Dates: May 3, 2023, and June 21, 2023
Venue: Lipschitzsaal, Mathezentrum, Endenicher Allee 60, 53115 Bonn
Wednesday, May 3
Wednesday, June 21
Abstracts
François Golse (Paris): Quantum Klimontovich Solutions and the Mean-Field Limit in Quantum Mechanics
Klimontovich observed that, if a system of interacting point particles satisfies the system of differential equations corresponding to Newton’s second law of motion written for each particle, the phase space empirical measure of this particle system is an exact solution in the sense of distributions of the Vlasov equation with the self-consistent, mean-field interaction potential. The purpose of this talk is (1) to define a quantum analogue of the notion of Klimontovich solution, and (2) to discuss applications of this notion to the mean-field limit in quantum mechanics. (Based on joint works with T. Paul and I. Ben Porath).
Matthias Aschenbrenner (Universität Wien): A transfer principle in asymptotic analysis
Hardy fields form a natural domain for a “tame” part of asymptotic analysis. They may be viewed as one-dimensional relatives of o-minimal structures and have applications to dynamical systems and ergodic theory. In this talk I will explain a recent theorem which permits the transfer of statements concerning algebraic differential equations between Hardy fields and related structures, akin to the “Tarski Principle” at the basis of semi-algebraic geometry, and sketch some applications, including to some classical linear differential equations. (Joint work with L. van den Dries and J. van der Hoeven.)
Magnus Goffeng (University of Lund): Counting negative eigenvalues with index theory
In this talk we will consider the Pauli Hamiltonian, a Schrödinger operator describing a spin particle moving in a magnetic field in two dimensions. The problem we will discuss is that of describing how many negative eigenvalues there are for its Neumann realization on a compact domain. We provide a sharp lower bound for the number of negative eigenvalues by means of a connection to Atiyah-Patodi-Singer’s index theorem for Dirac operators arising from an elementary integration by parts. We conclude a new asymptotic formula for the counting function of the semiclassical Landau-Neumann Hamiltonian. Joint work with Søren Fournais, Rupert Frank, Ayman Kachmar and Mikael Persson-Sundqvist.
Afonso S. Bandeira (ETH Zurich): Matrix Concentration and Free Probability
Matrix Concentration inequalities such as Matrix Bernstein inequality have played an important role in many areas of pure and applied mathematics. These inequalities are intimately related to the celebrated noncommutative Khintchine inequality of Lust-Piquard and Pisier. In the middle of the 2010's, Tropp improved the dimensional dependence of this inequality in certain settings by leveraging cancellations due to non-commutativity of the underlying random matrices, giving rise to the question of whether such dependency could be removed.
In this talk we leverage ideas from Free Probability to fully remove the dimensional dependence in a range of instances, yielding optimal bounds in many settings of interest. As a byproduct we develop matrix concentration inequalities that capture non-commutativity (or, to be more precise, ``freeness''), improving over Matrix Bernstein in a range of instances. No background knowledge of Free Probability will be assumed in the talk.
Joint work with March Boedihardjo and Ramon van Handel.