Gauss Mini Workshop

Enno Lenzmann (Universität Basel): Continuum Calogero-Moser Systems: Recent Results and Open Questions
This is an expository talk about a newly discovered class of completely integrable Hamiltonian PDEs, which arise as (formal) continuum limits of discrete classical Calogero-Moser systems. A particularly intriguing feature of these "continuum CS-systems" is their delicate interplay between scaling-criticality and their complete integrable nature. I will discuss recent results about global well-posedness and soliton resolution for the half-wave maps equation, which is a continuum spin CS-system with intriguing features. Parts of this talk is based on joint work with P. Gérard (Paris-Saclay).

Li Chen (Universität Mannheim): Mean-Field Control for Diffusion Aggregation system with Coulomb Interaction
The mean-field control problem for a multi-dimensional diffusion-aggregation system with Coulomb interaction (the so called parabolic elliptic Keller-Segel system) is considered. The existence of optimal control is proved through the Gamma-convergence of the control problem of a regularized particle control problem. There are three building blocks in the whole argument. Firstly, for the optimal control problem on the particle level, instead of using classical method for stochastic system, we study directly the control problem of high-dimensional parabolic equation, i.e. the corresponding Liouville equation of the particle system. Secondly, the strong propagation of chaos result for moderate interacting system is obtained by combining the convergence in probability and relative entropy method. Due to this strong mean field limit result, we avoid the compact support requirement for control functions, which has been often used in the literature. Thirdly, because of strong aggregation effect, additional difficulties arise from control function in the well-posedness theory, so that the known method for multi-dimensional Keller-Segel equation cannot be directly applied. Instead, we use a combination of local existence result and bootstrap argument to obtain the global solution in the sub-critical regime.

Benjamin Hinrichs (Universitäat Paderborn): Ground States of Spin Boson Models and Long Range Order in 1D Ising Models
In this talk we discuss the inherent connection of two distinct mathematical questions from separate physical backgrounds. On the one hand, the spin boson model is a model of a two-state quantum system linearly coupled to a bosonic quantum field and the existence of ground states therein is connected to the infrared catastrophe from a physical point of view. On the other hand, the Ising model is a classical model of statistical mechanics and it is frequently studied to understand the physical nature of phase transitions in magnetism. These seemingly very different phenomena are connected by a path integral formulation and we show how the Ising phase transition for long range interactions in one-dimension can be observed in the spin boson model. This talk is based on joint work with V. Betz, M. N. Kraft and S. Polzer.

Arnaud Triay (LMU München): The free energy of the dilute Bose gas at low temperature
We are interested in the spectral properties of the dilute Bose gas in the thermodynamic limit. More precisely, we justify the formula of the excitation spectrum computed by Lee, Huang and Yang in their celebrated article of 1957 by rigorously justifying the low density expansion of the free energy. This generalizes the results of Yau-Yin (2009) (and also Basti-Cenatiempo-Schlein 2021) for the upper bound, and Fournais-Solovej (2020-2023) for the lower bound of the ground state energy. This is based on joint works with F. Haberberger, C. Hainzl, P.T. Nam, B. Schlein and R. Seiringer.

Tobias König (Goethe-Universität Frankfurt): Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality
For the classical (i.e., first-order) Caffarelli-Kohn-Nirenberg (CKN) inequality, the non-degeneracy of minimizers and the symmetry-breaking phenomenon have been completely understood due to important results by Felli-Schneider (2003) and Dolbeault-Esteban-Loss (2016), respectively. In this talk, I will present some new results about a fractional-order variant of the CKN inequality, for which a full solution of the corresponding questions remains wide open. Precisely, we show non-degeneracy of minimizers for a certain parameter range and, as a consequence, obtain a quantitative stability inequality. Moreover, we exhibit a new region for which every minimizer must be radially symmetric. This is joint work with Nicola De Nitti (Lausanne) and Federico Glaudo (Princeton).

Jean-Claude Cuenin (Loughborough University): Spectral cluster bounds for orthonormal functions on compact manifolds with boundary
I will talk about L q bounds for spectral clusters (linear combinations of eigenfunctions) of the Laplace-Beltrami operator on a 2-dimensional compact manifold with boundary. Smith and Sogge (2007) proved optimal bounds for the case of a single cluster. In this talk, we will consider N orthonormal clusters and seek bounds on the L q norm of their square function (or density). This program was initiated by Frank and Sabin (2017), who extended the classical L q bounds of Sogge (1988) in the boundaryless case. The main challenge is to obtain an optimal dependence on N (the number of functions involved). The talk is based on joint work with Xiaoyan Su.