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 course will provide an introduction to the theory of rough paths. Loosely speaking, a rough path is a pair \((X_t,\mathbb{X}_t)\) which consists of a path \begin{equation}\label{path}X_t:=(X^1_t,\ldots,X^d_t)\in\textrm{C}^\alpha([0,T];\mathbb{R}^d)\end{equation} of low \(\alpha\)-Hölder regularity enhanced by its iterated integrals \begin{equation}\label{iterated} \mathbb{X}^{i,j}_t=:\int_0^tX^j_s\circ dX^i_s.\end{equation} Since the iterated integrals on the righthand side of the formula above are not classically defined if \(\alpha<\frac{1}{2}\), their values are instead postulated by the generally non-unique matrix \(\mathbb{X}_t\); such as in the case of a Brownian motion enhanced by its Itô or Stratonovich integrals.The foremost aim of the course will be to prove the well-posedness of rough differential equations \begin{equation}\label{eq} dY_t=f(Y_t)\circ dX_t,\end{equation} and, in particular, the continuity of the solution with respect to the driving noise \((X_t,\mathbb{X}_t)\) as measured by the rough path metric. We will furthermore prove a deterministic Itô formula and Doob-Meyer decomposition for rough paths. Additional topics may include the signature of a rough path and applications to stochastic partial differential equations.Date and time infoTuesday 10:30 - 12:00Keywordsrough path, rough differential equationPrerequisitescalculusAudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesThe course, while self-contained, will draw motivation and examples from probability theory and stochastic processes.
In this course we will consider stochastic scalar conservation laws of the type $$du + \sum_{i=1}^d \partial_{x_i} A^i(x,u)\circ d\beta^i_t =0,$$ for fluxes \(A\) being regular enough and \(\beta^i\) being independent Brownian motions.These equations appear as continuum limits of mean-field interacting particle systems subject to a common noise and as a simplified part of the mean field game system with common noise. We will first introduce a notion of an entropy solution to these equations and prove their well-posedness. In the case of spatially homogeneous fluxes $$du + \sum_{i=1}^d \partial_{x_i} A^i(u)\circ d\beta^i_t =0,\qquad (\star)$$ we will analyze the regularity of solutions by means of averaging Lemmata. In certain cases we will see that the regularity of solutions to \((\star)\) is higher than in the deterministic case.Date and time infoThursday 11:15 - 12:45KeywordsStochastic Partial Differential Equations, Stochastic AnalysisPrerequisitesbasic measure theory, functional analysis and probability theoryAudienceMSc 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
Benjamin Gess: Scalar conservation laws (5 lectures, 12.10. - 09.11.) László Székelyhidi: Basic properties of harmonic functions (5 lectures, 16.11. - 14.12.)Felix Otto: Regularity theory for elliptic equations (04.01. - 01.02.)ReferencesEvans: Partial differential equations, Chapter 3.4Perthame: Kinetic formulation of conservation laws, Chapter 3Evans: Partial differential equations, Chapter 3.2Krylov: Lectures on elliptic and parabolic equations in Hoelder spaces, Chapter 3Date and time infoThursday 09:00 - 11:00KeywordsGess: shocks, entropy solutions, Lax-Oleinik formula, kinetic solutionsSzékelyhidi: mean value property, maximum principle, unique continuation, Dirichlet principleOtto: Lp-theory (Calderon-Zygmund theory), Cα-theory (Schauder theory)PrerequisitesBasic analysis (Lebesgue integration, Lebesgue spaces, Gauss’s theorem etc.), characteristicsAudienceMSc students, PhD students, PostdocsLanguageEnglish
This lecture gives an introduction to how tools from algebraic geometry and topology can be used to tackle problems in data analysis and statistics.
The first half of the course will focus on what is known as Algebraic Statistics: see e.g., Pistone et al. (2001), Pachter and Sturmfels (2005), and Sullivant (2017). To set the scene, the first two lectures will illustrate some classical statistical theory such as linear models and Bayesian vs frequentist approaches to model selection on small-scale examples. We continue by treating topics such as exponential families and (Gaussian) graphical models and show how their properties can be naturally described using the language of (toric) ideals and varieties. We show that questions of equivalence for conditional independence models can be solved using polytopes.
The second half of the course will start with an intuitive introduction to topology, including homotopy equivalent spaces, homology groups, and homotopy groups. We will then move to the realm of data analysis: given only a dataset, i.e. a finite sampling from a space, what can we say about the space's shape (which may be reflective of patterns within the data)? To study the shape we consider different ways of building geometric objects (simplicial complexes) on point clouds and studying their properties. The main technique we cover is persistent homology; we describe its theoretical underpinnings and discuss examples of how it has been used on real-life data. Lastly, we explain how another popular method called 'mapper' works and show some of its applications.Date and time infoWednesday 13:30 - 14:30KeywordsAlgebraic Statistics, Topological Data AnalysisPrerequisitesUndergraduate degree in Mathematics, linear algebra, and probability theory.Undergraduate degree in Mathematics, linear algebra, and probability theory.AudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesAlmost all lectures will be self-contained.
In the neurosciences and other fields, structures and data are often represented as graphs (networks), simplicial complexes, hypergraphs and the like. In recent years many new tools from graph theory, spectral analysis, discrete geometry, topology etc. have been developed to deal with such structures. In this course, I shall introduce and apply the spectrum of such mathematical methods.ReferencesJ.Jost, Mathematical concepts, Springer, 2015AudienceGraduate students and postdocsLanguageEnglish
A Riemannian metric on an oriented surface, i.e. a two-dimensional oriented real manifold, turns the surface into a one-dimensional complex manifold.
Teichmüller space is a differential geometric approach to the classification of complex structures on the surface. Due to its invariances, namely conformal and diffeomorphism invariance the action functional of harmonic maps on surfaces serves as a tool to study Teichmüller space. After a brief introduction to Riemann surfaces we want to use the theory of harmonic maps to prove
the uniformization theorem: Every Riemann surface is the quotient of one of the three simply connected Riemann surfaces - Riemann sphere, the complex plane or the complex upper half plane.
Teichmüller theorem: The Teichmüller space is diffeomorphic to the space of holomorphic quadratic differentials on a given Riemann surface.
Here we will mostly follow the book "Compact Riemann surfaces" by Jürgen Jost. Depending on time and interest of the audience, we want to treat the Weil-Peterson geometry of Teichmüller space or give an outlook to the case of super Riemann surfaces.Date and time infoFriday 09:30 - 11:00KeywordsRiemann surfaces, Teichmüller space, harmonic mapsPrerequisitesbasic differential geometryAudienceMSc students, PhD students, PostdocsLanguageEnglish
The plan is to start the by now classical regularity theory for the Monge-Ampere equation, and optimal transportation, by L. Caffarelli from the 90s, and then to treat the partial regularity theory by Figalli and Kim, in order to come to a recent completely variational theory. If time permits, we also address the case of a general cost function.Date and time infoWednesday 09:15 - 11:00KeywordsRegularity theoryPrerequisitesThe arguments will be fairly self-contained, mostly relying on properties of convex functions and elementary arguments in PDE, like the maximum principle.AudienceMSc students, PhD students, PostdocsLanguageEnglish
The aim of this Reading Seminar is to discuss classical results and recent developments about random functions on the hypercube. They arise naturally in theoretical computer science and combina-torics, and in the last decade their general properties have been instrumental for new striking developments in statistical physics and percolation...Date and time infoFriday 15:15 - 17:00AudienceMSc students, PhD students, PostdocsLanguageEnglish
The lecture will be divided in two. In the first half we will discuss the complex representation theory of the general linear group, including Schur-Weyl duality and related Young tableaux combinatorics.
The second half will be an introduction to complex geometry and projective geometry. We will begin with first examples and Bezout’s theorem. Eventually, we will build up to vector bundles and explain how one uses them to solve enumerative problems. At the very end, the two lectures will merge by stating Borel-Weil-Bott theorem, which unites both subjects.Date and time infoThursday 15:15 - 16:30KeywordsGeomeric and combinatorial methods in the complex representation theory of semi-simple groups.PrerequisitesLinear algebraAudienceMSc students, PhD students, PostdocsLanguageEnglishRemarks and notesMost lectures will be self-contained, encouraging diverse participation!
