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!
Ergodic theory is the study of the qualitative properties of actions of groups on spaces. It is a very active area with many applications in physics, harmonic analysis, probability, and number theory. In this course, I will introduce some notations, examples about ergodicity, mixing, recurrence, ergodic decomposition, ergodic theorems,... More precisely, I am planning to teachErgodicity, Recurrence, Mixing. Invariant Measures for Continuous Maps. Conditional Measures and Algebras. Factors and Joinings. Structure of Measure Preserving Systems. Actions of Locally Compact Groups. Geodesic Flow on Quotients of the Hyperbolic Plane (if time allowed). TextbookM. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, Graduate Texts in Mathematics, 259. Springer-Verlag London, Ltd., London, 2011.Recommended reading Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory. M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981. Glasner, Ergodic Theory via Joinings. American Mathematical Society, Providence, RI, 2003. Petersen, Ergodic theory. Corrected reprint of the 1983 original. Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1989. Walters, An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79. Springer-Verlag, New York, Berlin, 1982. Date and time infoThursday 11.00 - 12.30KeywordsErgodic Theory, Recurrence, Mixing, Invariant MeasuresPrerequisitesYou should know basic Measure Theory and Functional AnalysisLanguageEnglish
Many dynamical systems are governed by Hamiltonian systems of ODEs. For some very simple systems such as two bodies interacting by gravitational forces or such as harmonic oscillators, it is possible to explicitly integrate the equations of motion. In this cases it is possible to find a system of coordinates (ρ, Θ) called the action-angle coordinates in which the dynamics reduces to ρ. = 0 and Θ. = constant. It is then natural to ask if for systems which are small perturbations of integrable systems, such as the three body problem, the same kind of simple dynamics persists. The classical Kolmogorov-Arnold-Moser (KAM) Theorem, asserts that indeed, for such small perturbations most of these invariant tori ρ = constant will survive and that moreover, on these tori, the dynamic is still very simple. Weak KAM theory, which was pioneered by Fathi and Mather, tries to understand what remains of this picture when the perturbations are not small anymore. This theory is intimately connected with the theory of homogenization for Hamilton-Jacobi equations developed by Lions-Papanicolaou-Varadhan. After giving a short introduction to the classical KAM theory, we will study the notion of viscosity solutions for Hamilton-Jacobi equations and their homogenization. Among the topics that will be covered, there will be:Aubry-Mather theory, Connection with optimal transport (following Bernard-Buoni),Aubry-Mather theory for minimal surfaces,Stochastic Weak KAM theory and applications to homogenization. Date and time infoThursday 13.30-15.00KeywordsDynamical systems, KAM theory, Hamilton-Jacobi equations, viscosity solutions, homogenization, Aubry-Mather theoryPrerequisitesYou should know about measure theory, basic ODEs
In this course, I shall describe the classical theory of the calculus of variations and relate it to modern mathematical developments. The concept of convexity will play an important role.ReferenceJ.Jost, X.Li-Jost, Calculus of Variations, Cambridge Univ.Press, 1999Further references will be given during the course.Date and time infoFriday 13.30 - 15.00KeywordsExtremals of variational integrals, fields, stability, convexityPrerequisitesAdvanced calculus and some basic knowledge of partial differential equationsLanguageEnglish
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
Lecturer: Jürgen Jost - FoundationsMonoids, Groups, Rings and their spectraPrinciples of Category TheorySimplicial Complexes, Homology and discrete Morse-TheoryLecturer: Peter Stadler - Graph LaplaciansSpectral Graph Theory and Random Walks on GraphsGraph Laplacians and Discrete Nodal Domain TheoremsCycle Bases, Dynamic Programming: Grammars and RecursionsLecturer: Andreas Thom - Limits of large networksRandom Graph ModelsSzemeredi's TheoremSubgraph DensitiesDate and time infoMondays 13:30 - 15:00
In the class we will discuss several advances topics in Real Analysis. Among the topics that will be covered there are:Abstract measure theory. Carathéodory construction of measures. Hausdorff Measures. Covering theorems and derivation of measures. Differentiability properties of Lipschitz functions. Kirszbraun extension theorem for Lipschitz functions. Area and Coarea formulas.Basic properties of Rectifiable sets.Date and time infoWednesday 11.15 - 12.45KeywordsReal Analysis, Measure and Integration Theory, RectifiabilityPrerequisitesAnalysis I, II, III and basics of functional analysisLanguageEnglish
Many partial differential equations have the structure of a gradient flow on an (infinte-dimensional) Euclidean space or Riemannian manifold. The gradient flow structure encodes the competition between a driving energy and the limiting dissipation (as modeled by the metric tensor). We will show in specific examples how such a gradient flow structure can be used in the analysis of the PDE. Specific examples could include:An existence result for a free boundary problem in solidification (Stefan problem)Convergence to a self-similar solution (porous medium equation) Coarsening (Cahn-Hilliard equation) Hydrodynamic limits (so-called Ginzburg-Landau model) Date and time infoTuesday, 09.00 - 11.00 (will start on November, 5th)KeywordsPDEs, gradient flow, Stefan Problem, porous medium equation, Cahn-Hilliard equation, Ginzburg-Landau modelPrerequisitesAnalysis, in particular vector calculus, elementary differential geometry, some familiarity with PDEsLanguageEnglish
In the class we will discuss several selected topics in the theory of partial differential equations. In particular, the leading theme of the course is the study of classical minimal surface theory, i.e. smooth surfaces in the 3-dimensional Euclidean space with vanishing mean curvature. Among the topics that will be covered there are:Minimal surface equation. Second variation formula, Morse Index and Stability. Weierstrass' representation. Berstein's theorem. Simons' inequality and curvature estimates. Conformal maps and Douglas-Rado solution to the Plateau problem.Date and time infoMonday 15.00 - 17.00; Friday 10.00 - 12.00KeywordsClassical minimal surfaces, conformal maps, curvature estimates, elliptic regularityPrerequisitesAnalysis I, II, III and basics PDELanguageEnglishRemarks and notesThe level of the class is suited for students from the 7th semester on up to PhD students: indeed, although the tools and the techniques of the course will be almost always elementary (basic analysis and PDEs), the results discussed will have several contact points with recent research developments.
