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!
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-borovik-korte-s26-join@mis.mpg.deAll meetings will take place in the Leon-Lichtenstein room (E2 10) between 13:00 and 15:00 (around 1.5 hrs to present followed by discussion). These will be live-streamed so that people who aren't in Leipzig can still attend, but we request that speakers be present in-person.Block I (22 June - 3rd July)Main reference: "Representation Theory: A First Course" by William Fulton and Joe Harris22 June (Smita Rajan): Finite Groups (Part I)26 June (Vincenzo Isoldi and Lauren Smyth): Lie groups and Lie algebras (Part II)29 June (Viktoriia Borovik): Representations of Lie algebras (Chapter 14 and selected examples from Part III)1 July (Otto Schmidt): Dynkin diagrams and complex Lie groups (Chapters 21 and 23)3 July (Gregory Li and Claire de Korte): Weyl character formula, real Lie algebras and Lie groups (Chapters 24 and 26)
Block II (20th July - 31st July)Main reference: "Computational Invariant Theory" by Harm Derksen and Gregor Kemper22 July (Otto Schmidt and Hongmiao Yu): Preliminaries (Chapters 1 and 2)24 July (Bernd Sturmfels and Daniele Taufer): Invariant theory of finite groups (Chapter 3)29 July (Julian Weigert and Leonie Kayser): Invariant theory of reductive groups (Chapter 4)31 July (Thiago Holleben and Hannah Friedman): Applications of invariant theory (selected sections of Chapter 5)
Block III (31st of August - 4th of September)Representation and Invariant Theory in PhysicsDate and time info22.06., 26.06., 29.06., 01.07., 03.07., 20.07.-31.07., 31.08.-04.09.
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-carmona-s26-join@mis.mpg.deA celebrated theorem by Lurie and Pridham states, roughly speaking, that over a field k of characteristic zero there is a correspondence between pointed formal deformation problems and differential graded Lie-algebras (dglas), implemented by associating with a nice enough dgla a deformation problem via the simplicial Maurer-Cartan functor. This deep result formalized Deligne-Drinfeld's derived deformation theory philosophy, and has been extended in several directions, such as relaxing the restriction on the characteristic, working over families, or even considering different parameters other than dg-Artin local k-algebras. The goal of this lecture series is to introduce the audience to this technology by focusing on operadic deformation theory. Time permitting, we will review work of Kontsevich, Merkulov, Tamarkin, Willwacher... on deformation quantization.Date and time info09/04, 22/04, 29/04, 06/05, 13/05, 27/05, 17/06 at 13:30KeywordsDeformation Theory, Maurer-Cartan Space, OperadsPrerequisitesBackground in Homological/Homotopical Algebra
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-fu-jay-s26-join@mis.mpg.deThis course provides an introduction to the dynamics of billiards in polygons and their connections with modern geometric structures. We begin with billiards in simple polygons, such as the square and rational triangles, and use them as guiding examples throughout the course. These systems naturally lead to the notion of translation surfaces, which offer a geometric framework for understanding polygonal billiards. Translation surfaces can be described from several complementary perspectives, and we discuss the equivalence between these viewpoints. A central theme of the course is the relation between translation flows and interval exchange transformations (IETs), which are specific functions from an interval to itself. Every directional flow on a translation surface can be encoded by the iterations of an IET, revealing how certain two-dimensional dynamical systems can be studied through one-dimensional systems. Conversely, IETs can be geometrically realized on translation surfaces. Keane's theorem provides a criterion for minimality of IETs and will be one of the main results discussed in the course. Renormalization is a key idea in dynamical systems: it is about transforming the initial dynamical system into another (in the same class) to get new information on the initial one. We introduce Rauzy induction as a renormalization procedure acting on the parameter space of interval exchange transformations. In parallel, we introduce parameter spaces of translation surfaces and the $GL^{+}(2,\mathbb{R})$-action, which gives a renormalization process for the directional flows. These two viewpoints provide complementary approaches to the study of translation surfaces, and we discuss how these actions can be used to investigate their dynamical behavior. The aim of the course is to present a coherent picture connecting polygonal billiards, flat geometry, and dynamical systems, while preparing the ground for further study in Teichmüller dynamics and related areas.Date and time infoWednesday, 15:15--17:15, weekly (start date to be confirmed).KeywordsPolygonal billiards; translation surfaces; interval exchange transformations; dynamical systems.PrerequisitesBasic background in complex analysis and topology. Familiarity with holomorphic functions, oriented two-dimensional manifolds, differential 1-forms, and basic notions of measure theory is assumed. Some familiarity with differential geometry or dynamical systems is helpful but not required. Necessary concepts will be introduced as needed.
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-jhirsch-s26-join@mis.mpg.deThe topics of the lectures can be chosen according to the interest of the participants. We started this term with the discussion of Savin's result on small perturbation of elliptic systems.Date and time infoThursdays, 9:15-10:45Keywordsmore recent developments in the field of analysis and PDE'sPrerequisitesPDE 1, PDE 2, measure theory would be of helpLanguageenglish
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-rebucci-zizza-s26-join@mis.mpg.deThe Boltzmann equation describes the time evolution of the particle density of a collisional rarefied gas in the phase space. In this series of lectures we will outline the derivation of the Boltzmann kinetic equation from a microscopic system of hard spheres for arbitrarily long times. This is based on the recent work of Deng, Hani and Ma (https://arxiv.org/abs/2408.07818).Date and time infoStarting from the week of 20th April, total of 4/5 lecturesKeywordsBoltzmann Equation, Long-time derivation, Lanford's theorem, Cluster ExpansionPrerequisitesNone
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-salguero-s26-join@mis.mpg.deFourier analysis has been a key tool in the study of partial differential equations since its first use in the heat equation. In the 1930s, Littlewood and Paley developed a powerful theory to decompose functions into spectrally localized pieces, which proved to be very effective in handling differential operators. But it was not until the early 1980s, with the pioneering work of J. M. Bony, that the Littlewood-Paley decomposition gave rise to a systematic framework for tackling nonlinear differential equations: paradifferential calculus. Since then, this machinery has proven highly successful in the analysis of nonlinear PDEs. In these lectures, we will introduce the Littlewood-Paley decomposition, Besov spaces, and the paraproduct decomposition (the essential ingredient in paradifferential calculus). Furthermore, we will discuss some applications of this theory to classical models in fluid dynamics.Date and time infoTuesdays at 10:00 starting from April 28KeywordsFourier analysis, Littlewood-Paley theory, Besov spaces, paradifferential calculus, PDEPrerequisitesBasic background in PDE and functional analysis is needed
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-rvl-s26-join@mis.mpg.dePart I: "The problem of global unique solvability of the Navier-Stokes equations"by Timofei Shilkin
May 08, 15, 22, 29
Part II: "Coarse Graph Theory" (TBC)by Sandra Albrechtsen and James Davies
Date: June 5, 10, 12, 17
When viewed from far away, many graphs and metric spaces appear to have a much simpler "coarse" structure which captures their large-scale geometry. This idea is formalised by the notion of quasi-isometries. For example, string graphs and complete Riemannian planes are both quasi-isometric to planar graphs. The blossoming area of coarse graph theory aims to analyse the coarse structure of graphs and metric spaces.
We give an introduction to coarse graph theory and discuss some recent results. Topics will include coarse Menger's theorem, asymptotic minors, string graphs, asymptotic dimension, and connections to Cayley graphs and Riemannian surfaces.
Part III:
Quasinormal modes of black holes in General Relativityby Marc Casals
Date: June 19, 26, July 3, 10
Field perturbations of a black hole spacetime can be modelled, during a phase of their evolution known as ringdown, by quasinormal modes: exponentially-damped oscillations whose frequencies and decay rates depend only on the parameters of the black hole. In the specific case of gravitational field perturbations, the Laser Interferometer Gravitational-Wave Observatory has detected waveforms emitted by black hole binary inspirals which were consistent with the theoretical quasinormal mode prediction during the ringdown. In this Ringvorlesung, we will first give a brief introduction to the theory of General Relativity and black holes. We will then study linear field perturbations of black hole spacetimes. After separation of variables, the field equations reduce to ordinary differential equations of Heun type and their analysis will enable us to model the quasinormal mode ringdown.
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-sodomaco-s26-join@mis.mpg.deAbstract
Many optimization problems can be represented by systems of parametric polynomial equations. The choice of parameters influences the geometry of the solutions associated with these polynomial systems. A sufficiently "generic'' or "random'' selection of parameters often results in solution sets that exhibit expected properties, such as the expected dimension or cardinality. In contrast, non-generic parameters give rise to algebraic subsets known as Data Loci. Each data locus is formed by imposing additional conditions on the solution sets.
In this course, we will provide an overview of the most relevant types of data loci in algebraic optimization, with a particular emphasis on data loci in distance optimization - a related field referred to as Metric Algebraic Geometry - and on the computation of Nash equilibria in Game Theory.
Each lecture will contain several examples and computations in Macaulay2 and Julia.
Lectures:
1) April 09 (10:00-12:00, G310): The intersection theory you need: polar classes and projective duality
2) April 13 (15:00-17:00, G3 10): Counting critical points
3) April 20 (10:00-12:00, G310): The first data loci: evolutes and distance discriminants
4) April 27 (10:00-12:00, G310): Offsets and distance polynomials
5) May 04 (10:00-12:00, G310): Relative polar classes and conditional data loci
6) May 07 (10:00-12:00, G310): Degrees of conditional data loci and Kalman varieties
7) May 18 (10:00-12:00, G310): Higher-order polar classes and higher-order distance loci
8) May 21 (10:00-12:00, G310): Data loci in Game Theory 1: Nash discriminants and resultants
9) May 28 (10:00-12:00, G310): Data loci in Game Theory 2: Nash LociDate and time info09.04., 13.04., 20.04., 27.04., 04.05., 07.05., 18.05., 21.05., 28.05. always 10:00-12:00 (except 13.04 at 15:00-17:00)
SubscriptionSubscription to the mailing list is also possible by sending an email with subject "subscribe" and empty email body to lecture-winter-s26-join@mis.mpg.deMuch appeal of polytope theory stems from the fact that convex polytopes are simultaneously geometric and combinatorial objects. In this lecture will highlight the geometric side of this interaction by following two threads.
On the one hand we study the metric properties of individual polytopes, such as edge lengths, angles, and volumes, the relations between these quantities as well as the constraints imposed on them by the combinatorics. Tools employed range from convex and hyperbolic geometry to spectral graph theory.
On the other hand we study the geometric and topological properties of entire realization spaces, that is, the ways in which a fixed combinatorial type can be realized or a given realizations can be deformed. Realization spaces are real algebraic sets, and we address topics such as universality, parametrization and rigidity.Date and time infoWednesday, ca 10am - 12am (running April 15 -June 17, no lecture on May 20 and (potentially) May 6)Keywordsconvex polytopes, realization spaces, metric geometry, rigidity, reconstructionPrerequisitesBasics of convex polytopes, face latticeRemarks and notesThis course is conceived as a follow up on the block course "Convex Polytopes" by Bernd Sturmfels (https://www.mis.mpg.de/de/events/event/ringvorlesung-2)
