Please find more information about the lectures at the detail pages.
For rooms at the MPI MiS please note: Use the entry doors Kreuzstr. 7a (rooms A3 01, A3 02) and Kreustr. 7c (room G3 10), both in the inner court yard, and go to the 3rd. floor. To reach the Leibniz-Saal (E1 05, 1st. floor) and the Leon-Lichtenstein Room (E2 10, 2nd. floor) use the main entry Inselstr. 22.
Please remember: The doors will be opened 15 minutes before the lecture starts and closed after beginning of the lecture!
The signature of iterated integrals was introduced by Chen in the early part of the 20th century. It maps smooth enough curves $$X: [0,T] \rightarrow \mathbb{R}^d$$ into an infinite collection of numbers \[ \int_0^T \int_0^{r_n} \int_0^{r_2} dX^{i_1}_{r_1} .. dX^{i_n}_{r_n}. \] We will coverHow analytic properties of integration translate to algebraic properties of the signature. This opens the door to studying the free associative algebra, the free Lie algebra and related Hopf algebraic concepts.How X is (almost) completely determined by its signature.How the signature has found application in stochastic analysis, via the theory of rough paths.How the signature is used in machine learning as a means for feature extraction.Date and time infoThursday, 09.00-10.30PrerequisitesBasic knowledge of real analysisAudienceMSc studens, PhD students, PostdocsLanguageEnglish
In this lecture we will focus on the porous medium equation. The porous medium equation arises, for example, as a model for the flow of an ideal gas in a porous medium. The degenerate nature of the diffusivity of this PDE leads to interesting effects such as limited regularity of solutions, finite speed of propagation, open interfaces, waiting times etc. We will first recall several applications of the porous medium equation, consider special (self-similar) solutions and highlight particular properties of solutions to degenerate PDE. Then, we will focus on the question of (optimal) regularity of solutions to porous media equations, which relies on methods from harmonic analysis (Fourier multipliers, interpolation theory, Besov spaces) which will be recalled in the course of the lecture.Date and time infoMonday 16:15 - 17:45PrerequisitesBasic PDE courses, functional analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish
Many data can be organized as networks, or more generally, as simplicial complexes or hypercomplexes, possibly weighted and/or directed.
We develop mathematical tools to analyze such structures systematically.
This is a continuation of the course from the last term, but newcomers are welcome, and the material will be organized accordingly.ReferencesJ.Jost, Mathematical concepts, Springer, 2015Date and time infoFriday 13:30 - 15:00AudienceMSc students, PhD students, PostdocsLanguageEnglish
The theory of J-holomorphic curves has been of great importance in symplectic geometry/topology as well as in geometric analysis. They are particular kind of harmonic maps from Riemann surfaces into almost complex manifolds. The Gromov-Witten invariants are defined on suitable moduli spaces of J-holomorphic curves and are used to study the geometry/topology of the target manifolds. There is a lot of similarities between the analysis of J-holomorphic curves and harmonic maps, and it is our goal to give an introduction to this theory, focusing on the analytical aspects.
This is a reading seminar. The main reference is the book “J-holomorphic curves and symplectic topology” by McDuff and Salamon. Each participant is encouraged to make some contribution by giving talks on certain topics in which she/he feels interested.Date and time infoWednesday, 13:00 - 14:30PrerequisitesBasic knowledge about Riemannian geometry and Riemann surfacesAudienceMSc students, PhD students, PostdocsLanguageEnglish
The Plateau problem has been one of the most influential geometric variational problems; for instance, this concerns the areas geometric analysis, geometric measure theory, elliptic partial differential equations, and differential geometry. It consists of finding the surface of least m dimensional area amongst all surfaces spanning a given boundary in Euclidean space.The lecture of consists of four parts: multilinear algebra (following [1]), basic geometric measure theory (Hausdorff densities, Hausdorff distance, Kirszbraun's extension theorem, tangent spaces and relative differentials for closes sets, area formula, rectifiable sets; following [1], [2], [3]), varifolds (rectifiable varifolds, first variation, radial deformations, and rectifiability theorem; following [4]), and a modern presentation of Reifenberg's approach to the Plateau problem (simplifying current research papers). In regard to the latter, our axiomatic treatment will allow the present lecture to focus on the analysis part of the problem.Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. https://dx.doi.org/10.1007/978-3-642-62010-2Ulrich Menne. Real analysis. Lectures notes, University of Potsdam, 2015.Ulrich Menne. Einführung in die Geometrische Maßtheorie. Lectures notes, University of Potsdam, 2015.William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417--491, 1972. https://dx.doi.org/10.2307/1970868Date and time infoMonday & Friday, 9:15 - 10:45PrerequisitesReal analysis (in particular, Hausdorff measure & Rademacher's theorem)AudienceDiploma & PhD studentsLanguageEnglishRemarks and notesLectures notes for the last part shall be created as the course progresses.
About this lectureDue to the rather broad spectrum of topics within the IMPRS, the curriculum consists of a core curriculum to be attended by all students and a variety of more specialized lectures and courses. The heart of our teaching program certainly is the Ringvorlesung. Each semester the Ringvorlesung focuses on one field and is usually delivered by scientific members of the IMPRS who introduce different approaches and visions within this field.
Topics
Speakers in this course are: Bernd Sturmfels (BS) and Mateusz Michałek (MM)
Description
This course offers an introduction to the concepts and techniques of Nonlinear Algebra. This subject covers tools for Mathematics in the Sciences that go beyond the familiar repertoire of Linear Algebra. The scope is aligned with the MPI group and the ICERM research semester that carry the same name.
Format
The course consists of 13 lectures. While there is a natural progression of topics, each lecture is self-contained and independent of the others. Notes will be posted. These include exercises. Informal afternoon sessions are dedicated to these exercises. Participants with different backgrounds are strongly encouraged to form study groups and to collaborate. Guests are most welcome to join in on individual Tuesdays.
Grading
Enrolled participants are expected to solve and hand in two or more of the exercises that are listed after each lecture. The hard deadline for all submissions is Tuesday afternoon in the following week.
Schedule
April 10: Polynomials, Ideals, and Gröbner Bases (BS) April 17: Algebraic Varieties (MM) April 24: Elimination and Implicitization (BS) May 8: Linear Spaces and Grassmannians (MM) May 15: Nullstellensätze (BS) May 22: Tropical Algebra (BS) May 29: Toric Varieties (MM) June 5: Tensors (MM + BS) June 12: Representation Theory (MM)June 19: Invariant Theory (BS) June 26: Semidefinite Programming (BS) July 3: Primary Decomposition (MM) July 10: Polytopes and Matroids (MM) Date and time infoTuesdays 10:00 - 12:00AudienceMSc students, PhD students, PostdocsLanguageEnglish
Stochastic homogenization is about the effective large-scale behavior of random heterogeneous media. I will mostly focus on the homogenization of elliptic equations in divergence-form. One focus is on a large-scale regularity theory, in particular Calderon-Zygmund theory (estimates in Lp-spaces) and Schauder theory (estimates in Hölder spaces).Date and time infoWednesday, 09:15 - 11:00PrerequisitesBasic knowledge in elliptic differential equations is helpful, but the course is self-contained in the sense that it provides an independent introduction into Calderon-Zygmund theory, for instance.AudienceMSc students, PhD students, PostdocsLanguageEnglish
Do invisibility cloaks only exist in the world of Harry Potter? Or is it maybe possible to realise such cloaks? Or can we prove that material behaviour of this type is impossible? Can one obtain information on the content of a box without opening it?
This lecture will deal with problems which are related to the above questions from a mathematically rigorous point of view. Studying the Calderón problem as a model problem we will acquaint ourselves with techniques and results on inverse problems. In general, these are problems in which one seeks to recover properties of a system by means of indirect measurements. This for instance finds important applications in medicine (e.g. in non-invasive strategies such as x-ray tomography) or geoprospection (oil discovery).
Mathematically, inverse problems often display strong "ill-posedness" features. This leads to interesting mathematical challenges, which we will discuss in this course. In this context, we will in particular deal with the following topics:Fourier series and Sobolev spaces.Uniqueness for the Calderón problem.Stability for the Calderón problem.The partial data problem.Cloaking.Date and time infoThursday 13:15 - 14:45PrerequisitesBasic knowledge from the courses "PDE" and "Functional Analysis I".AudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesIf desired it is possible to switch to German as the language of the lecture.
In this course, we will compare and contrast traditional techniques in stochastic PDE with more recent pathwise methodologies. Namely, the first part of the course will focus on the Bertini/Giacomin derivation of the KPZ equation from the fluctuations of an interacting particle system, where the solution is viewed as a Cole/Hopf transform of a linear, multiplicative stochastic heat equation.
The second part of the course will be devoted to a more direct, pathwise interpretation of the solution, working within the philosophy of rough paths and regularity structures.Date and time infoMonday 10:30 - 12:30PrerequisitesAnalysis, Probability and PDEAudienceMSc students, PhD students, PostdocsLanguageEnglish
The concept of matrix rank is fundamental in linear algebra, and the idea of low-rank matrix approximation has found many useful applications in modern applied mathematics. For instance it is used for data analysis and compression. The singular value decomposition is the central tool for this task and the starting point for a rich theory of low-rank matrices that combines linear algebra with analysis and geometry.
If one aims at a similar theory for higher-order tensors, one is faced with several options to generalize the notion of rank, and for any of these the low-rank approximation task is considerably harder. In this lecture we consider some of these generalizations, including low-rank tree tensor networks like the (hierarchical) Tucker format and the tensor train format. For them, the concept of higher-order singular value decomposition plays a similar role for low-rank approximation like the SVD in the matrix case. It is a main goal of this lecture to get familiar with this concept.
Additionally, some special topics of interest will be discussed, e.g., manifold structure of low-rank matrices, spectral norm of tensors, tensor products of operators, and motivating low-rank approximation tasks arising in high-dimensional scientific computing. Lectures will be as self-conatined as possible.Date and time infoThursday 11:00 - 12:30PrerequisitesBasic knowledge about linear algebra, analysis and functional analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish
There is a fundamental difference of approach between deductive (model-based) and inductive (data-based) reasoning in sciences. In this crush course we will focus on latter approach that is rarely used in pure mathematics yet might be useful in the applied research. We cover the following models and topics:General principles of data-driven approach: bottle-necks and advantages;Regression, Classification;Principle Component Analysis, clustering;Decision trees, random forests, gradient boosting;Neural networks, general principles and basic modelsAs we cover the most general and widely used models we will come back to the first topic revisiting it of a different level.Date and time infoFriday, 12.00-13.30PrerequisitesBasic knowledge about real world, Analysis, general understanding of Probability and StatisticsAudienceMSc students, PhD students, PostdocsLanguageEnglish
