Vorträge in der Woche 15.07.2024 bis 21.07.2024

Dienstag, 16.07.2024: Bruhat-Tits Buildings

Anton Deitmar

Donnerstag, 18.07.2024: Singularities in mean curvature flow

Dr. Stephen Lynch (Imperial College London)

We will discuss the flow of hypersurfaces in 4-dimensional space by their mean curvature. Compared with the case of surfaces in 3-space, many interesting new phenomena arise. In particular, many new kinds of singularities are expected to arise, but have not yet been constructed rigorously.

Donnerstag, 18.07.2024: Localization of Time Reversal Wannier Bases, an introduction

Gabriele Ciccarello (Sapienza Universitá di Roma)

We will look at 2D periodic hamiltonians that possess time reversal symmetry of odd type and at their Time-Reversal-Symmetric (TRS) Bloch bundles. More specifically the seminar will be an introduction to a conjecture that links the triviality of such a bundle to the localization properties of time reversal symmetric Wannier bases.

Donnerstag, 18.07.2024: Maximal surfaces and boosts in the Schwarzschild spacetime

Albachiara Cogo (Universität Tübingen)

In Minkowski spacetime there is a one-to-one correspondence between observers, boosts and complete spacelike maximal (vanishing mean curvature) surfaces by the renowned Cheng-Yau Bernstein type theorem. We address the question whether there is a correspondent generalization of this idea in non-flat spacetimes, starting from the Schwarzschild spacetime. As a first step in this direction, we will prove the existence of a maximal surface approaching a coordinate-dependent hypersurface related to a boost of Minkowski in the asymptotically flat end and discuss some of its properties. 

Donnerstag, 18.07.2024: A new Approach to the Anyon Hamiltonian via Boundary Conditions

Paul Vögele (Tübingen)

There already exist numerous mathematical descriptions of anyons. I present a new approach by investigating the topology of the unordered configuration space and its universal cover. The Hamiltonian can then be defined on a fundamental domain of this cover with suitable boundary conditions, making it self-adjoint.

Freitag, 19.07.2024: A Wasserstein perspective of Vanilla GANs

Lea Kunkel (Karlsruhe Institute of Technology )

Generative Adversarial Networks (GANs) have attracted much attention since their introduction by Goodfellow el al. (2014), initially due to impressive results in the creation of photorealistic images. Meanwhile, the areas of application have expanded far beyond this, and GANs serve as a prototypical example of the rapidly developing experimental and theoretical research area of generative models. The statistical literature focuses mainly on Wasserstein GANs and their generalizations, which allow for good dimension reduction properties. Statistical results for vanilla GANs, the original optimization problem, are still rather limited and require assumptions such as smooth activation functions and equal dimensions of the latent space and the ambient space. To bridge this gap, we draw a connection from vanilla GANs to the Wasserstein distance. In doing so, existing results for Wasserstein GANs can be extended to vanilla GANs. In particular, we obtain an oracle inequality for vanilla GANs in Wasserstein distance. The assumptions of this oracle inequality are designed to be satisfied by commonly used network architectures, such as feedforward ReLU networks. By providing a quantitative result for the approximation of a Lipschitz function by a feedforward ReLU network with bounded Hölder norm, we conclude a convergence rate for Vanilla GANs.