22-23

A two-layer quasi-geostrophic (2LQG) model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. We will discuss the problem of unique ergodicity and exponential convergence of transition probabilities as well as response theory for stochastic partial differential equations like the 2LQG model. 

In 1963, Lieb and Liniger formulated an exactly solvable model for interacting bosons in 1D. Thanks to its exact Bethe ansatz solution, the model and its generalizations soon became popular objects of study in mathematical physics. Later, when new techniques allowed for the creation of (quasi-)1D systems in the lab, Lieb and Liniger's model found experimental use and became even better known. 

In the meantime, the mathematical physics community had moved on and was rigorously studying interacting bosons in 2 and 3D. Without the availability of exact solutions, rigorous results were much more difficult to obtain, and a popular goal was the derivation of the ground state energy of Bose gases in various settings in 2 and 3D. Many of these efforts focused on the dilute limit, in which the density of the gas is very low.

Somehow, Bose gases in 1D were not studied in this way, probably because the Lieb-Liniger model provides an exactly solvable set-up. Nevertheless, we can use insights from the 2 and 3D approaches to prove results about the ground state energy of dilute Bose gases in 1D. In the talk, I will review the developments above and explain the new results. 

In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations. 

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain Ω, with both fast and slow boundary. For the random walks on Ω dual to SEP/SIP we establish a functional-CLT-type convergence to the Brownian motion on Ω with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions. We further show the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on Ω, and analyze their stationary non-equilibrium fluctuations.

arXiv:2112.14196, joint work with Lorenzo Portinale (IAM Bonn) and Federico Sau (Trieste)

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain Ω, with both fast and slow boundary. For the random walks on Ω dual to SEP/SIP we establish a functional-CLT-type convergence to the Brownian motion on Ω with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions. We further show the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on Ω, and analyze their stationary non-equilibrium fluctuations.

arXiv:2112.14196, joint work with Lorenzo Portinale (IAM Bonn) and Federico Sau (Trieste)

Quantum Mechanics was formulated in the years 1925/26 thanks mainly to the works of Heisenberg and Schrödinger . In the seminar, after a brief historical introduction, the fundamental ideas of these works will be discussed, keeping the technical aspects to a minimum. In particular, an attempt will be made to highlight the differences in the two approaches, which are based on different ways of conceiving the description of the physical world.