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!
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
Part I:
Lecturer: Nikolay Barashkov (May 6, 16, 30 and June 6)
Title: Optimal Control
Abstract: We will introduce the mathematical formulation of a determinstic and stochastic control problem, and discuss Bellmanns optimality principle and the connection with Hamilton Jacobi equations. Time permitting, we will also discuss the connection of Stochastic Control with Backwards SDE's.
Part II:
Lecturer: James Reed Farre (June 13, 20, 27, and July 4)
Title: Dynamics of the geodesic and horocycle flows
Abstract: We will study and survey some results in "homogeneous dynamics” concerning certain flows on finite volume quotients of PSL(2,R) by discrete subgroups, the geodesic and horocycle flows. There are interesting connections with number theory, topology, geometry, and dynamics, some of which we will explore after developing some basic geometry of the hyperbolic plane.Date and time infoMay 2, 16, 30 ; June 6, 13, 20, 27, and July 4, Time: 13.30-15.00
Homogenization concerns the large-scale behavior of the solutions of PDE with heterogeneous coefficients. This course will treat the model problem consisting of linear PDE in divergence form with stationary and ergodic (e.g. periodic, quasi-periodic, or random) coefficients. It consists of three parts. The first part will be a gentle introduction to the subject in the context of periodic coefficients, presenting the variational proof by G-convergence, explaining the role of correctors and the two-scale expansion, and briefly discussing boundary layer effects. The second part will present the qualitative large-scale regularity theory of Gloria, Neukamm, and Otto, which shows that averaging effects lead to better-than-expected regularity at large scales. The final part treats quantitative error estimates following the recent book of Armstrong and Kuusi: the main thrust of the work will lie in quantifying the variational argument from the first part of the course through a combination of PDE arguments and concentration inequalities.Date and time infoTuesday 14.00-15.30Keywordspartial differential equations, homogenization, regularity theoryPrerequisitesbasic PDE (e.g. Evans, up to and including Chapter 6), knowledge of probability is helpful later
The Ideal Incompressible Fluid is the most fundamental model of a continuous media. In this model, the configuration space of the fluid is the group D of volume- preserving diffeomorphisms of the flow domain M. D is an infinite-dimensonal manifold endowed with a weak Riemannian metric (the kinetic energy). The fluid flows (in the absence of the external forces) are the geodesics in this metric. There are two basic problems about the geodesics:
Find the geodesic trajectory for the given initial fluid configuration and velocity (the initial-value problem);
Find the geodesic for given fluid configurations at two different time moments, say at t=0 and t=1 (the 2-point problem).
In this course we consider both problems.
The course includes the following topics:2-point problem in 2-dimensional domain. The variational approach and its failure. Genaralized flows and generalized braids. Weak solutions of the variational problem, their regularity.
The category of quasi-ruled manifolds and quasi-ruled maps.
The quasi-ruled structure of the manifold D. Its possible significance for the solvability of the 2-point problem.
The geometrical nature of the exponential map Exp on D: Exp is a qiasi-ruled map.
The analytical nature of the geodesic exponential map Exp on the manifold D : Exp is an elliptic paradifferential operator of order zero. Open problems.Date and time infoThursdays, 11:00 am - 1:00 pmKeywordsGroup of diffeomorphisms, Euler equations, braids, variational methods, generalized flows and braids, quasiruled manifolds and maps, paradifferential calculusPrerequisitesjust the basics of Functional Analysis, Riemannian Geometry, and Measure TheoryAudiencegraduate students and above
Toric varieties form a well understood class of algebraic varieties. They often arise as the image of a monomial map. These lectures offer an introduction to basic toric geometry and its applications in mathematics and other sciences. Covered topics include toric ideals, semigroup algebras, Cox rings, discriminants and algebraic moment maps. We will also discuss standard constructions of toric varieties from cones, polytopes and fans, and the role of toric varieties in convex optimization, algebraic statistics, and equation solving.Date and time infosee https://simontelen.webnode.page/l/using-toric-geometry/Keywordstoric varieties, polytopes, fans, monomial algebras, Cox rings, discriminant, moment mapPrerequisitesBasic algebraic geometry: affine/projective varieties and their coordinate rings.Remarks and notesThere will be one lecture and one exercise session per week.
In this mini-course, we introduce persistent homology of a continuous function on a topological space. We define two descriptors, called the persistence Betti numbers functions and persistence diagrams and study the relations between them. In particular, we show that they are equivalent topological descriptions of the function and that one can be recovered from the other. We conclude by showing that persistent homology is a stable descriptor of a continuous function with respect to the uniform norm.
The lectures will cover the following topics:
May 29, 13-14:30. Persistent homology group, persistent Betti numbers function (PBNF), monotonicity of PBNF
June 12, 13-14:30. Discontinuities of PBNF, multiplicity
June 19, 13-14:30. Persistence diagram (PD), cornerpoints of PD
June 26, 13-14:30. Discontinuities of PD, local finiteness
July 3, 13-14:30. Representation theorem
July 5, 11-12:30. Bottleneck distance, stability theoremKeywordsPersistence diagram, persistence Betti numbers function, bottleneck distancePrerequisitesBasic notions of linear algebra and calculus are enough. Knowing the definition of homology may also help.AudienceAnyone who is interested in Topological Data AnalysisLanguageEnglish
