Course overview -- Sommersemester 2024

Monday,
2024-04-22
16:15-18:00
Raum 125, Ernst-Zermelo-Str. 1
Prof. Dr. Bernd Ammann (Regensburg): Minimal geodescis

A geodesic $c:\mathbb{R}\to M$ is called minimal if a lift to the universal covering globally minimizes distance. On the $2$-dimensional torus with an arbitrary Riemannian metric there are uncountably many minimal geodesic. In dimension at least $3$, there may be very few minimal geodesics. Let us assume that $M$ is closed. In 1990 Victor Bangert has shown that the number of geometrically distinct minimal geodesics is bounded below by the first Betti number $b_1$.

In joint work with Clara Löh, we improve Bangert's lower bound and we show that this number is at least $b_1^2+2b_1$.

The talk will have many ties to previous research done in Freiburg many years ago: to the research of Victor Bangert, to the Diploma thesis I have written in Freiburg in 1994 in Bangert's group, to the research of the younger Burago, when he was a long term guest in Freiburg and other aspects.

Monday,
2024-04-29
16:15-17:45
Raum 125, Ernst-Zermelo-Str. 1
Fabian Kertels: Multiplication of BPS states in heterotic torus theories

The space of states of an N = 2 superconformal field theory contains an infinite-dimensional subspace of Bogomol'nyi–Prasad–Sommerfield (BPS) states, defined as states with minimal energy given their charge. In particular, they arise in worldsheet theories of strings. In this setting, Harvey and Moore introduced a bilinear map on BPS states.

This talk presents a mathematically rigorous approach to this construction, which has been considered promising but not properly understood for almost 30 years now. The example used throughout is that of a heterotic string with all but four dimensions compactified on a torus. For this case, the BPS states were claimed to form a Borcherds–Kac–Moody algebra, as introduced in Borcherds' proof of the monstrous moonshine conjectures.

The first half of the talk, unfortunately, consists in pointing out problems with the proposed construction. The second half will provide more details on selected aspects, such as the existence of a finite-dimensional Lie algebra of massless BPS states.

Monday,
2024-05-27
16:00
Raum 125, Ernst-Zermelo-Str. 1
Jonas Schnitzer: Quantization of momentum maps and adapted formality morphisms

If a Lie group acts on a Poisson manifold by Hamiltonian symmetries there is a well-understood way to get rid of unnecessary degrees of freedom and pass to a Poisson manifold of a lower dimension. This procedure is known as Poisson-Hamiltonian reduction. There is a similar construction for invariant star products admitting a quantum momentum map, which leads to a deformation quantization of the Poisson-Hamiltonian reduction of the classical limit.

The existence of quantum momentum maps is only known in very few cases, like linear Poisson structures and symplectic manifolds. The aim of this talk is to fill this gap and show that there is a universal way to find quantized momentum maps using so-called adapted formality morphisms which exist, if one considers nice enough Lie group actions. This is a joint work with Chiara Esposito, Ryszard Nest and Boris Tsygan.

Monday,
2024-06-03
16:15-17:45
Raum 125, Ernst-Zermelo-Str. 1
Nadine Große: On local boundary conditions for Dirac-type operators

We give an overview on smooth local boundary conditions for Dirac-type operators, giving existence and non-existence results for local symmetric boundary conditions. We also discuss conditions when the boundary conditions are elliptic/regular/Shapiro-Lopatinski (i.e. in particular giving rise to self-adjoint Dirac operators with domain in $H^1$). This is joint work with Hanne van den Bosch (Universidad de Chile) and Alejandro Uribe (University of Michigan).

Monday,
2024-06-10
16:15-01:15
Raum 125, Ernst-Zermelo-Str. 1
Christian Ketterer: Ricci curvature, metric measure spaces and the Riemannian curvature-dimension condition

I explain idea of synthetic Ricci curvature bounds for metric measure spaces and one of their applications in Riemannian geometry.

Monday,
2024-06-17
16:15
Raum 125, Ernst-Zermelo-Str. 1
Jonah Reuß: A necessary condition for zero modes of the Dirac equation

We will state a necessary condition for the existence of a non-trivial solution of the Dirac equation, which is based on a Euclidean-Sobolev-type inequality. First, we will state the theorem in the flat setting and give an overview of the technical issues of the proof. Afterwards, we will consider and point out the main differences in the not necessarily flat setting. This talk is based on a work by R.Frank and M.Loss.

Monday,
2024-06-24
16:15
Raum 125, Ernst-Zermelo-Str. 1
Marius Amann: Fredholmness of the Laplace operator on singular manifolds with pure Neumann Data

Given a smooth Riemannian metric $h$ on $M$ one can consider the Laplace problem with pure Neumann data: Let $\Delta_h$ be the Laplacian given by $h$ and $n_h$ be the outer normal of $\partial M$. Exists an $u$ such that $(\Delta_hu,\partial_{n_h}u)=(F,G)$ for some given data $F$ and $G$. There is no well posedness to this problem on singular mandifolds in regular Sobolev spaces but during the talk I will introduce a scale of weighted Sobolev spaces such that it is Fredholm. In the second part of the talk I will give a formula for the Fredholm Index depending on the chosen weight function.

Monday,
2024-07-01
16:15
Raum 125, Ernst-Zermelo-Str. 1
Christoph Balcerzak : Seminarvortrag zur Masterarbeit: „Normalformen in der Poisson-Geometrie“

Gegeben ein Poisson-Hamilton-Raum mit hinreichend guter Gruppenwirkung lässt sich in einer Umgebung um die Impulsfläche der 0 mit einer Zusammenhangsform ein lokales Modell konstruieren, in welchem das System eine vereinfachte Gestalt annimmt.

Dafür werden wir den Rahmen der Poisson-Mannigfaltigkeiten verlassen und uns der Techniken einer Verallgemeinerung - den so genannten Dirac-Mannigfaltigkeiten - bedienen. Ich möchte diese im Rahmen des Vortrags diskutieren und schließlich die zentrale Idee des Beweises skizzieren.

Thursday,
2024-07-04
14:00-14:30
BBB
Misha Temkin, Dartmouth College, Hanover: tba
Thursday,
2024-07-04
15:30-16:00
BBB
Gayana Jayasinghe, University of Illinois Urbana Champaign: tba
Monday,
2024-07-08
09:00-09:30
BBB
Peter Kristel, Hausdorff Center for Mathematics, Bonn: tba
Monday,
2024-07-08
16:15
Raum 125, Ernst-Zermelo-Str. 1
Eric Trebuchon: Eine iterative Methode zur Lösung der Spin-Yamabe-Differentialgleichung

Wir werden die partielle Spin-Yamabe-Differentialgleichung untersuchen, die sich aus einem Variationsproblem ergibt, das die klassische Yamabe-Gleichung auf den Kontext von Spin-Strukturen erweitert. Unser Schwerpunkt liegt auf der Lösung der Spin-Yamabe-Gleichung auf einer Mannigfaltigkeit mit Rand unter bestimmten Randbedingungen. Der Ansatz beinhaltet die iterative Lösung einer Folge von einfacheren PDEs, um eine konvergente Folge zu konstruieren, deren Grenzwert die Spin-Yamabe PDE löst. Zu den wichtigen Annahmen gehören eine sich gut verhaltene Randbedingung und die Anforderung, dass der erste Eigenwert des Dirac-Operators ausreichend groß ist, um Konvergenz zu gewährleisten.

Tuesday,
2024-07-09
08:30-09:00
BBB
Jonathan Glöckle, Universität Regensburg: tba
Thursday,
2024-07-18
14:15-15:45
Raum 318, Ernst-Zermelo-Str. 1
Diarmuid Crowley, Melbourne: On the smooth classification of complete intersections

A complete intersection is a nonsingular complex projective variety formed as the intersection of a finite collection of hyper-surfaces.  Regarded as oriented smooth manifolds, the classification of complete intersections is a classical problem which has attracted the attention of many mathematicians.  It is organised by the “Sullivan Conjecture”.

In this talk I will review the history and recent progress on this problem, report on the verification of the Sullivan Conjecture in complex dimension 4 by myself and Nagy, and discuss the outlook for future work in higher dimensions.  This is part of joint work with Nagy.


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