One of the main challenges in studying non-self-adjoint operators is the lack of a spectral theorem. By applying Avila’s global theory, we derive new spectral and isospectral results for non-self-adjoint quasi-periodic Schrödinger operators. This presentation is based on joint work with Zhenfu Wang, Jiangong You, and Qi Zhou.

Describing the long-time dynamical behaviour of fermionic systems is a long-standing open problem. When the coupling constant $\lambda$ of the interaction is small, the dynamics of the system up to kinetic time $t \sim \lambda^{-2}$ is conjectured to be effectively governed by the Boltzmann-Nordheim kinetic equation. Considering a system of fermions on a cubic lattice at thermal equilibrium, we prove that the two-point time correlation function of the many-body quantum dynamics can be computed effectively using the collisional frequency of the Boltzmann-Nordheim collision operator.

Homogenization refers to the approximation of PDEs with rapidly oscillating coefficients with a nice (constant coefficient) PDE. Physically, we can think of a composite that is obtained by finely mixing together a "soft" and a "stiff" material. Mathematically, we can study the PDE $-div( a(x/\epsilon) grad u^\epsilon ) + u^\epsilon = f$, where the coefficient matrix is $a(y)$ is 1-periodic, takes values $c_{soft} I$ at the "soft" regions, and $c_{stiff} I$ at the "stiff" regions. We want to take \epsilon to 0. This talk focuses on the "high contrast" case $c_{soft} = \epsilon^2$ and $c_{stiff} = 1$. In other words, there is a loss of uniform ellipticity in \epsilon, and this poses fundamental mathematical challenges. To tackle this setting, I will explain the main ideas behind the operator theoretic framework developed by Cherednichenko, Ershova, and Kiselev (2020). The key object is that of a "boundary triple" in the sense of Ryzhov (2009). We obtain an effective limiting description with an $O(\epsilon^2)$ error in the operator-norm.

Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. Multi-dimensional coagulation equations have been used in recent years in order to include information about the system of particles which cannot be otherwise incorporated. Depending on the model, we can describe the evolution of the shape, chemical composition or position in space of clusters. In this talk, we focus on a model that is inhomogeneous in space and contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass-conserving solutions for a class of coagulation rates for which in the space homogeneous case instantaneous loss of mass occurs. This is based on a joint work with B. Niethammer and J J. L. Velázquez.