Talks at the mathematical institute

Hannes Jakob:

Failure of Approachability at the Successor of the First Singular for any Cofinality

Time and place

Tuesday, 6.5.25, 14:30-16:00, Seminarraum 404

Abstract

Wei Wei:

Geometric inequality related with \(\sigma_2\) curvature

Time and place

Tuesday, 6.5.25, 16:15-17:45, SR 404

Abstract

Jonah Reuß:

A note on the existence of nontrivial zero modes on Riemannian manifolds

Time and place

Tuesday, 13.5.25, 16:15-17:45, Seminarraum 404

Abstract

In this talk we consider nontrivial solutions, so called zero modes, to the Dirac equation on closed Riemannian manifolds. We will state and prove a necessary criterion for the existence of zero modes, which relates the norm of a certain vector field to the Yamabe constant of the manifold. In the end we will give some insight on a classification of manifolds on which this criterion is sharp.

Prof. Dr. Martin GROHE:

The Logic of Graph Neural Networks

Time and place

Thursday, 15.5.25, 15:00-16:30, Hörsaal 2

Abstract

Graph neural networks (GNNs) are deep learning models for graph data that play a key role in machine learning on graphs. A GNN describes a distributed algorithm carrying out local computations at the vertices of the input graph. Typically, the parameters governing this algorithm are acquired through data-driven learning processes.

After introducing the basic model, in this talk, I will focus on the expressiveness of GNNs: which functions on graphs or their vertices can be computed by GNNs? Understanding expressiveness will help us to understand the suitability of GNNs for various application tasks and guide our search for possible extensions.

Surprisingly, the expressiveness of GNNs has a clean and precise characterisation in terms of logic and Boolean circuits, that is, computation models of classical (descriptive) complexity theory.

Robin Riegel:

A Morse theoretical approach to the Chas-Sullivan product

Time and place

Monday, 19.5.25, 16:00-18:00, Seminarraum 404

Abstract

We will discuss how to build (and generalize) a Morse model for a fundamental operation in string topology, the Chas-Sullivan product on the free loop space of a closed manifold. This approach is based on the work of Barraud, Damian, Humilière and Oancea who introduced Morse Homology with differential graded coefficients which they show to be a particularly adapted framework to give a finite dimensional approach to study the homology of total spaces of fibrations over a closed manifold.

Mingwei Zhang:

Dirac-geodesics in surfaces

Time and place

Tuesday, 20.5.25, 16:15-17:45, Seminarraum 404

Abstract

In this talk, we consider the structure of Dirac-geodesics with curvature term in surfaces and give solutions on the 2-sphere and the hyperbolic plane, and then we give the structure of solutions in warped product spaces. Finally, we define the corresponding heat flow and prove the global existence and sub-convergence of the heat flow into any closed surfaces and space forms. This is a joint work with Prof. Q. Chen.

Prof. Dr. Sebastian Wartha:

Operations- und Zahlvorstellungen zu Beginn der Sekundarstufe

Time and place

Tuesday, 20.5.25, 18:30-20:00, Hörsaal 2

Abstract

Spätestens im Zeitalter von KI sei die Frage erlaubt, ob ein „Rechnenkönnen“ – unabhängig von Schulstufen – ein erstrebenswertes Ziel des Mathematikunterrichts sein sollte. Die Suggestivfrage verschärft sich, wenn das Rechnen losgelöst von Grundvorstellungen zu Zahlen und den sie verknüpfenden Operationen geschieht, was aber zahlreiche Studien und Erfahrungen von Lehrkräften berichten. Im Vortrag soll ein Überblick über ausgewählte Kompetenzen von Lernenden in Bezug auf deren Vorstellungen zu Zahlen (natürliche und positiv rationale) und zur Multiplikation mit Ihnen vorgestellt werden. Das ist die Grundlage für konstruktive Vorschläge für unterrichtliche Settings, in denen der Aufbau von Grundvorstellungen sowie die Kommunikation und Argumentation mit und über Zahlen und Operationen im Mittelpunkt steht.

Cagin Ararat:

Systemic Values-at-Risk: Computation and Convergence

Time and place

Monday, 26.5.25, 14:00-15:30, Seminarraum 232

Abstract

We investigate the convergence properties of sample-average approximations (SAA) for set-valued systemic risk measures. We assume that the systemic risk measure is defined using a general aggregation function with some continuity properties and value-at-risk applied as a monetary risk measure. Our focus is on the theoretical convergence of its SAA under Wijsman and Hausdorff topologies for closed sets. After building the general theory, we provide an in-depth study of an important special case where the aggregation function is defined based on the Eisenberg-Noe network model. In this case, we provide mixed-integer programming formulations for calculating the SAA sets via their weighted-sum and norm-minimizing scalarizations. To demonstrate the applicability of our findings, we conduct a comprehensive sensitivity analysis by generating a financial network based on the preferential attachment model and modeling the economic disruptions via a Pareto distribution.

Mingwei Zhang:

Stability of sharp spinorial Sobolev inequality on sphere

Time and place

Tuesday, 27.5.25, 16:15-17:45, Seminarraum 404

Abstract

In this talk we consider the sharp spinorial Sobolev inequality on S^n. From the variation point of view, this is a spinorial analogy of Yamabe problem. It is well known that the optimal Sobolev constant is the so-called Bär-Hijazi-Lott invariant which, as the Yamabe invariant, attains its maximum at round sphere. In this talk, we will prove on S^n that the Sobolev quotient being close to the optimal constant implies that spinor being close to an optimizer. Compared to the function case, the difficulty arises from the fact that the Dirac operator has unbounded spectrum both from above and blow. This is a joint work with Prof. Guofang Wang.

Ludger Overbeck:

Rough and path-dependent affine models and their path-valued interpretation

Time and place

Wednesday, 28.5.25, 16:00-17:30, Seminarraum 226 (HH10)

Abstract

We first extend results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel and inhomogeneous drift and diffusion coefficients and in the case of affine drift and variance we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a time homogeneous kernel of convolution type we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition the coefficients are affine, we prove that the conditional Fourier-Laplace functional is exponential-affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance. Secondly we investigate the equivalence between affine coefficients of a path dependent continuous stochastic differential equations and the affine structure of the log Fourier-transform. Applications in mathematical finance include e.g. delayed Heston model. In both cases the corresponding path processes are infinite-dimensional affine Markov processes. This is joint work (partially in progress) with Julia Ackermann (Wuppertal), Boris Günther (Gießen) und Thomas Kruse (Wuppertal).