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!
Date and time infoFridays, 14:00-15:30 on October 22, 17:00-18:30 starting October 29Keywordsquasi-stationary measures, quasi-ergodic measures, birth-death processes, Markov processes with killingPrerequisitesBasics of Markov processes
The idea of this reading group is to develop the basic notions of modern algebraic geometry, through the language of schemes.
We will follow closely the wonderful lecture notes by Vakil, Foundations of Algebraic Geometry.Date and time infoMonday, 11.15-12.45
This course will deal with the basic problem of understanding the structure (e.g. the geometry and topology) of the set of solutions of real polynomial equations with random coefficients. The simplest case of interest is the count of the number of real zeroes of a random univariate polynomial whose coefficients are Gaussian random variabl2s – this problem was pioneered by Kac in the 1940s. More generally algebraic geometers might be interested, for example, in the number of components of a random real plane curve, or in the expected number of real solutions of more advanced counting problems. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with random matrix theory and random fields.Date and time infoTuesdays, 11:15-12:45KeywordsAlgebraic Geometry, Probability, Expected Counting Problems, Average TopologyPrerequisitesBasic understanding of algebraic and differential geometry and probabilityLanguageEnglish
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 the generalization error and on the training of neural networks, i.e. non-convex optimization, and overparameterization.
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.Date and time infotent. Wednesdays 16:15-17:45
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 + II: Spectra of tensors and their applicationsSpeakers: André Uschmajew, Raffaella Mulas
There exist several possibilities for defining eigen- and singular values for tensors, based on critical points of multilinear forms or homogeneous equations. In this mini course we introduce some of these concepts, discuss their basic properties as well as their relation to low-rank approximation and hypergraph theory. Geometric questions and computational aspects will also be considered.
Part III: Symmetry enriched fractionalization in quantum matter with emergent Z_2 and U(1) gauge fields Speaker: Inti SodemannDate and time info6-8 lectures, on Wednesdays 2 pm, starting November 3, 2021Prerequisitesbasic knowledge in linear algebra, graph theory and optimization
We might address several topics: 1) Effective behavior of random media (stochastic homogenization), 2) Thermal fluctuations and nonlinearity (singular stochastic PDE), 3) Matching of Poisson point processes and coupling.Date and time infoTuesday, 09.15-11.00; first lecture on October 26, 2021Keywordsstochastic homogenization, singular stochastic PDE
This lecture series aim to offer a gentle introduction to the theory of algebraic operads and related topics, starting with the elements of the theory, and progressing slowly towards more advanced themes, including (inhomogeneous) Koszul duality theory, Gröbner bases, and higher structures. The course will consist of approximately 12 lectures, along with extra talks by willing participants, with the goal of introducing extra material to the course, and making them more familiar with the theory. In particular, participants will be encouraged to read (parts of) accessible research articles and present them during the last sessions of the lecture series.Date and time infoFriday, 09.00-10.30Keywordsalgebraic operads, Gröbner bases, Koszul duality, higher structuresPrerequisitesWe intend for the course to be as introductory as possible, so we will review any necessary material depending on the background of the participants, or provide necessary references. Nonetheless, some knowledge of basic homological algebra, algebraic topology, and representations of finite groups will be useful.AudienceThe course is aimed at advanced MSc students and PhD students.LanguageEnglish
Toric varieties form a well-understood and intensively studied class of algebraic varieties. They provide a rich source of examples and test cases for theorems and conjectures. Moreover, they have direct applications in physics and in polynomial system solving. For instance, compact, projective toric varieties are the natural generalization of projective space considered in the study of discriminants and resultants for sparse polynomials. The theory consists of a nice interplay between algebra, geometry and combinatorics.
In this course, we will start from embedded affine toric varieties via monomial maps to later discuss standard constructions of toric varieties from cones, fans and polytopes. We will motivate the theory by insights from sparse polynomial system solving, and (time permitting) present more advanced constructions such as the Cox ring and line bundles on toric varieties. Some important theorems and constructions that are featured include the orbit-cone correspondence, the Bernstein-Khovanskii-Kushnirenko theorem and the construction of a toric variety as a GIT (Geometric Invariant Theory) quotient.Date and time infoWednesday, 10.00-12.00Keywordstoric varieties, toric geometryPrerequisitesBasic algebraic geometry, at the level of introductory text books such as `Ideals, Varieties and Algorithms'.
