We consider determinantal point processes $\Lambda_\phi$ arising from generalized Fock spaces $\mathcal F_\phi$, defined by a doubling subharmonic weight $\phi$. We provide a complete characterization of when these processes are almost surely separated with respect to the euclidean metric on the plane. As a consequence, we also obtain a characterization of when these processes are almost surely interpolating for the classical Fock spaces. Furthermore, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $\Lambda_\phi$ with the Poisson process with the same first intensity. Additionally, we demonstrated that, with probability one, the point process $\Lambda_\phi$ fails to be separated with respect to the distance induced by the reproducing kernel $K_\phi$.

This is a joint work with X. Massaneda.

Kinetic theory describes gases and plasmas as large collections of interacting molecules, from a statistical point of view. Its most famous success, due to Boltzmann, is the first analytic derivation of the monotonicity in time of the entropy.

In most gases of practical interest, e.g. N₂, O₂ or CO₂, molecules are polyatomic: composed of multiple atoms. This geometrical structure induces rotational and vibrational effects, which are a challenge to include in the theory. We present a general framework for modeling polyatomic molecules in kinetic theory, based on elementary physical ideas and formulated in a measure-theoretic setting. We show that within this framework one may adopt different perspectives, each relevant for a different kind of problem (physical modeling, mathematical analysis or numerical simulation), clarifying and unifying the previous approaches in the literature.

We consider a class of Riesz transforms $R_V^a$ related to the Schrödinger operator $L = -\frac{1}{2}\Delta + V$ with $V$ being a non-negative potential. We will focus on $L^\infty$ boundedness of $R_V^a$ and give a specific integral condition on $V$ which provides the boundedness. We will also show that potentials with power or exponential growth satisfy this condition.

In this talk, I will discuss recent results regarding the intermediate nonlinear Schrödinger equation (INLS). Analytically, INLS is a one-dimensional completely integrable nonlinear Schrödinger equation with a cubic derivative nonlinearity and is $L^2$-critical. A limiting form of INLS is the continuum Calogero-Moser equation (CCM), which is also completely integrable. Interestingly, CCM keeps the Hardy space $L^2_+$ invariant, and, under this assumption, tools from complete integrability have recently resolved the well-posedness problem for CCM in $L^2_+$. I will discuss progress on the well-posedness for INLS and CCM (not relying on complete integrability), outside of the Hardy space and in low-regularity. Our approach combines a gauge transformation, bilinear Strichartz estimates and a refined decomposition for smooth solutions. This is based on joint work with A. Chapouto (CNRS, Monash) and T. Laurens (UW-Madison).