Vorträge in der Woche 03.07.2023 bis 09.07.2023

Montag, 03.07.2023: Inverse theorems for the Gowers Norm

Asgar Jamneshan (Koc University Istanbul)

Higher-order Fourier analysis, pioneered by Gowers' seminal work on Szemeredi's theorem, has evolved into a dynamic field that bridges various disciplines, including ergodic theory, additive combinatorics, harmonic analysis, and Lie groups. Its applications span significant problems in analytic number theory and theoretical computer science. At the core of this subject lies the Gowers norm. Understanding the qualitative and quantitative aspects of the inverse theory for the Gowers norm is of theoretical and practical interest. This talk provides an introduction to the Gowers norm and explores its inverse theory from an ergodic theoretical perspective. Additionally, time permitting, we will present our collaborative contributions to solving specific cases of the ergodic inverse problem

Montag, 03.07.2023: Inverse theorems for the Gowers Norm

Asgar Jamneshan (Istanbuhl)

Higher-order Fourier analysis, pioneered by Gowers' seminal work on Szemeredi's theorem, has evolved into a dynamic field that bridges various disciplines, including ergodic theory, additive combinatorics, harmonic analysis, and Lie groups. Its applications span significant problems in analytic number theory and theoretical computer science. At the core of this subject lies the Gowers norm. Understanding the qualitative and quantitative aspects of the inverse theory for the Gowers norm is of theoretical and practical interest. This talk provides an introduction to the Gowers norm and explores its inverse theory from an ergodic theoretical perspective. Additionally, time permitting, we will present our collaborative contributions to solving specific cases of the ergodic inverse problem

Montag, 03.07.2023: Inverse theorems for the Gowers Norm

Asgar Jamneshan (Koc University Istanbul)

Higher-order Fourier analysis, pioneered by Gowers' seminal work on Szemeredi's theorem, has evolved into a dynamic field that bridges various disciplines, including ergodic theory, additive combinatorics, harmonic analysis, and Lie groups. Its applications span significant problems in analytic number theory and theoretical computer science. At the core of this subject lies the Gowers norm. Understanding the qualitative and quantitative aspects of the inverse theory for the Gowers norm is of theoretical and practical interest. This talk provides an introduction to the Gowers norm and explores its inverse theory from an ergodic theoretical perspective. Additionally, time permitting, we will present our collaborative contributions to solving specific cases of the ergodic inverse problem.

Dienstag, 04.07.2023: Solving piecewise-smooth dynamical systems

Ernst Hairer (Universität Genf)

This talk considers piecewise-smooth dynamical systems and regularizations, where the jump discontinuities of the vector field are smoothed out in an eps-neighbourhood by using a continuous transition function. The limit solution (for eps to zero) of the regularized differential equation is studied on a formal level. In the case, where the discontinuous system has more than one solution, it permits to select the Filippov solution that is most meaningful. The analysis is more or less obvious in the case of a single discontinuity surface. Close to the intersection of two discontinuity surfaces, it is still possible to get a good understanding of the solution. The analysis is based on the hidden dynamics, and a classification of all possible situations can be obtained. The situation is much more complex close to the intersection of three discontinuity surfaces. Very little is known about the behaviour of the solution. A few results for particular situations will be presented. This talk is based on joint-work with Nicola Guglielmi at GSSI, L’Aquila.

Dienstag, 04.07.2023: Solid Groups II

Anton Deitmar

Mittwoch, 05.07.2023: Algebraic Curves, Computer Algebra and Integrable Systems

Türkü Celik (Istanbul)

Algebraic curves have significant applications in the study of the Kadomtsev-Petviashvili hierarchy, which is a universal integrable system that describes shallow water waves. The finite-genus quasi-periodic solutions of this system are expressed in terms of the Riemann theta function. These solutions, in turn, characterize the curves in the moduli space of abelian varieties, thus providing a solution to the Schottky problem. As the curve becomes more singular, the Riemann theta function degenerates, leading to soliton and rational solutions. To better understand this bilateral connection, we explore various perspectives in algebraic geometry, including transcendental and combinatorial approaches, while also utilizing mathematical software. Our study yields new results and insights, along with future directions for research in this field.

Donnerstag, 06.07.2023: Matroid invariants

Benjamin Schröter (Frankfurt)

Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid. In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., sparse paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new. However, this class appears naturally in this context. Moreover, (sparse) paving matroids are split. I will demonstrate the framework by looking at several examples. This talk is based on the preprint `Valuative invariants for large classes of matroids' which is joint work with Luis Ferroni.