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 first part of this course introduces the basic theory of finite random fields and corresponding geometric structures studied in information geometry. Topics include sampling algorithms and their convergence properties. Graphical models (an important class of random fields) will be discussed in more detail. The second part of the course deals with neural networks. We discuss Boltzmann machines (a kind of stochastic neural networks) and elaborate on related network architectures that have become prominent in machine learning applications (including restricted Boltzmann machines, deep belief networks, and deep Boltzmann machines). We will concentrate on geometric aspects of the different networks, addressing, in particular, the geometry of their parametrization and their expressive power.Date and time infoWednesday 10.15 - 11.45KeywordsInformation theory, Boltzmann machines, geometric aspects of networksAudienceMSc students, PhD students, PostdocsLanguageEnglish
In this series of lectures I will deal with the incompressible Euler equations
where D ⊂ ℝn, v : [0,T] × D → ℝn is the velocity and p : [0,T] × D → ℝn is the pressure of an incompressible nonviscous fluid moving inside a container with no external forces.
These equations, which are nothing but the Newton laws plus some additional structural hypotheses, are fundamental in the theory of fluid dynamics and were discovered by Euler in 1755. Notwithstanding this, many crucial questions concerning its solutions are still open and of great mathematical interest. In this course we will focus on the problem of existence and uniqueness, that surprisingly enough shows in this case two completely different behaviours at high and low regularity. Indeed, while in the class of classical solutions one can show (local in time) existence and uniqueness, in the class of weak solutions one can have, for a dense set of “wild” initial data, infinitely many solutions. In the first case, uniqueness can be easily deduced using the conservation of energy ∫|v(t)|2, while for such “wild” initial data one can produce infinitely many solutions having the same energy.
The part of the theory dealing with smooth solutions is classical and will be addressed in the first part of the course. The second part follows from a technique called convex integration, which has been first applied in this context and then developed in a series of papers by De Lellis and Székelyhidi. In the last part of the course we will explore convex integration starting from problems of other nature, namely the isometric embedding problem and differential inclusions, and once we will have singled out its main features in these simpler examples we will see how it applies to the Euler equations. In between, we will also mention and spend some time on the vorticity formulation of the 2D Euler equations and the vortex sheet problem.Date and time infoMonday 10.00 - 11.30 hKeywordsincompressible Euler equations, vortex formulation, nonuniqueness, convex integration techniquePrerequisitesODE's, Sobolev spaces, basic Fourier analysisAudienceMSc students, PhD students, PostdocsLanguageEnglish
In this course, I shall describe the conceptual structure of neuroscience and analyze it mathematically. On the way, a number of mathematical concepts and tools will be introduced and developed. Therefore, the course will be also of interest for students seeking an overview of mathematical methods.Date and time infoFriday 13.30 - 15.00 hKeywordsInformation theory, dynamical systems, neural coding, neural populations, synchronizationAudienceMSc students, PhD students, PostdocsLanguageEnglish
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.
Schedule
Dates: 13.10.2014, 20.10.2014
Topic: Metric spaces and geodesics
Lecturer: Jürgen Jost
Dates: 27.10.2014, 03.11.2014
Topic: Riemannian, Finsler manifolds
Lecturer: Hans-Bert Rademacher
Date: 10.11.2014
Topic: Symplectic manifolds and algebraic varieties
Lecturer: Jürgen Jost
Date: 17.11.2014
Topic: Basics of algebraic topology
Lecturer: Jürgen Jost
Dates: 24.11.2014, 01.12.2014
Topic: Notions of curvature in Riemannian and metric geometry (generalized sectional and Ricci curvature bounds)
Lecturer: Bernd Kirchheim
Date: 08.12.2014
Topic: Jacobi fields, comparison theorems
Lecturer: Hans-Bert Rademacher
Date: 15.12.2014
Topic: Analysis on metric spaces
Lecturer: Bernd Kirchheim
Date: 05.01.2015, 12.01.2015
Topic: Optimal transport
Lecturer: Bernd Kirchheim
Date: 19.01.2015
Topic: Closed geodesics
Lecturer: Hans-Bert Rademacher
Date: 26.01.2015
Topic: Fibre bundles, Dirac operators I
Lecturer: Jürgen Jost
Date: 02.02.2015
Topic: Fibre bundles, Dirac operators II
Lecturer: Hans-Bert RademacherDate and time infoMonday 13.15–14.45PrerequisitesAdvanced calculusAudienceMSc students, PhD students, PostdocsLanguageEnglish
In many applications, one has to solve a linear elliptic partial differential equation with uniformly elliptic coefficients that vary on a length scale much smaller than the domain size. We are interested in a situation where the coefficients are characterized in stochastic terms: Their statistics are assumed to be translation invariant and to decorrelate over large distances. As is known since more than forty years, the solution operator behaves - on large scales - like the solution operator of an elliptic problem with homogeneous deterministic coefficients!
A more recent insight is that, on large scales and with high probability, the regularity properties of solutions are very close to those of an equation with homogeneous coefficients, for instance in terms of Liouville-type statements. I will focus on this "random regularity theory", which turns out to be much stronger than the deterministic one in the class of uniformly elliptic coefficients, especially in case of systems.Date and time infoTuesday 09.15 - 11.00Keywordselliptic partial differential equations, random regularity theoryPrerequisitesThis course requires less technology in probability theory than it seems and will be self-contained in that respect. On the other hand, the course gives a good opportunity to recapitulate some classical techniques of elliptic regularity theory, like the approach to Schauder theory via Campanato spaces.AudienceMSc students, PhD students, PostdocsLanguageEnglish
