1- Metastability in statistical physics: exit event, Eyring-Kramers law, quasi-stationary distributions 

Illustration of metastability in a very (very very) simple case:

2- Spectral analysis, potential theory, and large deviations theory

3- Properties of processes arising from statistical physics (e.g. ergodicity and quasi-ergodicity)

4- Asymptotic analysis of singular perturbation problems

5- Mathematical analysis of neural networks

6- Narrow escape problem (a particle confined in a bounded domain with reflecting walls, except for small openings that allow escape)

Illustration of the Narrow escape problem in dimension 2 with two escape windows: