Vorträge in der Woche 25.11.2024 bis 01.12.2024

Dienstag, 26.11.2024: Multiple Rekurrenz aus funktionalanalytischer Sicht

Prof. Tanja Eisner (Leipzig)

Dienstag, 26.11.2024: Representations of p-adic groups.

Paul Vögele

Mittwoch, 27.11.2024: Cones of divisors for moduli spaces of polarized hyper-Kahler manifolds

Pietro Beri (Nancy)

We describe how to compute cones of Noether-Lefschetz divisors on orthogonal modular varieties, with a particular view towards moduli spaces of polarized K3 surfaces and hyperkähler manifolds. We then describe some geometric applications of these cone computations for these moduli spaces. This is a joint work with I. Barros, L. Flapan and B. Williams.

Donnerstag, 28.11.2024: Approach to aspects to Hamilton's conjecture via Potential Theory

Ariadna Léon Quiros (Universität Tübingen)

Ricci-pinched three-manifolds are manifolds with a lower bound on the Ricci curvature in terms of the scalar curvature and a positve constant, which is uniform for every point and direction of the manifold. This property provides some sort of isotropy to the manifold. From here, R. Hamilton formulated the Conjecture which states that when non-compact, connected and complete three-dimensional manifolds are Ricci-pinched, then they are flat. The first proofs of this conjecture were based on Ricci flow and dealt with several technicalities. Consequently, Huisken and Körber's proved the conjecture by using Inverse Mean Curvature Flow. In this talk, I will talk about a new approach to prove the conjecture which is based on potential theory. In addition, to implement it we have to apply some theorems used for the proof of the positive mass theorem by Bray, Kazaras, Khuri and Stern. At the end, I will also mention a method of proving Hamilton’s conjecture for three-manifolds with superquadratic volume growth via nonlinear potential theory (this is joint work with L. Benatti, F. Oronzio and A. Pluda).

Donnerstag, 28.11.2024: Macroscopic Thermalization for Highly Degenerate Hamiltonians

Cornelia Vogel (Tübingen)

An isolated macroscopic quantum system thermalizes if its initial state eventually reaches a suitable thermal equilibrium subspace and stays there for most of the time. For non-degenerate Hamiltonians, a sufficient condition for the thermalization of every initial state is an appropriate version of the eigenstate thermalization hypothesis (ETH). Shiraishi and Tasaki recently proved the ETH for a perturbation of the Hamiltonian of a large number of free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of the Hamiltonian. We point out that also for degenerate Hamlitonians, all initial states thermalize if the ETH holds for every eigenbasis, and we show that this is the case for free fermions in 1d. Additionally, we develop another strategy of proving thermalization by adding small generic perturbations to Hamiltonians for which it can be shown that one eigenbasis (but not necessarily all) fufills the ETH. This strategy applies to arbitrarily small generic perturbations of the Hamiltonian of free fermions in arbitrary spatial dimensions. This is joint work with Barbara Roos, Stefan Teufel and Roderich Tumulka.

Donnerstag, 28.11.2024: Semi-Classical limit in Quantum mechanics: Dynamics and Beyond

Dr. Shahnaz Farhat (Constructor University Bremen)

The aim of this talk is to explore the semi-classical limit in two distinct quantum mechanical models, both connected through the concept of Wigner measures. In the first part, I will discuss the dynamics of relativistic and non-relativistic charged particles interacting with a scalar meson field. Our main contribution is the derivation of the classical dynamics of a particle-field system as an effective equation of the quantum microscopic Nelson model, in the classical limit where the value of the Planck constant approaches zero ($\hbar \rightarrow 0$). We use a Wigner measure approach to study such transition. In the second part, I will focus on the many-body quantum Gibbs state of the Bose-Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of theorder of the average particle number. For this model, we prove an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The second part of the talk is a joint work with Professors Zied Ammari and Sören Petrat.

Freitag, 29.11.2024: The Eilenberg-MacLane Functor K(1,_) on the Category of Groups

Anastasios Papadopoulos