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 subdivided into two parts. In the first part, I will introduce basic information-theoretic quantities such as the entropy, the conditional entropy, the mutual information, and the relative entropy. I will highlight a measure-theoretic perspective, which provides strong tools for the treatment of information sources and information channels. Based on the developed information-theoretic quantities, I will present elementary results on the Kolmogorov-Sinai entropy of dynamical systems and prove the Shannon-McMillan-Breiman theorem. This theorem serves as a prerequisite for the second part of my course in which I will concentrate on information channels. I will introduce the transmission rate and the capacity of information channels. The central theorems of this part will be Shannon's celebrated coding theorems. I will develop Feinstein's fundamental lemma, which constitutes, together with the Shannon-McMillan-Breiman theorem, the main tool for the proofs of Shannon's coding theorems.ReferencesA. I. Khintchine. Mathematical Foundations of Information Theory. Dover, New York, 1958.Y. Kakihara. Abstract Methods in Information Theory. World Scientific, Singapore, 1999.P. Walters. An Introduction to Ergodic Theory. Springer, 1982.T. M. Cover, J. A. Thomas. Elements of Information Theory. Wiley, 2006.Date and time infoTuesday 11:00 - 12:30KeywordsPartial Differential Equations, Applications of PDEs in sciencePrerequisitesBasic knowledge in probability and measure theory is requiredAudienceMSc students, PhD students, PostdocsLanguageEnglish
The course is an introduction to the viscosity theory of first-order Hamilton-Jacobi equations with applications. A model example is the equation \begin{equation} \label{eq}\left\{\begin{array}{ll} u_t=H(\nabla u, x):=\lvert\nabla u\rvert^2+f(x) & \textrm{on}\;\;\mathbb{R}^d\times(0,\infty), \\ u=u_0 & \textrm{on}\;\;\mathbb{R}^d\times\left\{0\right\}.\end{array}\right. (1) \end{equation} The goals of the lecture series are fourfold: To explain the relationship between optimal control theory and solutions to convex Hamilton-Jacobi equations of the type (1). To show by explicit example and through an analysis of the associated characteristics that smooth solutions to equations like (1) do not exist in general, even for smooth Hamiltonians H. To present the viscosity formulation of equation (1), and to prove the fundamental results concerning the existence and uniqueness of viscosity solutions. To prove the periodic homogenization of equations like \begin{equation}\label{hom}\left\{\begin{array}{ll} u^\epsilon_t=H(\nabla u^\epsilon,\frac{x}{\epsilon}) & \textrm{on}\;\;\mathbb{R}^d\times(0,\infty), \\ u^\epsilon=u_0 & \textrm{on}\;\;\mathbb{R}^d\times\left\{0\right\},\end{array}\right. (2)\end{equation} for Hamiltonians H which are periodic in space, and to show under general assumptions that the homogenization occurs with an algebraic rate. Date and time infoThursday 16:00 - 17:30KeywordsFirst-Order Hamilton-Jacobi Equations, Viscosity Solutions, Optimal Control Theory, Periodic HomogenizationPrerequisitesCalculus, Measure Theory, Basic Functional AnalysisAudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThe course is self-contained, and assumes no previous knowledge of the aforementioned topics.
In this course we continue the introduction to stochastic partial differential equations, taking for granted the basics on Gaussian measure theory and semigroup theory introduced in the first part of the course. We will focus on semilinear parabolic problems driven by additive noise, such as stochastic reaction diffusion equations $$du = \Delta u\ dt + f(u)dt + dW_t$$ and stochastic Navier-Stokes equations $$du = \Delta u\ dt - (u\cdot\nabla u)u\ dt-\nabla p\ dt + dW_t,\quad \textrm{div}\ u =0,$$ where \(W\) is an infinite-dimensional Wiener process. Such SPDE can be treated as stochastic evolution equations in some infinite-dimensional Banach space and they already form a rich class of problems with many interesting properties. Furthermore, this class of problems has the advantage of allowing to completely pass under silence many subtle problems arising from stochastic integration in infinite-dimensional spaces. After having established the basic well-posedness results we will investigate questions on long-time behavior, ergodicity and random dynamics, e.g. the existence of random attractors. Towards the end of the course an introduction to the recent theory of paracontrolled distributions will be given.Date and time infoWednesday 11:00 - 12:30KeywordsStochastic Partial Differential Equations, Stochastic AnalysisPrerequisitesbasic measure theory, functional analysis and probability theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
We discuss stochastic models of neuronal activity, like the Poisson neuron, and of synaptic transmission and weight changes, like STDP. The course will also develop the necessary mathematical methods from stochastic processes, dynamical systems and information theory.Date and time infoFriday 13:30 - 15:00Keywordsneural networks, stochastic processes, dynamical systems, information theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
Real algebraic geometry is the study of semialgebraic sets, i.e. sets described by polynomial equations and inequalities, and the behaviour of polynomial functions on those sets. A classical and fundamental result is the solution of Hilbert's seventeenth problem by Emil Artin stating that every rational function that is globally nonnegative can be written as a sum of squares of rational functions. In the second part we will look at recent applications of real algebra related to semidefinite programming and polynomial optimization.Date and time infoFriday 11:15 - 12:45KeywordsReal algebraic geometry, semialgebraic sets, Hilbert's seventeenth problem, semidefinite programming, polynomial optimizationPrerequisiteslinear algebra, basic knowledge in ring and field theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
Culminating in Hairer's regularity structures, there has been much recent progress in developing a robust solution theory for nonlinear stochastic partial differential equations (SPDEs). This progress is inspired by Lyons' treatment of stochastic ordinary differential equations, which is much more deterministic than Ito's approach. The main deterministic ingredient can be seen as an extended Schauder theory, so a maximal regularity theory for constant-coefficient parabolic equations in Hölder spaces, where polynomials are supplemented by more general, "rough" functions. The sole stochastic ingredient is to give an "off-line" sense to a finite number of singular products of rough functions and their distributional derivatives. So far, this approach has been limited to SPDEs where the leading-order part is the constant-coefficient diffusion operator. In this course, we will present a treatment of the quasi-linear SPDE $$ \partial_tu+a(u)\partial_x^2u=\sigma(u)f $$ with a noise \(f\in C^{\alpha-2}\) on the parabolic Hölder scale. Provided \(\alpha>\frac{2}{3}\) (which includes white noise in time) and giving an "off-line" sense to products of the form \(v(\cdot,a_0)\partial_x^2v(\cdot,a_0')\) with \(v(\cdot,a_0)\) solving the constant-coefficient SPDE \(\partial_tv-a_0\partial_x^2v=f\), we obtain a (small-data) solution theory \(C^\alpha\) for u. Loosely speaking, we extend the treatment of the singular product \(\sigma(u)f\), in the spirit of Gubinelli, to the product \(a(u)\partial_x^2u\), which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. Next to treating a wider class of non-linear equations, the merit is that we introduce some simpler tools. More specifically, we treat the singular product \(a(u)\partial_1^2u\) by controlling the commutator \([a(u),(\cdot)_\epsilon]\partial_x^2u\) of multiplication with the first factor a(u) and convolution \((\cdot)_\epsilon\), where the family of symmetric convolution operators \(\{(\cdot)_\epsilon\}_\epsilon\) satisfies a semi-group property and respects the parabolic scaling. Controlling such commutators is reminiscent of the DiPerna-Lions theory for rough transport equations. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory. This is joint work with Hendrik Weber.Date and time infoThursday 09:15 - 11:00Keywordsnonlinear SPDEs, Schauder theoryAudienceMSc students, PhD students, PostdocsLanguageEnglish
We will discuss several topics in global Riemannian geometry. One can compare the geometry of a Riemannian manifold with a lower or upper bound for the sectional curvature with the geometry of a space of constant sectional curvature. One also obtains rigidity results characterizing constant curvature metrics. These methods can also be used to estimate the number of short closed geodesics on spaces of positive curvature.Date and time infoThursday 13:15 - 14:45KeywordsSectional curvature, closed geodesics, Morse theory, injectivity radiusPrerequisitesBasic knowledge about differential geometry and topologyAudienceMSc 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
Lecturers: Rainer Verch, Klaus Kroy, Bernd Rosenow
Date:2016: Wednesday 09:15 - 10:45, 2017: Tuesday 13:30 - 15:00
Tutorials I:
Date: on 5.12.2016, 12.12.2016, 19.12.2016 and 09.01.2017
Tutor: Gianmaria Falasco
Tutorials II:Date: on 18.01.2017, 25.01.2017, and 01.02.2017
Tutor: Heinrich-Gregor ZirnsteinDate and time infoWednesday 09:15 - 10:45KeywordsMathematical Cosmology, Equilibrium and Nonequilibrium Brownian Motion, Topological Phases and AnyonsAudienceMSc students, PhD students, PostdocsLanguageEnglish
