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WS 2024/25

30.10.2024 um 17:15 Uhr in Raum 69/117

Prof. Dr. Andrew Baker (University of Glasgow)

An Algebraic Perspective on the Steenrod Algebra

06.11.2024 um 17:15 Uhr in Raum 69/117

Prof. Dr. An Chen (Universität Ulm)

Optimal Payoffs under Smooth Ambiguity

We study optimal payoff choice for an investor in a one-period model under smooth ambiguity preferences, also called KMM preferences as proposed by Klibanoff et al. (2005). In contrast to the existing literature on optimal asset allocation for a KMM investor in a one-period model, we also allow payoffs that are non-linear in the market asset. Our contribution is fourfold. First, we characterize and derive the optimal payoff under KMM preferences. Second, we demonstrate that a KMM investor solves an equivalent problem to an investor under classical subjective expected utility (CSEU) with adjusted second-order probabilities. Third, we show that a KMM investor with exponential ambiguity attitude implicitly maximizes CSEU utility under the ‘worst-case’ second-order probabilities determined by his ambiguity aversion. Fourth, we reveal that optimal payoffs under ambiguity are not necessarily monotonically increasing in the market asset, which we illustrate using a log-normal market asset under drift and volatility uncertainty. (Joint work with Steven Vanduffel and Morten Wilke)

13.11.2024 um 17:15 Uhr in Raum 69/117

Prof. Mireille Boutin (TU Eindhoven)

Discrete Object Recognition and Symmetry Detection

How do you determine if two objects have the same shape? How do you reconstruct a room from echoes? How do you find HAZMAT signs in a picture?  How do you reconstruct 3D objects from a movie? These questions boil down to the problem of characterizing the orbits of the action of a Lie group on a manifold.  In this talk, I will discuss how to use invariants to solve such problems. In particular, I will discuss how to recognize configurations of points up to a rigid motion and relabeling using the "bag of distances,"  the related problem of reconstructing the shape of a room consisting of planar walls from the echoes heard by four microphones held in a rigid configuration on a drone, and how to detect symmetries in an object on an image using a pyramid of moment invariants.

27.11.2024 um 17:15 Uhr in Raum 69/117

Dr. Aida Maraj (MPI-CBG Dresden)

Understanding Multivariate Gaussian Models via Toric Geometry

Algebraic geometry has recently provided a new approach to advancing problems in multivariate Gaussian models. This is achieved by identifying Gaussian distributions with symmetric matrices and analyzing the polynomials that vanish on these matrices, known as ideals. The talk will focus on Brownian motion tree (BMT) models, a type of Gaussian model used in phylogenetics. BMT models have a hidden toric geometry, which we use to provide formulas on the maximum likelihood degree and its dual. Finally, the need to classify statistical models with toric geometry motivates us to introduce the symmetry Lie group of an ideal. This group can detect when an ideal is toric under a linear transformation. No prior knowledge of toric ideals or BMT models is required.

04.12.2024 um 17:15 Uhr in Raum 69/117

Prof. Dr. Alexander Drewitz (Universtität Köln)

A Journey Through Percolation - From Independence to Long-Range Correlations

Percolation models have been playing a fundamental role in statistical physics for several decades by now. They had initially been investigated as a model for the gelation of polymers during the 1940s by chemistry Nobel laureate Flory and Stockmayer. From a mathematical point of view, the birth of percolation theory was the introduction of Bernoulli percolation by Broadbent and Hammersley in 1957, motivated by research on gas masks for coal miners. One of the key features of this model is the inherent stochastic independence which simplifies its investigation, and which has lead to deep mathematical results. During recent years, there has been a growing interest in investigating percolation models with long-range correlations, aiming to capture a more realistic and complex scenario.  We will survey parts of the development of percolation theory, and then discuss some recent progress for the Gaussian free field with a particular focus on the understanding of the critical parameters in the associated percolation models. 

11.12.2024 um 17:15 Uhr in Raum 69/117

Prof. Dr. Michael Walter (Ruhr-Universität Bochum)

(Hidden) Symmetries of Computational Problems

 Many computational problems have underlying (and sometimes hidden) symmetries. Revealing these symmetries can be essential for finding faster algorithms and obtaining structural insight. I will give a gentle introduction to these connections, survey some applications (from combinatorics and statistics all the way to complexity and quantum information), and sketch how optimization in curved spaces (which arise naturally from non-commutative symmetries) has recently led to significant progress.

18.12.2024 um 17:15 Uhr in Raum 69/117

Prof. Holger Brenner (Universität Osnabrück)

Module Schemes in Invariant Theory

Let G be a finite group acting linearly on the polynomial ring with invariant ring R. We assign, to a linear representation of G, a corresponding quotient scheme over Spec R, and we show how to reconstruct the action from the quotient scheme. This works in particular in the case of a reflection group, where Spec R itself is an affine space, in contrast to the Auslander correspondence, where one has to assume that the basic action is small, i.e. contains no pseudo reflection. These quotient schemes exhibit rich geometric features which mirror properties of the representation. In order to understand the image of this construction, we encounter module schemes (a forgotten notion of Grothendieck), module schemes up to modification and fiberflat bundles.

08.01.2025 um 17:15 Uhr in Raum 69/117

Dr. Valentin Katter (Universität Bielefeld)

Large Language Models (LLMs) im Mathematikunterricht: Potenziale und Herausforderungen

Seit der Veröffentlichung von ChatGPT im November 2022 haben sich die mathematischen Leistungen von LLMs erheblich verbessert. Diese Entwicklung eröffnet vielfältige Möglichkeiten für den Mathematikunterricht. In diesem Vortrag werden die mathematischen Fähigkeiten von ChatGPT auf ihre Relevanz für die schulische Praxis hin analysiert. Darauf aufbauend werden verschiedene Einsatzmöglichkeiten von LLMs in der Unterrichtsplanung beleuchtet – von der Entwicklung von Lehrmaterialien über die Erstellung von Arbeitsblättern bis hin zur Ausarbeitung detaillierter Unterrichtsentwürfe. Ein besonderes Augenmerk liegt auf der Frage, welche neuartigen Aufgabenformate durch den Einsatz von ChatGPT im Unterricht eingesetzt werden können. Hierbei wird insbesondere die Fähigkeit des Modells hervorgehoben, Geometriesoftware durch natürliche Sprache zu bedienen. Anhand einer Fallstudie mit Lehramtsstudierenden der Universität Bielefeld wird aufgezeigt, wie diese mithilfe von ChatGPT und Wolfram Alpha den Verlauf von Funktionsgraphen durch verbale Beschreibungen rekonstruieren.

15.01.2025 um 17:15 Uhr in Raum 69/125

Prof. Dr. Detlef Müller (Christian-Albrechts-Universität zu Kiel)

On Lp-Bounds for Wave Equations in Riemannian and Sub- Riemannian Settings

Around 1980, Miyachi and Peral independently proved sharp Lp-bounds for solutions to the classical wave equation (∂2 t − ∆)u = 0 (1) on Rd. Later, in 1991, Seeger, Sogge and Stein were able to extend the corresponding estimates to wide classes of Fourier integral operators (short: FIOs) by means of a new method. Their results immediately imply short-time analogues of the estimates by Miyachi and Peral for the wave equation (1) on any Riemannian manifold M (with ∆ then denoting the Laplace-Beltrami operator on M ), as it is known that the corresponding wave propagators can be represented by means of FIOs, up to smoothing errors. For many reasons, one is also interested in extending these results to settings of sub-Riemannian manifolds. The foundations of sub-Riemannian geometry had been laid around 1939 through eminal works by Chow and Rashevskii, and it remains a highly active area to date. Also in such geometries (which show important “non-commutativity” features) there exists an analogue of the Laplace-Beltrami operator, which, however, typically is no longer elliptic. For a long time, this fact, among others, seemed to preclude any possible approach via FIOs. In my talk I shall first briefly discuss the key idea of the approach by Seeger, Sogge and Stein in the classical setting of Rd, before concentrating on a recent joint work with A. Martini, in which we have eventually been able to indeed devise an FIO based approach to Miyachi-Peral type estimates for the wave equation on 2-step Carnot groups (such nilpotent Lie groups provide important models of sub-Riemannian manifolds).

22.01.2025 um 17:15 Uhr in Raum 69/125

Dr. Michael Quellmalz (TU Berlin)

High-Dimensional Radial Kernels via Slicing

05.02.2025 um 16:15 Uhr in Raum 69/125

Prof. Dr. Anne Wald (Universität Göttingen)

Inverse Problems Related to Biophysical Experiments

Many processes in cells are driven by the interaction of multiple proteins, for example cell contraction, division or migration. The shape of a cell and its dynamics is largely determined by the cytoskeleton, a network of various protein filaments. Myosin motor proteins are able to bind to and move along filaments by converting energy into motion, which creates dynamic networks and yields the basis for cell deformation. A major goal in biophysics is to gain information on the mechanical properties of cells or, in general, active matter. One key quantity are the forces that are generated due to the activity. In this talk, we take a look at two inverse problems arising in biophysics, where such forces are to be determined from indirect measurements. One is traction force microscopy, a popular method to reconstruct traction forces exterted by a cell on an elastic substrate. A further experiment aims at finding the forces driving fluid flow inside an actomyosin droplet, which is a model system for active matter. We will take a look at the underlying physical models, the mathematical analysis of the inverse problems as well as numerical results for experimental data.

This is an Osnabrücker Maryam Mirzakhani Lecture