a critical value J_c, then the ground state is homogeneous and ferromagnetic. As the critical value is approached from smaller values of J, it is believed that the ground state consists of a periodic array of stripes (d=2) or slabs (d=3), all of the same size and alternating magnetization. In this talk I present a rigorous proof that the ground state energy per site converges to that of the optimal periodic striped/slabbed state, in the limit that J tends to the ferromagnetic transition point. While our theorem does not prove rigorously that the ground state is precisely striped/slabbed, it does prove that in any suitably large box the ground state is striped/slabbed with high probability. Joint work with E. H. Lieb and R. Seiringer.
All Gibbs states of the supercritical q-state Potts model on Z2 are convex combinations of the q pure phases; in particular, they are all translation invariant. We recently proved this theorem with Hugo Duminil-Copin (Geneva), Dima Ioffe (Haifa) and Yvan Velenik (Geneva). I will explain the basic concepts underlying this result and present the heuristics of the proof, which consists of considering the model in large finite boxes with arbitrary boundary condition, and proving that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature beta > beta_c(q)=log(1+sqrt(q)).
The Kardar-Parisi-Zhang (KPZ) universality class includes directed polymers in random media in 1+1 dimension. According to the universality conjecture, for any finite temperature, the fluctuations of the free energy (e.g. for point-to-point) directed polymers is expected to be distributed as the (GUE) Tracy-Widom distribution in the limit of large system size. This distribution arose first in the context of random matrices.
Detailed results as the fluctuation laws for models in the KPZ were, until recently, available only for "zero temperature models". We consider two models at positive temperature, a semi-discrete and the continuum directed polymer models, and determine the law of the free energy fluctuations. Results for the stationary KPZ solution will be also mentioned.
This talk is based on a joint work with Alexei Borodin and Ivan Corwin http://arxiv.org/abs/1204.10242, their previous work http://arxiv.org/abs/1111.4408, as well as a work in progress with them and Balint Veto.
In a celebrated paper, Smirnov proved that critical site-percolation on the regular triangular lattice has a non-trivial, conformally invariant scaling limit and that this can be used to derive for instance the value of critical exponents. The argument is unfortunately very specific to this particular lattice, and so far has not been generalized to any other natural case — in particular, percolation on $\mathbb Z^2$ is much beyond reach of current methods. I will present one direction in which the proof can be extended into a non-trivial class of models that somehow interpolate between the triangular lattice and general planar cases.
Preferential attachment networks are a class of dynamical networks in which new nodes are connected to existing nodes with a probability proportional to an increasing function of their degree. Asymptotically the degree distributions of these networks follow a power law with tunable power law exponent tau. In this talk I survey recent results, obtained in collaboration with Steffen Dereich and Maren Eckhoff, on robustness and vulnerability of these networks. In particular we show that in the regime 2
When one removes a sector from a circular sheet of paper and glues the edges back together, the resulting shape is approximately conical. Despite the simplicity of the setup, a proof of this 'fact' starting from basic models for thin elastic sheets has not been found yet. In this talk we investigate the scaling of the elastic energy with the sheet thickness $h$ for this setting. Our main innovation is the identification of a suitable technical assumption to work with. Namely, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a diffeomorphism of some subset of the tangent space at 0 with the whole manifold. Under this assumption, we will show that the elastic energy per unit thickness scales with $h^2|\log h|$, which is the scaling of a (regularized) cone.
We study the model of random walk in random environments under the ballisticity condition (T') of Sznitman in dimension four and higher. We prove a version of a local CLT for the model and also the existence of an equivalent measure which is invariant with respect to the point of view of the particle. This is a joint work with Noam Berger and Moran Cohen.
By analogy with the Liouville measures constructed by Duplantier and Sheffield in the case of the Gaussian Free Field, we construct measures associated to a collection of nested conformal loops. Then, we study the extremal process associated to the maximal conformal radius. We show that it gives a decorated Poisson point process, similarly to what is already known in the branching Brownian motion case, and to what is conjectured for the GFF.
When one removes a sector from a circular sheet of paper and glues the edges back together, the resulting shape is approximately conical. This might seem very obvious, but a rigorous proof is lacking. In this talk we investigate the scaling of the elastic energy with the sheet thickness $h$ in this setting. Thinking of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space, we will assume that the exponential map at the origin (the center of the sheet) supplies a diffeomorphism of some subset of the tangent space at 0 with the whole manifold. Under this assumption, I will show that the elastic energy per unit thickness scales with $h^2|\log h|$, which is the scaling of a (regularized) cone.
Magnetization dynamics in ferromagnetic materials at low temperatures can be described by the Landau-Lifshitz-Gilbert (LLG) equation. The classical LLG model holds for constant temperatures that are sufficiently far from the Curie temperature. Due to increasing miniaturization of magnetic devices and recent applications, such as the thermally-assisted magnetic recording, it has become necessary to account for temperature effects in the model. We review two most common approaches currently used for the modelling of thermally activated dynamics in ferromagnetic materials. The first approach includes thermal fluctuations via a random term and leads to a stochastic mesoscopic model, the so-called Stochastic-LLG equation. The second approach includes temperature effects via an additional term in the LLG equation and leads to a deterministic macroscopic model. We propose structure preserving finite element based numerical approximations for the respective equations and show numerical experiments to demonstrate some interesting features and connections between the two models.
We will be interested in the distribution of complex zeros of the random polynomials of the form $$ P_n(z) = \sum_{k=0}^{n} f_{k,n} \xi_k z^k, $$ where $f_{k,n}$ are deterministic complex coefficients and $\xi_1,\xi_2,\ldots$ are independent identically distributed random variables such that $\mathbb E \log |\xi_k|
In a joint work with Christophe Sabot (2011), we show an explicit link between the Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, and the supersymmetric hyperbolic sigma model developed by Zirnbauer in 1996, through another continuous-time process conceived by Werner in 2000, the Vertex-reinforced jump process (VRJP), favouring sites with more local time.
This correspondence enables to show Diaconis' conjecture that ERRW is strongly recurrent in any dimension for large reinforcement (in fact on any graph of bounded degree), and that it is transient in dimension larger than 3, using localisation/delocalization techniques of Disertori, Spencer and Zirnbauer (2010).
Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed?
In three or more dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-defined "component" structure in our random porous set.
Based on joint work with Frank den Hollander.
Given a sequence of undirected connected graphs $G_n$ of $n$ vertices with bounded degrees and weights and an additional sequence of killing parameters $c_n$, one can define a sequence $\mu_n$ of natural non-normalized measures on non-trivial loops of $G_n$. Denote by $Q_n$ the transition matrix associated with $G_n$. Assuming that the empirical distributions $\nu_n$ of the eigenvalues of $Q_n$ converge, we determine the limit of the $\mu_n-$proportion of \{loops which cover every vertex\} as $n\rightarrow\infty$. Let $\mathcal{L}_n$ be the Poissonian "loop soup" with intensity $\frac{\mu_n}{n}$. A a corollary, we determine the limit law of the number of the loops covering the whole graph. As two concrete examples, we consider the closed balls in a regular graph and the discrete tori.
In contrast to inert matter, biological systems are characterized by structurally complex network architecture with domineering long range interactions. This leads, even in the absence of ATP, to highly unusual mechanical properties in both statics and dynamics. In this talk we discuss the simplest system of this type representing a segment of a muscle fiber and systematically study its peculiar thermodynamics and kinetics. Among the main macroscopic effects exhibited by our prototypical model are: the non-equivalence of soft and hard devices, the possibility of negative stiffness, coherent internal fluctuations at fixed loading and the presence of long living non-equilibrium steady states. Most intriguing, there is evidence that muscles are internally tuned to perform close to a critical point. We show that in the presence of a correlated noise, representing ATP, the system can generate active elasticity.
We show that the electric resistance between the origin and the n-th generation of a critical oriented branching random walk trace in dimensions d < 6 is at most O(n^{1-alpha}) for some universal alpha > 0. As a corollary, the mean exit time of the simple random walk on the trace from a ball in the graph distance is polynomially smaller than in d > 6, answering a question of Barlow, Jarai, Kumagai and Slade. (Joint work with Asaf Nachmias)
Consider super-critical bond percolation on $\mathbb{Z}^2$ (the square lattice). We prove the Cheeger constant of the super-critical percolation cluster restricted to finite boxes scales a.s to a deterministic quantity. This quantity is given by the solution to the isoperimetric problem on $\mathbb{R}^2$ with respect to a specific norm. We also prove every set achieving the Cheeger constant converges to the deterministic Wulff crystal, with respect to the same norm. Joint work with Marek Biskup, Oren Louidor and Ron Rosenthal.
I will discuss some properties of timelike extremal surfaces in the flat Minkowski space. I will also present a notion of weak solution, in the spirit of the varifold theory in the Euclidean space.
We study the limiting spectral distribution (LSD) of Wigner matrices whose entries come from a stationary process. We provide an explicit description of the moments of the limiting measure. The description of the LSD can also be made via its Cauchy transform which is characterized as the solution of a functional equation. In two special cases, we get a neat description of the LSD - one as a free product convolution of two distributions, and the other one as a dilation of the Wigner semicircular law.
This is a joint work with Arijit Chakrabarty and Deepayan Sarkar.
Recently there has been some interest in the deduction of the variational principles of non-linear elasticity from statistical mechanics. A powerful tool that is relevant in such derivation is the so-called renormalization group technique. In order to simplify the renormalization group technique a decomposition of the field is needed: the so-called Finite Range Decomposition. In this talk I will discuss how to deduce the finite range decomposition for the non translation invariant case and show haw this may be employed to prove the strict convexity of the free energy near the origin. If time permits, I will also discuss a general representation formula for the free energy.
Many spatial interacting models are in high-dimension expected to behave similar to a corresponding simple model without interactions, such as simple random walk and branching random walk. In this talk I introduce the concept of mean-field behavior and discuss a method to prove it. We apply these ideas to key models in statistical physics, namely, self-avoiding walk, percolation, lattice trees and lattice animals.
We focus on a novel method call Non-backtracking lace expansion, which can be used to study nearest-neighbor models in dimensions above, but close to, the upper critical dimension above which mean-field behavior is expected.
In this talk I will describe some free boundaries arising in the description of regions where the motion of thin fluid layers becomes slow. These regions are often characterized by the onset of oscillations in the in the interfaces. Classical examples include the description of fluid layers attached to a moving wall. I will describe the free boundary problem arising in the description of the regions where a liquid moving in a bumpy interface accumulates. In the two-dimensional case, this problem reduces to a boundary value problem for a fourth order ordinary differential equation whose solution can be obtained using methods of dynamical systems.
(Joint work with C. Cuesta).
Among the basic results of random matrix theory, the circular law is in the same time one of the simplest to state, and one of the hardest to prove. In this talk, we will present various models related to the circular law, in connection with recent advances.
After an introduction to Poissonian ensembles of Markov loops, we study their clusters in several situations: On the line, on the complete graph, and on d-dimensional lattices.
Aldous introduced a growth model for the binary tree where clusters freeze as soon as they become infinite. Benjamini and Schramm showed that such process does not exist on the triangular lattice. To circumvent the problem of non-existence, van den Berg, de Lima and Nolin modified the process so that the clusters freeze as soon as their diameter reach N, the parameter of the model. They raised the question whether the probability that a given vertex eventually freezes in the N-parameter model tends to 0 as N goes to infinity. We answer this question in the affirmative by showing that for all k>0, in the N-parameter frozen percolation process the probability that all the frozen clusters in a box of size kN form close to time 1/2 tends to 1 as N goes to infinity. We also highlight some connections between frozen percolation processes and forest fire models.
We consider a continuous time random walk on the lattice $\mathbb{Z}^d$ in an environment of random conductances, $\mu_{x,y}$. The law of the environment is assumed to be ergodic with respect to space shifts with $\mathbb{P}[0 < \mu_{x,y} < \infty] = 1$. In this talk, I will explain how a quenched invariance principle can be established under suitable moment conditions. A key ingredient in the proof is to establish the sub-linearity of the corrector by means of Moser's iteration scheme. This is joint work with Sebastian Andres (Univ. Bonn) and Jean-Dominique Deuschel (TU Berlin).
Many properties of electrons in crystalline solids are described by Schroedinger operators with potentials whose periodicity reflects the periodic structure of the crystal. The construction of generalized eigenfunctions with exponential decay is crucial, both from a theoretical and from a numerical viewpoint. Through the Bloch-Floquet transform a band structure for the spectrum of the Hamiltonian naturally appears (Bloch bands) and in presence of a spectral gap, typical of insulators and semiconductors, to the relevant family of bands there is an associated Bloch bundle, whose rank is the number of relevant bands.
A natural basis for simple or multiband systems is given by composite Wannier functions, whose existence together with their exponential localization can be obtained using geometric obstruction theory whenever the Hamiltonian satisfies the time reversal symmetry (which ensures the triviality of the associated Bloch bundle and the existence of a smooth trivializing frame). This result largely extends partial results in case of a simple band or in one dimension.
In the talk we describe how to construct exponentially localized composite Wannier functions by variational methods, minimizing the localization functional introduced by Marzari and Vanderbilt. This functional can be transformed in one of calculus of variations type for maps from a torus to the unitary group. To study both existence and regularity of minimizers (and the corresponding localization property for the Wannier functions), it is very useful to exploit the connection with the theory of harmonic maps between Riemannian manifolds and the regularity theory for the corresponding elliptic systems. Here we aim to show existence and exponential localization for maximally localized Wannier functions (minimizers), in presence of the time reversal symmetry.
(joint with Gianluca Panati)
We investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices. The crystal lattice is a generalisation of the square lattice, the triangular lattice and the hexagonal lattice. As we shall observe, the quasi-linear parabolic equation in the limit is defined on a flat torus and depends on the local structure of the crystal lattice. The scaling limit is constructed by using a discrete harmonic map. We formulate the local ergodic theorem on the crystal lattice by introducing the notion of local function bundle, which is a family of local functions on the configuration space. The ideas and methods are taken from the discrete geometric analysis to these problems. Results we obtain are extensions of ones by Kipnis, Olla and Varadhan to crystal lattice cases.
