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!
This course is broadly concerned with differential equations driven by signals of low regularity. The prototypical example of such an equation is $$d Z_t = \sigma ( Z_t ) d X_t,$$ where \(Z\) is the function we are looking for, and \(X\) is a function of low regularity (a "rough signal"). When \(X\) has Hölder regularity in \((1/2,1]\), a "classical" solution theory is known since the work of Young (1936). When \(X\) is a Brownian motion (whose Hölder regularity is slightly below \(1/2\)), solution theories based on Ito (1942) or Stratonovich (1964) integration are available.We are interested in the general case where \(X\) may have arbitrarily low regularity: in this case, Terry Lyons' rough path theory (1997) gives a general, pathwise, solution theory; the only requirement is that we should beforehand give a meaning to the collection of iterated integrals of \(X\) against itself.The theory of rough paths lies at the intersection of analysis, probability, and algebra; the purpose of this course is to expose some of its (beautiful) ideas. The first few lectures will be dedicated to an elementary introduction to the analytical part of the theory in the case where \(X\) has Hölder regularity in \((1/3, 1]\), up to a well-posedness statement for the equation above. The rest of the course will be organized in function of the interests of the audience.Date and time infoThursday, 10.30-12.00 (Caution: Different room on October 19: G310)Keywordsrough path theory; differential equationsPrerequisitesNone
The aim of the course is to present some classic results in geometric measure theory and to introduce some of the basic tools used in the field.
The topics will tentatively include the following (and can be modulated depending on the audience): Hausdorff measure and dimension; Rademacher's theorem and rectifiable sets; tangent measures and Marstrand's density theorem; the Fourier transform of measures, Frostman's lemma, and Marstrand's projection theorem; Besicovitch's projection theorem and Kakeya sets.Date and time infoMonday, 15.30-17.00, starting October 23KeywordsGeometric measure theory; rectifiable sets; Hausdorff dimensionPrerequisitesReal analysis; Some familiarity with basic measure theory is useful
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
Samantha Fairchild: Riemann Surfaces and Algebraic CurvesStefan Hollands: Concept of entropyThe concept of entropy originally emerged from considerations about physical processes involving the exchange of "heat". Boltzmann's idea to view entropy as a measure of the number of microscopic configurations of a system given certain global parameters in effect started the entire subject of statistical physics. Boltzmann's viewpoint broadened the scope of this concept from a relatively narrow physical context to basically any system, physical or mathematical, with a probabilistic description. As such it has found applications in many areas of the natural sciences and mathematics including chemistry, economics, (quantum-) information theory, theory of networks, social sciences, ... In this Ringvorlesung, I will introduce the concept of entropy and outline some of its properties and applications.Alexander Kreiß: Mathematical StatisticsDate and time infoFriday 11.00-12.30 (Part I), Friday 12.00-13.30 (Part II), Friday 09.15-10.45 (19.1., 26.1., 2.2., 9.2.: Part III)KeywordsRiemann surfaces and Algebraic Curves, Entropy,AudienceIMPRS studentsLanguageEnglish
The nonabelian Hodge correspondence is a deep and beautiful theory that identifies three moduli spaces: the Betti moduli space \(\mathcal{M}_{\text{B}}\) of surface group representations, the de Rham moduli space \(\mathcal{M}_{\text{dR}}\) of flat connections and the Dolbeault moduli space \(\mathcal{M}_{\text{Dol}}\) of Higgs bundles, whose objects on the surface appear to arise from very different contexts. The moduli spaces themselves are constructed as GIT quotients based on the geometric invariant theory pioneered by Mumford. The correspondence between these moduli spaces is the cumulative work of Corlette, Donaldson, Hitchin and Simpson built on that of many mathematicians and the machinery involved to prove these results lies in the intersection of algebraic geometry, complex geometry, geometric analysis and gauge theory.The goal of this reading seminar is two-fold: Understand the workings of geometric invariant theory and how they are applied in the construction of the moduli spaces involved in the nonabelian Hodge correspondence. Understand the geometric properties of objects in the moduli spaces and without going into too much nasty detail, sketch the correspondence between the moduli spaces for \(G = \mathrm{GL}(n, \mathbb{C})\).This topic lies in the intersection of many aspects of mathematics and may yield something useful for everyone in spite of different background and interests. For those more geometrically minded who are working with the moduli spaces, we provide the algebro-geometric foundation of their construction, while for those more algebraically minded, these are excellent examples of GIT and testing ground for your understanding. We aim to divide the topics in a way that strikes a balance between algebra and geometry, thereby bringing forth interaction and collaboration.Date and time infoThursdays, 11:00-12:30, starting 23.11.23KeywordsGeometric Invariant Theory, GIT, Moduli problems, Nonabelian Hodge correspondence, character variety, flat connections, Higgs bundlesPrerequisitesBasic knowledge in Algebraic Geometry and Differential Geometry is useful but not required.
Date and time infoMonday 2-3.30 and Friday 2-3.30KeywordsRiemannian metrics and their properties, curvatures, short introduction to geometric analysisAudiencePhD students, postdocs, advanced master students
The lecture “Mathematics of Machine Learning” serves as an introduction to analyzing common numerical problems appearing in modern machine learning applications. We will particularly focus on supervised learning and the dynamics of stochastic gradient descent. This involves the analysis of stochastic processes in discrete time, whose behavior is closely linked to deterministic, as well as stochastic, differential equations. We derive convergence rates for the classical Robbins-Monro algorithm and its Ruppert-Polyak smoothing and analyze the effect of adding inertia (momentum) to the dynamical system. Other possible topics include (stable) central limits theorems, Multilevel Monte Carlo and reinforcement learning. While the general techniques for the asymptotic analysis of stochastic processes are also introduced, proper basic knowledge of probability theory (including martingale theory) is required.Date and time infoWednesday, 16-18KeywordsStochastic Gradient Descent, Stochastic Approximation, Ruppert-Polyak averaging, Supervised learningPrerequisitesBasic knowledge of probability theory (including martingale theory)Remarks and notesThis is a hybrid lecture with the possibility to participate online. It will be held at the University of Bielefeld.
The aim of this minicourse is to give an introduction to anti-de Sitter geometry and anti-de Sitter manifolds, focusing especially on the case of dimension 3 and on its deep relations with hyperbolic geometry in dimension 2 and Teichmüller theory. The main emphasis will be on a particular class of Lorentzian (2+1)-manifolds called "globally hyperbolic maximal Cauchy compact anti-de Sitter 3-manifolds". We will describe properties of both their internal geometry (e.g. their convex core and the structure of its boundary) and of the structure of their associated deformation spaces (see e.g. Mess' classification result).Date and time infoLate January/Early FebruaryKeywordsAnti-de Sitter space and its isometries, globally hyperbolic spacetimes, geometric structures, hyperbolic surfaces, Teichmüller spacePrerequisitesExperience with classical tools of Differential Geometry (Riemannian metrics, Gaussian and sectional curvature, geodesic and metric completeness) will be necessary. Some familiarity with the geometry of hyperbolic surfaces (the hyperbolic plane and its isometries, ) will be extremely helpful.LanguageEnglish
The existence of closed geodesics on compact Riemannian manifolds can be shown using Morse theory on the free loop space. On one hand there are results about the existence of infinitely many closed geodesics. On the other hand one can give existence results for short closed geodesics in case of positive curvature with length estimates.Date and time infoTuesday, 9:15 - 10:45KeywordsRiemannian manifold, Morse theory, free loop space, curvaturePrerequisitesdifferential geometry, algebraic topology
A recent area at the interface of probability/combinatorics and topology/geometry has been random topology. The idea is to understand the topological properties of random geometric and combinatorial structures. We will explore the topology and geometry of different random simplicial and cubical complex models. Many ideas developed for random graph models will be extended to higher dimensional notions, for example, connectedness can be thought of as Betti so one can ask about cycles in terms of Betti 1. We will explore two models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also study a more combinatorial model, the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube. We will also examine the properties of the
Linial-Meshulam model for simplicial complexes, an extension of the Erdos-Renyi random graph model. We will discuss the notion of a giant component for the Linial-Meshulam as well as the properties of minimum spanning cycles. On the road, we will understand some algorithms that were developed and used for analyzing and doing computational experiments on these models. The probabilistic method and ideas such as branching processes will be used extensively.Date and time infoWednesday, 13:30-15:00, starting January 10, 2024
We introduce various discrete Ricci curvature notions under the unified framework of gradient estimates for the heat equation. We discuss the corresponding notions of Ricci flows. Assuming Ricci lower bounds, we prove eigenvalue estimates, diameter bounds, concentration of measure, Li-Yau inequalities and Log-Sobolev inequalities.Date and time infoFriday 15.45-17.15KeywordsRicci curvature, Markov chains, Gradient estimatesPrerequisitesBasic linear algebra and analysis should be sufficientLanguageEnglish
In the field of Real Algebra, the main basic objects are orderings of algebraic structures. We will particularly focus on ordered fields (examples are the rational and real numbers) and see that the set of sums of squares plays a special role in this theory. In particular, we will discuss Artin's solution of Hilbert's 17th problem. We will study the real spectrum in detail.Date and time infoTuesday, 13:15 to 14:45KeywordsReal Algebra, Geometry, Hilbert's 17th problemPrerequisitesPrior knowledge in abstract algebra is sufficient. Basics in algebraic geometry and commutative algebra are helpful but not necessary.AudienceDiploma students, PhD students
Typical topics include: basic definitions, algebraic graph theory, bases and cycle bases, connectivity, vertex coloring, isomorphisms, and planar graphs.
This year the special lecture and the associated practical course will be on mathematical phylogenetics. For the seminar each participant has to present a self chosen original scientific article relating to the topic of graph theorie/networks.See all dates:09.10-27.11.2023, Monday 14:00 - 17:00Special Lecture: 04.12-18.12.2023, Monday 14:00 - 17:00Praktikum/Practical Course: 03.01.2024 - 16.01.2024, every day, core times 10:00 - 15:00Seminar: between 17.01.2024 - 02.02.2024, exact dates and times to be esthablished after registrationDate and time info09.10-27.11.2023, Monday 14:00 - 17:00 (see abstract for more)Keywordsalgebraic graph theory, bases and cycle bases, connectivity, vertex coloring, isomorphisms, planar graphsPrerequisiteslinear algebra or a lecture on math for the natural sciences
The theory of D-modules is the algebraic theory of systems of linear PDEs. We will discuss the Weyl algebra, and learn how to view PDEs as ideals in this algebra. We will learn about regular holonomic systems and computational techniques to solve them. Our main example will be that of A-hypergeometric systems. If time, we will discuss b-functions, integration of D-modules, and restriction of D-modules.Date and time infoMonday, 13.00-14.30, last lecture November, 27thKeywordsD-modules, computation, groebner bases, integrals, A-hypergeometric systems, linear partial differential equations, algebraPrerequisitesBasic knowledge of rings and ideals is necessary. A student should ideally have taken an introductory ring theory course. Knowledge of Groebner bases in the commutative algebra setting would be helpful but not necessary.
