Talks at the mathematical institute

Charlotte Bartnick:

Definable groups in differentially closed fields of positive characteristic

Time and place

Tuesday, 1.7.25, 14:30-16:00, Seminarraum 404

Abstract

Definable groups in theories of fields can often be described in terms of algebraic groups. For example, by a result of Pillay, every definable group in a differentially closed field of characteristic 0 embeds into an algebraic group.

In this talk, we show that the same holds true for positive characteristic. In fact, methods used by Delon and Bouscaren to describe groups in separably closed fields of finite degree of imperfection can be generalized to several theories of fields in positive characteristic with extra structure. Before outlining the proof for differentially closed fields of positive characteristic, we will first introduce the theory and the properties that are used in the proof.

Prof. Dr. Barbara Niethammer:

Free boundary problems in models for cell polarization

Time and place

Thursday, 3.7.25, 15:00-16:30, Hörsaal 2

Abstract

Cell polarization denotes the rearrangement of certain substances on the membrane of a cell in response to an external chemical signal. It is a crucial ingredient in many biological processes, such as for example the motion of cells. Starting from a bulk-surface reaction-diffusion system for several protein densities describing this process, we rigorously derive nonlocal free boundary problems that allow for a relatively simple characterization of polarization. For these limit systems we prove global stability of steady states and characterize the parameter regime for the onset of polarization. We also discuss some aspects of regularity in time of the free boundary.

(joint work with A. Logioti (Stuttgart), M. Röger (Dortmund) and J. Velazquez (Bonn))

Simon Ebert:

A local perspective on manifolds with a Lie structures at infinity

Time and place

Monday, 7.7.25, 16:15-17:45, Seminarraum 404

Abstract

A Lie structure at infinity on a smooth manifold can be seen as the extension of the tangent bundle to a Lie algebroid on the closure of the respective manifold. This allows us to extend a Riemannian metric on the manifold up to infinity. In the talk, I will present the notion of a Riemannian manifold with Lie structure at infinity following Ammann, Lauter & Nistor (2004) and focus on the role of the Lie algebroid. I will use a local approach to show that the Riemannian metric blows up towards the boundary. Furthermore, I will show exemplarily how a Lie structure at infinity on a manifold yields a compactification when linking the Riemannian metric and the corresponding metric space.

Yohance Osborne:

Stabilized finite element approximation of Mean Field Game Partial Differential Inclusions

Time and place

Tuesday, 8.7.25, 14:00-15:00, Seminarraum 226, HH10

Abstract

In 2006, Lasry and Lions introduced Mean Field Games (MFG) as models for Nash equilibria of differential games of stochastic optimal control that involve a very large number of players. MFG equilibria are typically described by nonlinear forward-backward systems where a Hamilton--Jacobi--Bellman (HJB) equation, satisfied by the value function of a representative player, is coupled with a Kolmogorov--Fokker--Planck (KFP) equation for the player density. A range of applications of MFG have since appeared, such as in finance, economics, pedestrian dynamics, optimal transport, smart grid management, and traffic flow.

A typical assumption in the literature on MFG asserts the differentiability of the Hamiltonian in the HJB equation of the MFG system, which leads to an unambiguous advection in the KFP equation of the system that is based on the derivative of the Hamiltonian. However, it is known from applications of optimal control that optimal controls are not always unique, such as being bang-bang for example, thereby leading to HJB equations with convex but non-differentiable Hamiltonians.

In this talk I will introduce an extension of the MFG system to the case where the optimal controls for players are non-unique and the Hamiltonian of the MFG system is non-differentiable, where the KFP equation is generalized to a partial differential inclusion (PDI) based on selections of the subdifferential of the Hamiltonian. This results in a new class of models called Mean Field Game Partial Differential Inclusions (MFG PDI). I will introduce a monotone stabilized finite element discretisation of the weak formulation of time-dependent MFG PDI with Lipschitz, convex but (possibly) non-differentiable Hamiltonians. I will present theorems on the well-posedness of the discretisation and its strong convergence to weak solutions of the MFG PDI in the joint limit as the time-step and mesh-size vanish. The talk will be concluded with discussion of a numerical experiment for an MFG PDI with non-smooth solution. This talk is based on my doctoral thesis which was supervised by Dr. Iain Smears.

Xuwen Zhang:

Bounding the area of submanifolds with prescribed boundary in terms of its curvature energy II

Time and place

Tuesday, 8.7.25, 16:15-17:45, Seminarraum 404

Abstract

This time we continue the last talk and sketch the proof of the main results.

S. Hornuss:

Characterization and Modelling of epoxy resin and its curing process

Time and place

Tuesday, 15.7.25, 14:15-15:45, SR 226 (HH10)

Abstract

Dr. Michael Bürker :

„Die Mathematik ist die Sprache der Natur“ – Spannende Aufgaben mit historischem Hintergrund

Time and place

Tuesday, 15.7.25, 18:30-20:00, Hörsaal 2

Abstract

In der Auseinandersetzung mit historischen Problemen der Mathematik können Schülerinnen und Schüler die Wirkungskraft mathematischer Argumentationen erfahren: Welche Argumente haben Mathematikerinnen und Mathematiker früherer Zeit genutzt, denen unsere technischen Hilfsmittel nicht zur Verfügung standen? Wie konnten beispielsweise die Zeitgenossen von Euklid den Erdumfang und die Entfernung zum Mond bestimmen? Im Vortrag werden einige spannende Aufgaben mit historischem Hintergrund vorgestellt. Dabei geht der Referent auf die Frage ein, an welchen Stellen Verständnisschwierigkeiten zu erwarten sind und welche Hilfe man den Lernenden anbieten könnte. Bei den im 20. Jahrhundert vieldiskutierten „unanschaulichen“ Effekten der Speziellen Relativitätstheorie wird im Vortrag gezeigt, wie die durch reine Logik geprägte Argumentation mit Mitteln der Schulmathematik veranschaulicht werden kann.

Julia Flach:

Swiss Re - Mathematik trifft Rückversicherung

Time and place

Tuesday, 22.7.25, 14:15-15:45, SR 226 (HH10)

Abstract

Matthias Aschenbrenner:

A transfer principle in asymptotic analysis

Time and place

Friday, 25.7.25, 10:30-12:00, Seminarraum 404

Abstract

Hardy fields form a natural domain for a “tame” part of asymptotic analysis. They may be viewed as one-dimensional relatives of o-minimal structures, and have applications to dynamical systems and ergodic theory. In this talk I will explain a theorem which permits the transfer of statements concerning algebraic differential equations between Hardy fields and related structures, akin to the “Tarski Principle” at the basis of semi-algebraic geometry, and sketch some applications, including to some classical linear differential equations. (Joint work with L. van den Dries and J. van der Hoeven.)

Alexis Anagnostakis:

General Diffusions on Metric Graphs as limits of Time-Space Markov Chains

Time and place

Friday, 25.7.25, 12:00-13:00, Seminarraum 404

Abstract

We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph \(\Gamma\). The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of \(\Gamma\), with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the \(p\)-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. Additionally, (i) we present a method for constructing numerically efficient approximations with optimal convergence rates and (ii) provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results.

Benjamin Brotz:

Relativistic \(\delta\)-shell interactions

Time and place

Monday, 28.7.25, 16:15-17:45, Seminarraum 404

Abstract

In quantum mechanics, the practice of coupling Hamiltonians with singular potentials supported on lower-dimensional subsets of the ambient space is a well-established concept. For instance, the physicists Ralph Kronig and William Penney had already considered Schrödinger operators coupled with \(\delta\)-point potentials in their 1931 work, "Quantum mechanics of electrons in crystal lattices.“

However, one should remember that such \(\delta\)-potentials are merely idealized representations of real-world physical phenomena. Therefore, understanding how these operators can be rigorously approximated by more regular potentials is important for their physical interpretation and mathematical validity.

In this talk, we will present a result by Albert Mas and Fabio Pizzichillo which shows that the Dirac operator on \(\mathbb{R}^{3}\) coupled with suitable short-range potentials converges in the strong resolvent sense to the Dirac operator coupled with a \(\delta\)-potential supported on the boundary of a \(C^{2}\) domain. Following this, we'll discuss a possible generalization of this result to more general spinor bundles.

Kang Zuo:

Loci of non-rigid families of varieties in corresponding moduli spaces

Time and place

Tuesday, 29.7.25, 16:00-17:30, Seminarraum 125

Abstract