In this seminar I will describe the derivation of certain laws that rule the space-time evolution of the conserved quantities of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a model for which we can obtain a collection of (fractional) reaction-diffusion equations given in terms of the regional fractional Laplacian with different types of boundary conditions.

Spaces equipped with a decomposition into even dimensional cells arise naturally in many areas of mathematics. This talk will survey some of these spaces, some conjectures about analogues of them in equivariant and motivic homotopy theory, and some applications.

Kinetic theory and the associated Boltzmann transport equations are one of the few tools which allow to bridge the orders of magnitude from the scale of microscopic dynamical models to their macroscopic transport properties. Although it is often restricted to models whose dynamics are weak perturbations of constant velocity motion, many physical systems offer examples of such behaviour. One case which can be controlled mathematically, is a rarefied gas of hard spheres with elastic collisions, in the Boltzmann-Grad scaling limit. Other less obvious examples are given by weakly perturbed wave motions, including weakly interacting quantum particles and weakly nonlinear discrete wave equations.

In this talk, we discuss mathematical evidence which support the validity of these kinetic theory approximations, as well as their known limitations and modifications. One key step is starting the microscopic dynamics with initial data which is random and sufficiently "chaotic". The importance of this assumption is highlighted in the dynamical formation of a Bose-Einstein condensate: the macroscopic correlations appearing together with the condensate are not compatible with the standard assumptions, and indeed modified kinetic equations are expected to be needed after the formation. This picture is corroborated by the work of Escobedo and Velázquez who prove that the related bosonic kinetic theory has solutions which blow up in a finite (kinetic) time.