The Teichmüller space of a surface parametrizes (marked) conformal structures. It covers the moduli space of Riemann surfaces and carries many interesting structures in its own right. The uniformization theorem allows to identify the Teichmüller space with the space of hyperbolic structures, and to embed into the space of representations of the fundamental group of the surface into SL(2,R).
Higher Teichmüller spaces are generalizations of Teichmüller space in the context of Lie groups of higher rank such as SL(n,R). They are related to Higgs bundles, bounded cohomology, dynamics, as well as cluster algebras or total positivity. The talk will provide an introduction to higher Teichmüller spaces and showcase some of the connections.
Threshold-linear networks (TLNs) are commonly-used rate odels for modeling neural networks in the brain. Although the nonlinearity is quite simple, it leads to rich dynamics that can capture a variety of phenomena observed in neural activity: persistent activity, multistability, sequences, oscillations, etc. Here we study competitive threshold-linear networks, which exhibit both static and dynamic attractors. These networks have corresponding hyperplane arrangements whose oriented matroids encode important features of the dynamics. We will show how the graph associated to such a network yields constraints on the set of (stable and unstable) fixed points, and how these constraints affect the dynamics. In the special case of combinatorial threshold-linear networks (CTLNs), we find an even stronger set of "graph rules" that allow us to predict emergent sequences and to engineer networks with prescribed dynamic attractors.
Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance.
PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA.
In this talk I will first give an introduction to persistent homology; I will then discuss some applications, and the theoretical challenges in the multi-parameter case.
No prior knowledge on the subject is assumed.
We will present a brief introduction to phylogenetic tree models. These are generalisations of Markov chains used in phylogenetic. We will focus on their algebraic side, mostly on so called group-based models and their relations to toric geometry.
The talk is devoted to sharp Hardy inequalities for Laplace (type) operators on weighted graphs. We give sufficient criteria for the existence of Hardy weights in terms of positive superharmonic functions. The method is based on a discrete version of the supersolution construction which has been studied before in the context of elliptic Schrödinger operators on continuous spaces. We also intend to shed some light on optimality criteria and say a word or two about the proofs.
Joint work with Matthias Keller and Yehuda Pinchover.
We prove the Gruenbaum conjecture, i.e. for a 2-dimensional simplicial complex embedding in $R^4$ on T triangles and E edges, we have $T < 4E$. We also extend this to higher dimensions and manifolds. The provided bounds are tight. The methods are perhaps surprisingly, not topological.
Counting the number of eigenvalues of the Laplace operator on a closed Riemannian manifold is an old problem. Its solution is the so called Weyl law which gives an asymptotic count of these eigenvalues. If the Laplace operator acts on a non-compact space, e.g., the modular surface, the situation is more difficult and has applications in number theory and automorphic forms. I want to explain some recent and current work on some variant of the Weyl law for families of operators which has applications to the distribution of Hecke eigenvalues in number theory.
We give diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result is using only curvature and maximal vertex degree. Rigidity of this diameter bound characterizes the hypercube. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians.
The strong Arnold conjecture states that every Hamiltonian symplectomorphism on a closed symplectic manifold $M$ has as many fixed points as the number of critical points of a smooth function on $M$. In my talk I shall survey progress in the strong Arnold conjecture including recent results obtained by Kaoru Ono and myself in 2015 concerning the strong Arnold conjecture on compact nonsimply connected manifolds.
The moduli space of compact Riemann surfaces of genus 1 is equal to the basic locally symmetric space $SL(2, Z) \ H^2$, the quotient of the upper halfplane $H^2$ by the modular group $SL(2, Z)$. Therefore, the moduli spaces $M_g$ of compact Riemann surfaces of genus $g$ share the common root with locally symmetric spaces and other similarity. In this talk I will discuss some results on the geometry, topology and analysis of the moduli spaces partially motivated by this analogy.
In this talk I will talk about uniformly convex metric spaces and give some properties of those spaces. The main focus is on a notion of weak sequential convergence and on the so-called co-convex topology. Those two notions of weak convergence agree with the usual weak topology on Banach spaces. However, they might not agree in general: On CAT(0)-spaces one can show that convergence in the co-convex topology is weaker than the weak sequential convergence. I will construct a CAT(0)-space whose co-convex topology is not Hausdorff showing that the two topology are distinct.
In the end generalized barycenters of measures will be introduced. With such a notion one can proof a variant of the Banach-Saks property, i.e. every bounded sequence has a subsequence weakly sequentially converging to some point such that the sequence of (generalized) barycenters of Cesàro means strongly converges to the same limit point. Here the Cesàro means of a sequence are defined as the partial sums over the delta measures supported on the sequence.
The "fibering problem" asks whether a given manifold M is a fiber bundle over another given manifold B. We will survey on some classical and recent results, focusing on the relation of this problem to algebraic K-theory.
The talk will give an introduction to the spectral theory of quotients of semisimple Lie groups by lattices and of the associated locally symmetric spaces. Classic examples are provided by the lattices ${\rm SL} (n, \mathbb{Z})$ (and their finite index subgroups) in the semisimple Lie groups ${\rm SL} (n, \mathbb{R})$. Of particular interest is the case of noncompact quotients (which includes this series of examples). Here, even the existence of infinitely many eigenvalues in the discrete spectrum is not clear (and has been established only in the last decade). There are (to a large part conjectural) structural relationships between these spectra for different groups, and connections to number theory. An important tool is the trace formula introduced by Selberg (1956) and developed by Arthur (since 1978), which in this case is the result of a regularization procedure. We will discuss asymptotic counting problems for the discrete spectrum (Weyl's law and limit multiplicities).
In this talk, I will explain an optimal dimension-free upper bound for eigenvalue ratios $\lambda_k/\lambda_1$ of the Laplacian on a closed Riemannian manifold with nonnegative Ricci curvature. This is achieved by borrowing tools from theoretical spectral clustering algorithm analysis in computer science. I will further discuss several of its applications, including improving higher-order Buser inequality, higher-order Gromov-Milman inequality and multi-way isoperimetric constant ratios estimate. Its extension to compact finite-dimensional Alexandrov spaces with nonnegative curvature affirms a recent conjecture of Funano and Shioya. On finite discrete graphs, it also has very interesting applications.
We will survey a number of important results on spectral radii and principal eigenvectors of adjacency matrices of graphs that appeared within the last ten years. Particular focus will be put on proof techniques and open problems in the field.
Experimental geometry is an exploration of geometric form aimed at generating understanding and insight. This talk gives an overview of some of my projects that unite under the idea of experimental geometry as a route to exploring soft matter physics. I will explore the link between two-dimensional hyperbolic tilings and self assembled miktoarm star copolymers, periodic filament entanglements and skin swelling, as well as geometric insight into foam rheology.
Supersymmetry is traditionally thought of as an extension of the classical Lie-algebras describing the isometries, or conformal isometries, of Minkowski space and its cousins in different metric signatures. The corresponding algebraic structure is conveniently formalized by the concept of a super-, or graded-, Lie-algebra. The superalgebras relevant for physical applications were classified in the late 70's by Nahm, using results of Kac.
One may also consider supersymmetric extensions of the algebras of conformal isometries of more general manifolds. The corresponding algebraic structure was proposed by us to be that of a ``conformal Killing superalgebra''. I report on their definition and classification. Such algebras arise as symmetries of classical field theories such as Yang-Mills-type theories on certain manifolds. One may ask to what extent these symmetries are still realized in the corresponding quantum theories. I give precise criteria in the context of a particularly interesting field theory.
Joint work with Paul deMedeiros.
Asymptotic expansions of heat traces appear naturally in the study of spectral invariants and their variations. In this talk I consider a surface with cusps and a conformal variation of the metric. The conformal factor does not necessarily have compact support but should decay at infinity. I consider the Laplace operators associated to these metrics and the corresponding heat operators. If the conformal factor decays at infinity, the difference of the heat operators is trace class. I will give explicit conditions on the decay of the conformal factor that, in addition, allow the existence of an asymptotic expansion of this trace (up to certain order) for small values of t.
Weyl geometry constitutes an intermediate link between conformal and (pseudo-) Riemannian geometry. Although proposed by H. Weyl in 1918 with a unified field theory in mind, it reappeared in gravity theory in the second half of the 20th century and may be of interest for present day reflections on foundational questions in physics. In the talk I intend to discuss (shortly) the following points:Introduction to Weyl geometry and a slight generalization of Einstein gravity (WOUD gravity) built upon it in the 1970s by Omote, Utiyama, and DiracRelationship to Jordan-Brans-Dicke gravity and the question of frame choiceStructural similarity of the electroweak extended WOUD Lagrangian with the Higgs potential and some related questions,and another look at cosmological building from the vantage point of the Weyl geometric (and WOUD gravity) extension of Riemannian geometry (and Einstein gravity).
This talk will consider a 4-dimensional manifold M admitting a metric g of arbitrary signature and with Levi-Civita connection D. Then one allows g' to be any other metric (of arbitrary signature) on M with Levi-Civita connection D' and assumes D and D' to be projectively related, that is, D and D' have identical families of (unparametrised) geodesics. The question then is to establish the relationships between D and D', between g and g' and also between the signatures of g and g'. (It is mentioned here that the situation when (M,g) is an Einstein space is known and this fact will be briefly recapped.) The results to be described were achieved in collaboration with David Lonie and Zhixiang Wang in Aberdeen.
The techniques will involve holonomy theory and the finding of the subalgebras o(4), o(1,3) and o(2,2) (representing g) in a convenient form. The case when g is of Lorentz signature is of particular importance in Einstein's general theory of relativity and the situation for vacuum and cosmological metrics will be briefly discussed. If time permits, some other related remarks on the connection between the symmetries of g and g' will be made and also on (the apparently unrelated problem of) a converse to Weyl's conformal theorem.
In this talk, I will present some of the most relevant geometrical results of Leibniz's studies on Analysis Situs, as well as his new conception of geometry of a science of space. I'll also sketch how these mathematical studies influenced, and were in turn influenced by, some parts of Leibniz' metaphysical views.
In this talk I will first outline a new construction of the Fukaya-Seidel category, which is associated to a symplectic manifold equipped with a compatible almost complex structure J and a J-holomorphic Morse function. Then this construction will be applied in an infinite dimensional case of holomorphic Chern-Simons functional. The corresponding construction conjecturally associates a Fukaya-Seidel-type category to a smooth three-manifold.
We prove that a finitely generated group contains a sequence of non-trivial elements which converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian. As a consequence, we show that a finitely generated group satisfies Chu duality if and only if it is virtually abelian.
We investigate lightlike hypersurfaces of indefinite Sasakian manifolds, tangent to the structure vector field $\xi$ and whose screen distribution is integrable. A geometric configuration of the screen distribution is established. We show that any totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface is an extrinsic sphere.
In this talk, I explain how to generalize the analytic torsion by adding a flux form motivated from physics. The twisted torsion is invariant under the deformation of the metric as the classical Ray-Singer torsion. It is also related to generalised geometry and T-duality.
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian systems on $S^1 x D^2$. In this 3-dimensional setting we can think of flow-lines of the Hamilton equations as closed braids in the solid torus $S^1 x D^2$. In the spirit of positive braid classes and flat-knot types we define braid classes and use Floer’s variational approach on these spaces to define a chain-complex and the associated Floer homology invariants. This yields a Morse type theory for braided solutions of the Hamilton equations and in particular for periodic points of area-preserving diffeomorphisms of the 2-disc. The ideas presented here carry over to general two dimensional compact symplectic manifolds (M, $\omega$) (with or without boundary). In the case of the 2-disc we prove a reduction result with respect to Gar- side’s normal form for braids and conjugacy classes of braids.
I will associate to every symplectic manifold of dimension at least four a small preadditive category of open strings and will explain how Lagrangian Floer theory gives rise to a functor with target the homotopy category of bounded chain complexes of open strings (and source a category of Lagrangian conductors).
It is a report on work in progress, joined with D. Burago and S. Ivanov. We consider compact length spaces which admit intrinsic isometries to Euclidean d-space. The class of these spaces is quite rich, it includes all d-dimensional Riemannian and sub-Riemannian manifolds, Euclidean polyhedra and more. The main result roughly states that the class of these spaces coincides with class of inverse limits of d-dimensional Euclidean polyhedra.
Joint work with A. Berarducci and M. NovagaSome existence problems concerning subsequences with special properties, in a context of dynamical systems, ask for special intersection lemmas in measure theory. The archetype of this situation is the recurrence theorem of Poincare', and the Borel-Cantelli lemma. I will discuss some of these intersection problem. For instance, in the simplest form, we have:PROBLEM. Let ${X_ij}$ be a double sequence of masurable subsets in a probability space Omega, with indices over all pairs $i0$. Is there an increasing sequence of numbers $i_0$ These problems may be restated as percolation problems on infinite random graphs. In particular, given the parameter $\lambda$, we look for sharp estimates on the probability of percolation, that is, for instance, in the above mentioned example, estimates on the measure of the event:${x \in \Omega$: there exists a sequence $i_0
Using the link between the SFT of a prequantization space and the GW potential of its symplectic base, it was pointed out by Eliashberg in his ICM 2006 talk that the rich algebraic formalism of SFT sheds new light on the deep relations between Gromov-Witten theory and integrable systems. In this talk I will generalize the discussion from circle bundles to general contact manifolds, where the key point is the invariance of the algebraic structures under choices of contact form, cylindrical almost complex structure and abstract perturbations. As a concrete example beyond the circle bundle case I will introduce the SFT of a closed geodesic, which turns out to be nontrivial enough to serve as a toy model for problems and questions.
It is a report on work in progress, joined with D. Burago and S. Ivanov. We consider compact length spaces which admit intrinsic isometries to Euclidean d-space. The class of these spaces is quite rich, it includes all d-dimensional Riemannian and sub-Riemannian manifolds, Euclidean polyhedra and more. The main result roughly states that the class of these spaces coincides with class of inverse limits of d-dimensional Euclidean polyhedra.
This is joint work with Kai Cieliebak. We study Morse functions on noncompact manifolds whose moduli spaces of gradient flow lines for each action window are compact up to breaking. For each action window we get a Morse homology group. There are three ways to define the full Morse homology. One is via Novikov sums and the other two via inverse and direct limits. We study the relation between these three definitions.
Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L=\int_{\Sigma}\frac{1}{\cos\alpha}d\mu$ in the class of symplectic surfaces. It is $\cos3\alpha H=(J(J\nabla\cos\alpha)^\top)^\bot$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of the symplectic critical surfaces.
The concept of categorification is quite old. It refers to the fine resolution of global invariants, usually by suitable (co)homologies. It has been revived recently with a spate of new invariants of low dimensional manifolds with close connections to physical field theories. We will discuss some examples of this for invariants of three-manifolds with or without links, such as the Casson invariant and the Jones polynomial.
Riemannian symmetric spaces admit representations as coset spaces of Lie groups and, in some cases, admit totally geodesic embeddings into those groups. Algebraic decompositions of the groups induce decompositions of the symmetric spaces. In this lecture we discuss homogeneous Poisson structures on symmetric spaces which are related to the Iwasawa, Bruhat, and and Birkhoff decompsositions. In the classical case of the complex grassmannian, the Lie group of interest is the special unitary group SU(n) and the relevant decompositions are of its complexification, SL(n,C). The Iwasawa decomposition is that given by the factorization resulting from the Gram-Schmidt process and the Birkhoff decomposition is that arising from the matrix factorization resulting Gaussian elimination. We describe the geometry of the symplectic foliation of these Poisson structures and exhibit momentum maps for natural torus actions on the symplectic leaves. Through Duistermaat-Heckman techniques these momentum maps have found application to the computation of integral formulas for the diagonal distribution of the invariant measure of a compact symmetric space. This is joint work with Doug Pickrell.
Most of the known examples of closed manifolds with nonnegative sectional curvature admit an isometric group actions of cohomogeneity one. On the other hand, not all cohomogeneity one manifolds admit invariant metrics of nonnegative curvature. In this talk, we shall discuss existence and non-existence results for nonnegatively curved metrics under the additional hypothesis that there is a totally geodesic principle orbit.
Let $(M,\omega)$ be a symplectic manifold and let $\phi:M\to M$ be a Hamiltonian diffeomorphism. During the last twenty years, symplectic topologists intensively studied the following two questions:1. How many fixed points does $\phi$ at least have?2. Given a Lagrangian submanifold $L$ in $M$, how many intersection points do $L$ and $\phi(L)$ at least have?Consider now a coisotropic submanifold $Q$ in $M$. A point $x$ in $Q$ is called a leaf-wise fixed point of $\phi$ if $x$ and $\phi(x)$ lie in the same isotropic leaf of $Q$. Generalizing the above questions, one may ask if there is a lower bound on the number of leaf-wise fixed points of $\phi$. The main result of this talk is that under suitable assumptions on $M,\omega,Q$ and $\phi$ such a bound is given by the sum of the $\Z_2$-Betti numbers of $Q$.
This is joint work with Kai Cieliebak. We define Floer homology for a Lagrange multiplier functional first studied by Rabinowitz and compute it in particular situations. As an application of these computations we obtain obstructions for exact contact embeddings.
We introduce and analyze generalized Ricci curvature bounds for metric measure spaces $(M,d,m)$, based on convexity properties of the relative entropy $Ent(. | m)$. For Riemannian manifolds, $Curv(M,d,m)\ge K$ if and only if $Ric_M\ge K $ on $M$. For the Wiener space, $Curv(M,d,m)=1$.
One of the main results is that these lower curvature bounds are stable under (e.g. measured Gromov-Hausdorff) convergence.
Moreover, we introduce a curvature-dimension condition CD$(K,N)$ being more restrictive than the curvature bound $Curv(M,d,m)\ge K$. For Riemannian manifolds, CD$(K,N)$ is {equivalent} to $\mbox{\rm Ric}_M(\xi,\xi)\ge K\cdot |\xi|^2$ and $\mbox{\rm dim}(M)\le N$.
Condition CD$(K,N)$ implies sharp version of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows to construct canonical Dirichlet forms with {Gaussian upper and lower bounds} for the corresponding heat kernels.
Leonid Polterovich exhibited a beautiful Lagrangian torus in T*S2 and asked whether this torus is Hamiltonianly displaceable. In joint work with Urs Frauenfelder we prove that its Lagrangian Floer homology does not vanish, indeed equals the singular homology of the torus. In particular, this gives a negative answer to Polterovich's question. In the talk I will describe the construction of the Lagrangian torus and present the computation of the Lagrangian Floer homology which is based on an symmetry argument.
This talk does not assume familiarity with all details of Floer homology. A basic exposure to the ideas of Floer theory is sufficient.
Symplectic Field Theory defines invariants of contact manifolds by counting holomorphic curves in their ``symplectizations'' and related symplectic cobordisms. In the case of contact 3-manifolds (the lowest nontrivial dimension), one can define special versions of SFT that count only embedded curves, and in particular curves that come in non-intersecting families: these tend to form foliations transverse to the Reeb vector field, thus giving strong constraints on the Reeb dynamics. They also satisfy some remarkable compactness properties, which are as yet only partially understood. I will outline the basic theory of holomorphic foliations and explain some partial compactness results, suggesting that a theory based on counting such curves may miraculously avoid the transversality problems that cause such analytical headaches in more general SFT.
After a short introduction to globally hyperbolic Lorentzian manifolds we discuss the analysis of normally hyperbolic operators on such manifolds: Local versus global existence of fundamental solutions, Green's operators, Cauchy problem. Then we show how to quantize such fields.
In my talk, I will introduce a Yang-Mills functional in a purely symplectic context. As we will see, a suitable notion of a symplectic Ricci tensor will come into play. I will study first properties of this symplectic Yang-Mills functional und explain relations to the question of finding preferred symplectic connections.
I present results of my article math.DG/0506230, in which is prove a compactness result for positive special legendrian submaniofolds. I also show how this result may be used in the study of Weingarten hypersurfaces: ie. hypersurfaces whose different curvatures satisfy linear relations.
In the mid eighties S.T. Yau conjectured that a Fano manifold would admit a K.E. metric provided the variety is Stable. The precise stability condition at the time was not clear. Over the past few years, mainly through the work of Gang Tian, a precise conjecture-and several theorems-have appeared. This talk will focus on some of these developements: First we will recall Mumfords' Geometric Invariant theory of Chow and Hilbert points, G. Tians' K and CM stability, and finally the relationship of these to the K-Energy map of Mabuchi, the (classical) Futaki invariant, the generalized Futaki invariant of Ding and Tian, and K-stability as it appeared in work of Simon Donaldson.
This is joint work with Gang Tian.
We will sketch a new approach to the problem of computing the intersection theory of the moduli space M of stable bundles on a Riemann surface. This proceeds by realizing intersections of natural classes on M as intersections on a different space, a Grothendieck Quot scheme compactifying holomorphic maps from the Riemann surface to a Grassmannian. This Quot scheme is endowed with a natural group action and a virtual fundamental class compatible with the action. Thus one can compute intersection numbers on it using the virtual localization formula of Graber and Pandharipande.
I will review some natural problems in real algebraic geometry, differential geometry and geometric group theory where answers are provided by very rapidly growing functions. The most rapidly growing functions appear when one attempts to classify all sequences of Betti numbers of finitely presented groups (our recent joint work with Shmuel Weinberger).
I will talk about upper bounds for the smallest length of minimal geodesic nets, geodesic loops and closed geodesics on a closed Riemannian manifold. These estimates will be in terms of the volume or the diameter of a manifold. I will also discuss some curvature-free upper bounds for the smallest area of a minimal surface in a closed Riemannian manifold, and, more generally, for the smallest volume of a minimal submanifold.
A Calabi-Yau orbifold is locally modeled on C^n/G where G is a finite subgroup of SL(n, C). One way to handle this type of orbifolds is to resolve them using a crepant resolution of singularities. We use analytical techniques to understand the topology of the crepant resolution in terms of the finite group G. This gives a generalization of the geometrical McKay Correspondence.
We shall first discuss solutions of heat equation with plurisubharmonic initial data on complete Kähler manifolds with nonnegative curvature. Then we shall apply the results to study structure of this class of manifolds. We shall also discuss some properties of KählerRicci flow on complete Kähler manifolds with nonnegative curvature and its relation to the uniformization conjecture by Yau which states that a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to ${{\mathbb C}^{n}}$.
Floer theory associates to a symplectic manifold $(M,\omega)$ and a Hamilton function $H:S1\times M\rightarrow \mathbb{R}$ the symplectic invariant $\mathrm{HF}_*(H)$, namely Floer homology groups. We address the question how to assign to a symplectic map $f:(M,\omega)\rightarrow(N,\sigma)$ an induced homomorphism on Floer homology, and propose a construction for a large class of such maps. Inspection of special embeddings leads to new results on the topology and geometry of Lagrangian submanifolds.
I will introduce fermion representations (related to holomorphic vector bundles on the Riemann surface). Based on this example, I'll explain how to modify the theory of highest weight representations in order it would become applicable to affine Krichever-Novikov algebras.
Each convex smooth curve in the plane has at least 4 points where its curvature has a local extremum; if the curve is generic then it has an equidistant with at least 4 cusps. Arnold formulated, using the language of contact topology, two conjectures that generalize these classical results to cooriented fronts in the plane (fronts are curves that are allowed to have certain singularities). These conjectures has been recently proved by the speaker and P. E. Pushkar'. The formulations of these conjectures and the ideas of the proof will be the subject of the talk.
Let Y be a compact Riemann surface with curvature -1 and the associated Laplace-Beltrami operator D. Let ${f_i}$ be an orthonormal basis in $L^2(Y)$ consisting of eigenfunctions of D with the corresponding eigenvalues ${m_i}$. We prove that the $L^4$-norm of $f_i$ is bounded by a constant independent of the eigenvalue $m_i$. We discuss some application of this result to the spectrum of D. The proof is based on ideas from representation theory of the group $SL(2,R)$ and we plan to explain this connection from scratch. The result is a joint work in progress with J. Bernstein.
Starting with a real manifold B whose transition maps are integral affine linear transformations, one can easily construct either a symplectic manifold fibred by tori over B or a complex manifold fibred by tori over B. Furthermore, these fibrations are naturally dual to each other. This exhibits a simple version of Mirror Symmetry as predicted by Strominger, Yau and Zaslow.
The above situation is too simple, though. One does not get interesting examples because the fibrations obtained using this method do not include singular fibres. To get interesting examples --such as Mirror Symmetry for complete intersections in toric varieties-- one should study affine manifolds with singularities. In this case it is much more difficult to construct symplectic or complex manifolds.
In this talk I shall focus on the `symplectic side' of Mirror Symmetry. I will explain a method to construct interesting examples of Lagrangian 3-torus fibrations of compact, simply connected, symplectic 6-manifolds, starting from suitable affine manifolds with singularities.
In this talk I will retell the latest (not even published yet) result, joint with F.Musso, on factorization of matrices with polynomial entries (how to reduce polynomial Darboux matrices to linear ones). Some new examples of linear Darboux matrices and solutions of the quantum Yang-Baxter equation will be given.
I will discuss a model generalizing classical solitonic hierarchies. It has many common features with Kadomtsev-Petviashvili hierarchy but is essentially simpler. Although a priori this model is not connected with any spectral problem, its reductions introduce an interesting generalization of finite-gap potentials of classical spectral problems.
The spectacular rigidity phenomena for symplectic mappings discovered in the last two decades show that certain things CANNOT be done by a symplectic mapping. The aim of this talk is to show that certain other things CAN be done by symplectic mappings. To this end we describe some elementary and explicit symplectic embedding constructions.
The talk will consist of two parts: the first one will deal with some topics about the singular Yamabe problem, more explicitly, of how to construct a metric of constant positive scalar curvature that is singular along a specified set of certain types, of dimension smaller than (n-2)/2. In second part I will consider some fully non-linear equations coming from conformal geometry that generalize scalar curvature, and in particular, obtain a dimension estimate for this singular set. If time permits, I will mention some relations between these two problems.
This talk will start with the infimum definition of quasi-local mass and its more obvious properties. Assuming a natural but possibly deep conjecture on the nature of the infimum, reduces the calculation of quasi-local mass to solving an elliptic boundary value problem of Ricci type. I will describe what little is known about this system, and recent attempts to numerically compute solutions in the static axially symmetric case (joint work with Marsha Weaver).
In 1928, at the Mathematical Congress in Bologna, Veblen devoted his talk to the problem of classifying invariant operators between "geometric quantities". In modern terms these are tensor fields, connections and similar objects. Eventually, it was conjectured that there is only one (type of) unary invariant differential operators.
I will retell Grozman's list of binary invariant differential operators. The answer is remarkable: there are no operators of order >3, all operators of orders 3 and 2 are compositions of 1st order operators (bar one exception), and the 8 series of first order operators determine a Lie superalgebra structure on their domain. One of these superalgebras is the well-known Poisson algebra. I will explain how to get this result and say what is known about its generalization: differentional operators invariant with respect to the group of symplectomorphisms.
Students and other listeners are most welcome to take up open problems.
In several recent works, Hamiltonian Floer theory has been used to calculate the Hofer-Zehnder capacity in new cases. However, these methods are limited by the fact that standard Floer homology can only see one homotopy class of periodic orbits at a time.
In this talk, based on a joint project with Viktor Ginzburg and Basak Gurel, I will describe a method which should allow us to deal with this problem. It uses the notion of Branching Floer homology, an extension of Hamiltonian Floer homology in which periodic orbits in different homotopy classes are allowed to interact via the Floer differential.
We show that on a compact rank one manifold of nonpositive curvature the volume of spheres (hence also that of balls) has an exact asymptotic; it is purely exponential, and the growth rate equals the topological entropy.
We study the evolution of a closed, convex hypersurface in R^(n+1) in direction of its normal vector, where the speed equals a positive power k of the mean curvature. We show that the flow exists on a maximal, finite time interval, and that, approaching the final time, the surfaces contract to a point.
The motion of a unit charge on a Riemannian manifold (N,g) subject to a magnetic field 𝜎 can be described as the Hamiltonian flow of the metric Hamiltonian (p,q) ↦ 1/2 |p|2 on the twisted cotangent bundle (𝓣 * 𝓝, 𝜔𝜊 + π*𝜎) where 𝜔𝜊 is the standard symplectic form and 𝜎 (the magnetic field) is a closed 2-form on N. In contrast to the geodesic flow, the dynamics of a charge in a magnetic field depends on its energy.
We shall explain two recent results on closed trajectories of a slow charge.Given any magnetic field 𝜎 ≠ 0, for a dense set of sufficiently small energies the corresponding energy level carries a closed orbit projecting to a contractible trajectory on N.If 𝜎 is exact (i.e., the magnetic field has a potential), then almost every sufficiently low energy level carries a closed orbit projecting to a contractible trajectory on N. While the proof of a) relies on results from Hofer-geometry, the proof of b) uses an explicit isomorphism between the Floer homology of (𝓣 * 𝓝, 𝜔𝜊 + π*𝜎) and the Morse homology of 𝓣 * 𝓝.
This is joint work with Urs Frauenfelder (Hokkaido University).
A classical theorem in differential geometry states that if Σ ⊂ R3 is a compact connected surface without boundary and all points of Σ are umbilical, then Σ is a round sphere and therefore its second fundamental form A is a constant multiple of the identity. In a joint work with Stefan Müller we give a sharp quantitative version of this theorem.
More precisely we prove the existence of a universal constant C such that Building on this we also show that Σ is W2,2 close to a round sphere. Both estimates are optimal. Indeed, for p<2, one can exhibit smooth compact connected surfaces with arbitrarly small LP norm of the traceless part of A and which are close to the union of two distinct round spheres.
Our proof uses: Ideas of Müller and Sverak to ensure the existence of a conformal parameterization of Σ which enjoys good bounds; Codazzi equations and elliptic PDE techniques to estimate the Marcinkievicz norm ∥A - Λ∥L2,∞(Σ); Elementary algebraic computations combined with Wente-type estimates on skew-symmetric quantities to improve the L2,∞ bound to the desired L2, estimate.
We study stability of non-compact gradient Kähler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kähler potential of the soliton will converge to the original soliton under Kähler-Ricci flow as time tends to infinity. To obtain this result, we construct appropriate barriers and introduce an Lp-norm that decays for these barriers with non-negative Ricci curvature. This is joint work with Albert Chau.
In this note we prove certain necessary and sufficient conditions for the existence of embedding of statistical manifolds. In particular we prove that any smooth statistical manifold can be embedded into the space of all probability measures on a finite set. As a result we get positive answers to the question of Amari on the existence of embedding of exponential families and to the Lauritzen question on realization of statistical manifolds as statistical models.
We introduce a natural dynamical system on configurations of points on a sphere, based on projective geometry and represented by a rational map. In the case S2 the map is holomorphic. The map has interesting universality type properties which shed light on the structure of the space of rational maps, equivariance properties which relate it to moduli spaces and to metric properties such as equi-distribution (electrons on a sphere, but wrt log-potentials). Variations on the construction give interesting algorithms for dealing with rational maps. Computer experiments played a significant role in the initial stages of this work, and should be necessary in its subsequent development.
The Double Bubble Theorem says that the familiar double soap bubble is the least-area way to enclose and separate two given volumes of air. I'll discuss the problem and recent proof (with Hutchings, Ritore', and Ros, Annals of Math., March 2002), including a new second variation formula, for singular surfaces.
We talk about a new scalar curvature related to Schouten tensor in conformal geometry. We address a generalized Yamabe problem for this new scalar curvature, various geometric inequalities and topological implications of positivity of this new scalar curvature.
We use Morse theory and a particular homotopy theoretic tool - a variant of the Hopf invariant - to discuss a method of detection of periodic orbits of hamiltonian flows.
We establish the most precise asymptotic formula ever found for the number of periodic orbits for the geodesic flow, counted by homotopy. We prove it for every compact manifold of nonpositive curvature with rank one.
This extends a celebrated result of Fields medalist G.A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper.
While proving this result, we also manage to carry out Margulis' construction of the measure of maximal entropy without requiring strong hyperbolicity.
We present minimal harmonic embeddings of the sphere 𝕊5 into the quaternionic projective plane ℍℙ2 and of the sphere 𝕊11 into the octonionic projective plane 𝕆ℙ2 that represent generators of the homotopy groups π5(ℍℙ2) ≈ ℤ2 and π11(𝕆ℙ2) ≈ ℤ24, respectively. The embeddings parametrize singular orbits of isometric cohomogeneity one actions. In the case of the complex projective plane the analogous singular orbit is the quadric which represents twice a generator of π2(𝕆ℙ2). The opposite singular orbits in the three cases are the totally geodesic submanifolds ℝℙ2 ⊂ 𝕆ℙ2, 𝕆ℙ2 ⊂ ℍℙ2, and ℍℙ2 ⊂ 𝕆ℙ2, respectively. The related Hopf fibrations 𝕊2 → ℝℙ2, 𝕊5 → 𝕆ℙ2, and 𝕊11 → ℍℙ2 are realized in the projective planes by intersections of the singular orbits with projective lines. We also show that the above mentioned orbits together with the projective lines provide all orbits that are diffeomorphic to spheres and represent non-trivial elements in the corresponding homotopy groups.
(joint work with A. Rigas, Campinas)
We give rigorous construction of the S1-equivariant Dirac operator (i.e., Dirac-Ramond operator) on the space of (mean zero) loops in ℝd and compute its equivariant L2-index. Essential use is made of infinite tensor product representations of the Canonical Anticommutation Relations algebra.
This work originated in the try to understand the results from B. Kostant's recent paper on a "cubic Dirac operator" from the point of view of differential geometry, and in particular the well-established Dirac operator techniques one has therein. At the same time, we became interested in invariant metric connections whose torsion (viewed as a (0,3)-tensor) is a 3-form as a means to construct solutions to the common sector of type II string theories. It turns out that the former is helpful for the latter, with the concept of connections with totally skew symmetric torsion on some special homogeneous spaces as the unifying principle.
We prove a version of the $L^2$-index Theorem of Atiyah which uses the universal center-valued trace instead of the standard trace. We construct for $G$-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring $zz subset Lambda^G subset qq$ obtained from the integers by inverting the orders of all finite subgroups of $G$.We use these two results to show that the Baum-Connes Conjecture implies the modifiedTrace Conjecture which says that the image of the standard trace $K_0(C^*_r(G)) o r$ takes values in $Lambda^G$. The original Trace Conjecture due to Baum and Connespredicted that its image lies in the additive subgroup of $r$ generated by the inverses of all the orders of the finite subgroups of $G$, and has been disproven by Ranja Roy recently.
Let p: M → B be a family of compact manifolds, and let F → M be a flat vector bundle. The higher torsion invariants τ (M/B;F) by Igusa-Klein and Τ (M/B;F) by Bismut-Lott are both generalisations of the classical Franz-Reidemeister torsion. These invariants detect homeomorphic, but not diffeomorphic bundles with the same given fibre X and base B, e.g. if X is an odd-dimensional spere. We establish a relation between τ (M/B;F) and Τ (M/B;F) in the case that there exists a function f: M → ℝ that is Morse on every fibre of p.
Let (,) be a non-degenerate symmetric product of signature (p,q) on a vector space V and let R be an algebraic curvature tensor. Assume q is at least 5. Let R(*) be the associated skew-symmetric curvature operator. Assume R(*) has constant rank 2 on the space like 2 planes. We classify these tensors and show these tensors are geometrically realizable in this context by hyperplanes in flat space of signature (p,q+1) or (p+1,q). We also classify the algebraic curvature tensors of constant rank whose complex Jordan form is constant. This is joint work with Tan Zhang and generalizes previous results from the Riemannian to the pseudo Riemannian setting.
The parabolic geometries are curved deformations of (real) homogeneous spaces G/P, such that the complexification is a parabolic subgroup in a semisimple Lie group. The general ideas go back to Cartan's 'generalized spaces'. Examples involve the conformal, projective, almost quaternionic, and CR geometries. The first talk will present an introduction, survey of recent existence results, and a more detailed explanation of new techniques leading to an effective calculus for invariant operators. The explicit construction of curved analogues of the famous Bernstein-Gel'fand-Gel'fand resolutions for all these geometries (without any use of its representational theoretical version) will be presented as the first major application of our new approach.
The second talk will focus on the parabolic geometries with irreducible tangent bundles. In this case, all the standard geometrical structures known from conformal Riemannian geometries admit nice general counterparts and, in particular, there is a more straightforward explicit construction of the operators from the above mentioned resolutions based on finite dimensional representation theory. It is remarkable that the closed formulae for all these operators of a given order do not depend on the choice of the structure groups.
An orthogonal representation of a compact Lie group K on a euclidean vector space V is called polar if there exists a linear subspace ("section") which intersects each orbit perpendicularly. Examples are the isotropy representations of Riemannian symmetric spaces G/K where the section is any maximal flat subspace. J. Dadok has shown a converse statement: any polar representation is orbit equivalent to the isotropy representation of a Riemannian symmetric space, i.e. there exists an orthogonal Lie triple R : V ⨯ V ⨯ V → V such that the K-orbits on V agree with the orbits of the orthogonal automorphism group of (V,R). In fact, Dadok obtains this result by classifying all polar representations. We give a direct proof for the case where the cohomogeneity is bigger than 2: We construct the Lie triple product R on V from the submanifold geometry of the K-orbits.
(joint work with E.Heintze)
