Seminários e cursos curtos

The Kerr de Sitter geometry models a rotating black hole in an expanding universe. I will review its stability properties in the context of the Einstein vacuum equations with positive cosmological constant, and present recent progress on the non-linear stability problem for the cosmological region.

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 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.