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 goal of this lectures is to give a proof of the local rigidity theorem for lattices of semisimple Lie groups. Let be a simple Lie group not locally isomorphic to . Let be a cocompact lattice in , i.e. a discrete subgroup such that is compact. The theorem states that the inclusion of in is an isolated point of the character variety . We will present a beautiful proof due to Andre Weil that you can find in the book "Discrete subgroups of Lie groups" of Raghunathan. During the proof, we will develop tools that allow to compute for a linear representation of . The lecture will mainly be about flat vector bundles and bundle-valued differential forms, which we will cover from scratch.KeywordsDifferential geometry, symmetric spaces, discrete subgroups of Lie groupsPrerequisitesWe will only need basic facts from from Lie theory and differential geometry.Remarks and notesThere will be notes on my website.
The course aims to introduce the study of real algebraic sets and semialgebraic sets from a topological and differential point of view. After reviewing the basic techniques in real algebra, we discuss the Tarski-Seidenberg theorem, showing that the projection of a semialgebraic set is semialgebraic. We then prove a semialgebraic version of Sard's theorem of critical values, and Hardt's semialgebraic triviality.
We apply the previous results to the study of real algebraic sets, bounding the number of their connected components (Harnack's theorem for curves), and more generally their Betti numbers (Thom-Milnor bound). We conclude by discussing the existence of algebraic models for compact real manifolds, i.e. the Nash-Tognoli theorem.
The text that will be used as a reference, and a more detailed program of the course, can be found at https://lorenzobaldi.github.io/teaching/Date and time infoTuesday 11-13, from mid October to mid FebruaryKeywordsReal Algebraic Geometry; Semialgebraic Geometry; Differential and Topological properties of Real Algebraic Sets; Algebraic Models of ManifoldsPrerequisitesBasic knowledge in topology and differential geoemtry. Beckground in basic algebraic geometry and commutative algebra is helpful but not necessary.LanguageEnglishRemarks and notesDuring the course, exercise sheets on related topics not discussed in the lectures, or detailing proofs not completely detailed, will be provided.
The goal of this course is to give the audience an idea what kinds of mathematical problems are being studied in general relativity (GR), along with some intuition for these problems.
In the first part of the course, I will introduce the basic concepts and notions of GR.
In the second and main part of the course, I will descend to spherical symmetry (which has the advantage of removing many of the technical difficulties one typically encounters in GR while still allowing for a rich variety of phenomena), give a completely self-contained development of the theory in spherical symmetry, and prove the spherically symmetric analogues of some of the more recent advances in the field.
In the last part of the course, if enough interest remains, I will then give an introduction to some of the actual methods used when studying GR without symmetry.Date and time infoTuesday or Wednesday 11 am.Keywordsgeneral relativity, initial value problem, black hole formation, black hole stability, cosmic censorship conjecturesPrerequisitesBasic familiarity with Differential Geometry will be helpful but not necessary. Familiarity with PDE will also be helpful but even less necessary.AudiencePeople with an interest in what relativity is about
This will be a hybrid lecture/reading seminar on the arithmetic geometry of Feynman integrals.
Perturbative quantum field theory is a framework with which physicists understand, predict and compute the probabilities of quantum interactions. A key tool in this is the concept of a Feynman integral. These integrals have remarkably deep number theoretic and arithmetic geometric connections to the theory of motives and periods, arising from the values of Feynman integrals. In this course, we will study this connection between Feynman integrals and motives from a computational point of view.KeywordsFeynman integrals, arithmetic geometry, number theoryPrerequisitesNo background in physics is assumed. Knowledge of algebraic varieties will be beneficial.LanguageEnglish
We will potentially consider the following topics
Introduction to Pari/GP
Basic algorithms in number theory
Modular forms
Arithmetic of elliptic curves
L-functions and zeta functions
Computational algorithms for some number theory problems
KeywordsComputational, analytic number theory, modular forms, elliptic curves, L functionsPrerequisitesComfort with programming in python/C++/Haskell/Mathematica. Background on complex analysis and basics of algebra will be assumed.LanguageEnglish
Algebraic analysis investigates linear PDEs and their solution functions by algebraic methods. The main actor is the Weyl algebra, denoted D. It is a non-commutative ring that gathers linear differential operators with polynomial coefficients. The theory of D-modules provides deep classification results of linear PDEs, structural insights into problems in the sciences, as well as new computational tools. Functions that can be encoded by an annihilating D-ideal are called holonomic, and these are ubiquitous in the sciences.
The course is hands-on: the focus lies on the introduction of concepts from algebraic analysis and utilizing them for solving problems arising in applications, such as a systematic study of Feynman integrals from particle physics.
Many physical problems that are described through partial differential equations can also be seen as variational problems, in which we need to find an energy minimisers. Due to natural considerations, physical quantities often satisfy some differential side-constraints; for instance the velocity field of an incompressible fluid is divergence-free. As a consequence, the study of variational problems under such constraints is rather important.
In this lecture, I will attempt to give an overview over different topics connected to linear differential constraints. In particular, topics treated in this lecture (might) include:a (very soft) look at some algebraic properties of differential operators, Korn-type inequalities;A-quasiconvexity and weak lower-semicontinuity;Young measures;different notions of convexity for functions and for sets;some simple schemes of convex integration connected to A-free problems;A-free measures;regularity theory.
Especially topics 5-7 are quite flexible and the detail in treatment might depend on the interest of the audience.KeywordsA-quasiconvexity, notions of convexity, lower semicontinuity, differential inclusions, convex integration, A-free measuresPrerequisitesCourses in Analysis up to Functional Analysis, Linear Algebra 1 & 2
This term, the Ringvorlesung is offered by Peter Smillie, Felix Otto, and Daniel Roggenkamp. Topics of the three parts are:
Part I (Peter Smillie): Harmonic maps
Abstract: A map between Riemannian manifolds that minimizes (or is a stationary point for) total (Dirichlet) energy is called a harmonic map. This is about the most natural PDE arising in Riemannian geometry, and several ubiquitous ideas in geometric analysis were first discovered in the study of harmonic maps. They have found many applications outside of geometric analysis, including in algebraic geometry and manifold topology. And they remain a hot subject, for instance in the context of the non-abelian Hodge correspondence.
In these lectures, I will explain just enough differential geometry to prove some foundational existence, uniqueness, and rigidity theorems for harmonic maps, and then highlight some of their applications to other fields.
Part II (Felix Otto): Convection-Enhanced Diffusion
Part III (Daniel Roggenkamp): Defects and higher categorical structures in quantum field theories
In the past decade, following breakthrough works by Hairer and by Gubinelli, Imkeller and Perkowski, there has been rapid progress in obtaining a notion of solution for singular SPDEs; which are roughly speaking PDEs with a random forcing of low enough regularity that the PDE is not classically well-posed in any space of distributions and instead requires the use of probabilistic data for a suitable "renormalisation".
In a series of challenging works lying at the intersection of analysis, probability and algebra, Hairer's theory of regularity structures has been extended to a systematic machinery treating general "subcritical" (or "super-renormalisable") semilinear singular SPDEs. The purpose of this course is to give a more digestible introduction to Hairer's approach. In early lectures, in order to expose the ideas leading to the general machinery, we will develop a small parameter solution theory for some relatively simple singular SPDEs. The topics of later lectures will be fixed according to audience interest.KeywordsSPDE, Renormalisation, Regularity StructuresPrerequisitesA basic knowledge of spaces of test functions and distributions (as in e.g. Rudin's Functional Analysis Chapter 6).
This weekly lecture series will cover topics at the intersection of mathematics and machine learning, running from November 4, 2024, to March 24, 2025.
Topics include:
Deep learning (brief introduction)
Convolutional neural networks
Autoencoders
LLMs for code generation
Foundations of feature learning
Geometric and topological deep learning
Dynamical systems and machine learning
Varieties and machine learning, with applications to computer vision and biology
Lecturers: Paul Breiding, Marzieh Eidi, Jan Ewald, Robert Hasse, Parvaneh Joharinad, Duc Luu, Guido Montufar, Nico Scherf, Diaaeldin Taha, Angelica Torres.
Organizers: Parvaneh Joharinad, Diaaeldin Taha.Keywordsmathematics and machine learningAudienceStudents and researchers in Math and/or CSLanguageEnglish
