Vorträge in der Woche 28.07.2025 bis 03.08.2025
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Dienstag, 29.07.2025: Convergence of normalized Betti numbers: a brief history and recent developments
Giacomo Gavelli
Abstract: Given a sequence of Riemannian manifolds of finite volume, one can study the asymptotic behavior of the normalized k-th Betti numbers, that is Betti numbers divided by the volume. If the sequence of manifolds converges to a common universal cover in some geometric sense, one expect the answer to be related to the geometry of such cover. In particular, following the seminal "Lück's approximation Theorem", one expects the sequence to converge to the L2-Betti numbers of X. In this talk we give a brief history of results concerning convergence of normalized Betti numbers to the L2-Betti numbers of the universal cover and discuss the most recent developments.
| Uhrzeit: | 14:15 |
| Ort: | C9A03 |
| Gruppe: | Oberseminar Analysis und Zahlentheorie |
| Einladender: | Deitmar |
Donnerstag, 31.07.2025: Boosted maximal hypersurfaces in Schwarzschild spacetime
Albachiara Cogo (Universität Tübingen)
In Minkowski spacetime, there is a one-to-one correspondence between inertial observers, Lorentz boosts and entire spacelike maximal hypersurfaces –namely, those with vanishing mean curvature– established by the renowned Cheng–Yau Bernstein-type theorem. The question of whether a similar correspondence exists in non-flat spacetimes was raised by Bartnik in 1988. In this talk, we present one of the main results from my PhD thesis, addressing a first step in this direction. We demonstrate the existence and discuss properties of an entire maximal hypersurface approaching a coordinate-dependent boost in the asymptotically flat ends of the maximally extended Schwarzschild spacetime. This analytically consists in finding complete, noncompact solutions with specific prescribed asymptotics to the maximal surface equation, a geometric quasilinear elliptic PDE. While such an existence problem is straightforward in Minkowski spacetime, it presents significant challenges in the Schwarzschild setting.
| Uhrzeit: | 14:00 |
| Ort: | N14 C-Bau (Morgenstelle) |
| Gruppe: | Promotionsvortrag im Fachbereich Mathematik |
| Einladender: | MNF Der Dekan |
Freitag, 01.08.2025: Mean curvature flows of higher codimension
Dr. Stephen Lynch (King’s College)
Many fascinating phenomena occur when a submanifold of higher codimension is evolved by its mean curvature vector. In this more general setting much of the structure of hypersurface flows is absent e.g. embeddedness and mean-convexity fail to be preserved. However Andrews and Baker discovered a family of quadratic curvature pinching conditions which are preserved by the flow. I will describe recent developments concerning the singularities of flows with this kind of pinching, from joint works with Huy Nguyen.
| Uhrzeit: | 14:00 |
| Ort: | N14 C-Bau (Morgenstelle) |
| Gruppe: | Geometric Analysis Miniconference |
| Einladender: | Cederbaum, Huisken |
Freitag, 01.08.2025: Yamabe metrics of Sobolev regularity
Dr. Rodrigo Avalos (Universität Tübingen)
In this talk, we examine the Yamabe problem for rough Riemannian metrics with limited Sobolev regularity. This analysis is motivated by the growing interest in low-regularity aspects of scalar curvature, including recent developments in low-regularity positive mass-type theorems and the study of rough initial data for the Einstein equations. In this rough setting, in particular for Yamabe positive metrics, the Yamabe problem requires developing new elliptic theory for the conformal Laplacian, including a fine blow-up analysis of its Green function. The aim of this talk is to motivate, contextualize, and present these results. Time permitting, we will also discuss applications to a broader low-regularity program for conformally covariant geometric equations.
| Uhrzeit: | 15:30 |
| Ort: | N14 C-Bau (Morgenstelle) |
| Gruppe: | Geometric Analysis Miniconference |
| Einladender: | Cederbaum, Huisken |