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!
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.
Topics
Part I: Tobias Ried: Multimarginal Optimal Transport and Applications
Multimarginal optimal transport is a natural generalisation of the classical optimal transport problem. Recently, it received quite some attention due to its applications in density functional theory and statistical learning. In the four lectures I want to introduce the problem, sketch the main existence results and structural properties of optimisers, and highlight some of the applications. The focus will be more on conceptual questions instead of full proofs, and making connections to the many areas of mathematics and applications that multimarginal optimal transport appears in.
Dates: 24.10., 1.11. (Tuesday due to Public holiday!), 7.11. (in G3 10), 14.11
Part II: Florentin Münch: Discrete Ricci Curvature
The lecture aims to give an overview about discrete Ricci curvature notions, particularly, Forman, Ollivier, Bakry Emery, and entropic curvature. We will use a unified approach to the four curvature notions using the Bochner formula and gradient estimates for the heat equation. The main goal is to give a good intuition about the curvature, and to this end, we will calculate the curvature of many examples, and particularly, your favorite graphs.
Dates: 21.11., 28.11. (in G3 10), 5.12., 12.12.
Part III: Dejan Gajic and Stefan Czimek: The Einstein Equations: Initial data and dynamics
Dates: 06., 13., 20., and 27. February 2023Date and time infoMondays 10.30-12.00 (please note special dates and rooms at the lectures)AudienceIMPRS students and othersLanguageenglish
This course is designed so someone can understand the definitions of standard data science terms, and the associated mathematical terms. We also give proofs of how commonly used techniques in data science work along with implementing algorithms and examples with a computer program.
We start by covering Linear algebra and Probability necessary for the course, and then proceed in 4 different topics: network analysis, machine learning, topological data analysis, and then low rank matrices and tensor.Date and time infoTuesdays 11:15-12:45 and Wednesdays 15:15-16:45KeywordsLinear algebra, probability theory, network analysis, machine learning, topological data analysis, tensorsAudienceMasters level course, but still of interest to PhD students and postdocs wanting to become familiar with data science terms
In this online course we aim to get acquainted with some of the recent progress in the mathematical understanding and theory of machine learning. Particular emphasis will be laid upon overparametrization, implicit bias and aspects of unsupervised learning. The aim of the course is to get an overview of a series of recent articles and the methods developed therein. The course is directed to non experts with a solid background in mathematics aiming to get an idea of recent progress in the mathematics of machine learning. Active contribution in form of a presentation of a research paper (to be chosen) is required for each participant.During the first five sessions of this course, an introduction to the field of (supervised and un-supervised) machine learning will be given.Date and time infoWednesdays 16:15-17:45
The Boltzmann Equation is a differential-integral equation, describing how the distribution of velocities in a dilute gas evolves over time. This minicourse will focus on the spatially homogeneous case, where the theory has connections to many different areas of analysis and probability, and we will discuss aspects of the well-posedness theory, the derivation from a stochastic many-particle system, and relaxation to equilibrium.Date and time infoWednesdays, 14:00-15:30KeywordsBoltzmann Equation, Kinetic Theory, Mean-Field LimitsPrerequisitesBasic knowledge of probability and analysis
Date and time infoFriday, 2-3.30pm (Attention: On 14.10. in room A3 01)KeywordsBasic concepts of network analysis, examples of data. discrete curvatures of graphs and hypergraphs, Laplacian spectraPrerequisitesMathematical abilityAudiencePhD and master students, researchers
Nonlinear potential theory studies properties of solutions of nonlinear elliptic equations in analogy to the study of solutions to Laplace equation in harmonic analysis. In recent years, this perspective has gained increased interest for questions regarding the regularity theory of such equations and has been used to prove a number of surprising and sharp regularity statements.
We will first explore important notions of nonlinear potential theory such as capacity, the comparison principle and polar sets. We will then use these tools to prove a-priori estimates, culminating in a nonlinear-Stein theorem - solutions to a general class of nonlinear elliptic PDEs have continuous gradient if the data is in L(n,1).Date and time infoThursdays, 09.00-10.30Keywordsnonlinear elliptic PDE, potential theory, Wolff potentialsPrerequisitesKnowledge of basic properties of Sobolev spaces is assumed. Knowledge of basic elliptic regularity theory will be useful, but is not essential.
A recent area at the interface of probability/combinatorics and topology/geometry has been random topology. The idea is to understand the topological properties of random geometric and combinatorial structures. We will explore the topology and geometry of different random simplicial and cubical complex models. Many ideas developed for random graph models will be extended to higher dimensional notions, for example, connectedness can be thought of as Betti so one can ask about cycles in terms of Betti 1. We will explore two models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also study a more combinatorial model, the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube. We will also examine the properties of the
Linial-Meshulam model for simplicial complexes, an extension of the Erdos-Renyi random graph model. We will discuss the notion of a giant component for the Linial-Meshulam as well as the properties of minimum spanning cycles. On the road, we will understand some algorithms that were developed and used for analyzing and doing computational experiments on these models. The probabilistic method and ideas such as branching processes will be used extensively.Date and time infoTuesdays, 13:30-15:00
We will continue the lecture series from last semester. There will be a lot of repetition of the contents of last semester, so that it should be easy for the new audience to follow along. The planned subjects areGradient estimates
Eigenvalue estimates
Homology
Volume growthDate and time infoThursday, 15.00-16.30
This is a specialized course in Algebraic Geometry. In a long series of papers starting from the eighties, Shigeru Mukai provided several constructions of Fano threefolds, K3 surfaces and canonical curves in some special types of Grassmannians, which we will call Mukai Grassmannians. Apart from the elegance of the constructions itself, these models were used in the classification of prime Fano threefolds and the proof of the unirationality of the moduli spaces of curves in low genus. Moreover, they give explicit equations for canonical curves and K3 surfaces.
Through the course we will survey the known Mukai Grassmannians and their connection with classification problems in Algebraic Geometry, together with some applications.Date and time info12 lectures on Tuesdays 10:00 - 11:30 and Thursdays, 10:30 - 12:00 starting September 20
The Hilbert scheme of a given projective variety is the scheme that parameterize all of its closed subschemes. As such it is ubiquitous in Algebraic Geometry, with connections with other areas of Mathematics and the Sciences. One of the celebrated results of Grothendieck was the construction of this scheme. The goal of this reading group will be to study the construction of the Hilbert scheme following a mostly explicit approach, explore its properties and give some examples and applications.
The schedule of the talks is the following
March 14Lecture 1 15:00-16:00, Speaker: Maximilian Wiesmann. Title: Introduction to Hilbert Schemes
Lecture 2 16:10-17:10, Speaker: Emeryck Marie. Title: Construction of the Hilbert Scheme
March 16Lecture 3 11:00-12:00, Speaker: Alexander Elzenaar. Title: Connectedness
Lecture 4 13:30-14:30, Speaker: Leo Kayser. Title: Smoothness
March 2111:00-12:00 Exercises/Questions/Discussion
March 23Lecture 5 11:00-12:00, Speaker: Javier Sendra-Arranz. Title: Examples and pathologies
Lecture 6 13:30-14:30, Speaker: Barbara Betti and Stefano Mereta. Title: Multigraded Hilbert scheme
March 24Lecture 7 10:00-11:00, Speaker: Henry Robert Dakin. Title: Hilbert scheme of points on surfaces
Lecture 8 11:10-12:10, Speaker: Pierpaola Santarsiero and Casabella. Title: Applications of Hilbert Schemes
March 2811:00-12:00 Exercises/Questions/DiscussionKeywordsAlgebraic Geometry, Hilbert Schemes, Commutative AlgebraLanguageEnglish
Date and time infoWednesday 11-13 and Thursday 15-17PrerequisitesLinear and abstract algebraLanguageDeutsch (voraussichtlich -- evtl. Englisch möglich)
The course will be given as a topics/overview course with a lot of technical details omitted, in order to cover a maximal range of aspects. The lecture series will end with an optional one day in person seminar by students. Everyone is invited to attend either part.Date and time infoFriday 09.15 -- 10.45 CET, starting Oct 28.KeywordsMonge-Kantorovich problem, Kantorovich duality, Otto Calculus, Gradient Flows and JKO scheme, Ricci Curvature and Optimal Transport, Schrödinger Problem and Entropic Relaxation, Sinkhorn Iterative Scaling Algorithm, Diffusion Generative Models, Quantum Optimal Transport
