Particle Systems & Stochastic PDEs: analysis of models in fluctuating hydrodynamics and neuroscience

In this presentation I give an overview of my research area, and discuss my accomplishments and future directions of investigation. My work so far has investigated the interplay between PDEs and particle systems through the study of two classes of models. The first class is the so-called generalized Dean-Kawasaki equation coming from areas like fluctuating hydrodynamics, mean-field theory and stochastic geometry. The second class is a generalized model for interacting neurons encompassing many of those commonly used in computational neuroscience, in particular in the study of grid cells. For the first model I focus more on the PDE side, while for the second I concentrate on the particle side, where I have given my contributions, respectively. For both models I also present aspects of current and future investigation. Finally, I will discuss other research directions to be pursued in the coming period.