Regular lectures Winter semester 2024-2025: MPI MIS


01.10.24	Jacques Audibert Local rigidity of lattices SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24jacques-join@mis.mpg.deThe goal of this lectures is to give a proof of the local rigidity theorem for lattices of semisimple Lie groups. Let \(G\) be a simple Lie group not locally isomorphic to \(\textrm{SL}_2(\mathbb{R})\). Let \(\Gamma\) be a cocompact lattice in \(G\), i.e. a discrete subgroup such that \(G/\Gamma\) is compact. The theorem states that the inclusion of \(\Gamma\) in \(G\) is an isolated point of the character variety \(\textrm{Hom}(\Gamma,G)/G\). 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 \(H^1(\Gamma,V)\) for \(V\) a linear representation of \(G\). 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. Subscribe
01.10.24	Leonhard Kehrberger Mathematical Problems in General Relativity and Black Holes SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24leonhard-join@mis.mpg.deThe 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 Subscribe
01.10.24	Abhiram Mamandur Kidambi Arithmetic of Feynman Integrals SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24abhiram-join@mis.mpg.deThis 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 Subscribe
01.10.24	Abhiram Mamandur Kidambi Computational number theory - I SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24abhiram1-join@mis.mpg.deWe 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 Subscribe
01.10.24	Anna-Laura Sattelberger Introduction to Algebraic Analysis SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24annalaura-join@mis.mpg.deAlgebraic 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. Subscribe
01.10.24	Stefan Schiffer Linear differential constraints in the Calculus of Variations SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24stefan-join@mis.mpg.deMany 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 Subscribe Slides
01.10.24	Rhys Steele An Introduction to Singular Stochastic PDEs via Regularity Structures SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24rhys-join@mis.mpg.deIn 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). Subscribe

01.10.24

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24jacques-join@mis.mpg.deThe goal of this lectures is to give a proof of the local rigidity theorem for lattices of semisimple Lie groups. Let \(G\) be a simple Lie group not locally isomorphic to \(\textrm{SL}_2(\mathbb{R})\). Let \(\Gamma\) be a cocompact lattice in \(G\), i.e. a discrete subgroup such that \(G/\Gamma\) is compact. The theorem states that the inclusion of \(\Gamma\) in \(G\) is an isolated point of the character variety \(\textrm{Hom}(\Gamma,G)/G\). 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 \(H^1(\Gamma,V)\) for \(V\) a linear representation of \(G\). 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.

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01.10.24

Leonhard Kehrberger

Mathematical Problems in General Relativity and Black Holes

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24leonhard-join@mis.mpg.deThe 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

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01.10.24

Abhiram Mamandur Kidambi

Arithmetic of Feynman Integrals

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24abhiram-join@mis.mpg.deThis 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

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01.10.24

Abhiram Mamandur Kidambi

Computational number theory - I

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24abhiram1-join@mis.mpg.deWe 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

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01.10.24

Anna-Laura Sattelberger

Introduction to Algebraic Analysis

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24annalaura-join@mis.mpg.deAlgebraic 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.

Subscribe

01.10.24

Stefan Schiffer

Linear differential constraints in the Calculus of Variations

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24stefan-join@mis.mpg.deMany 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

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Slides

01.10.24

Rhys Steele

An Introduction to Singular Stochastic PDEs via Regularity Structures

SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture24rhys-join@mis.mpg.deIn 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).

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Regular lectures Winter semester 2024-2025

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