To start, I will discuss Dirichlet's proof of infinitude of primes in an arithmetic progression.  This leads up to the study of special values of L-functions and their arithmetic properties.  If time permits, I will try to explain some conjectures and philosophy in this direction.

Take a finitely-generated group of (analytic) circle diffeomorphisms. Since the times of Poincaré we know that any such action admits either a finite orbit, or a Cantor minimal set, or the action is minimal on all the circle. But what else can be said on such a group?

In this direction, there are well-known questions due to Sullivan, Ghys and Hector: assuming that such an action is minimal, is it necessarily Lebesgue-ergodic? If there is a Cantor minimal set, is it necessarily of a zero Lebesgue measure?

Our results provide a positive answer to the latter question, in some cases allow to resolve the former one and, more generally speaking, give some kind of understanding how a general characterization of an action can look like. This is a joint project with B. Deroin, D. Filimonov, and A. Navas.

I will discuss natural energy functionals related to the
existence of holomorphic structures on vector bundles and show how
inauspicious Hodge data implies blow up of minimizing sequences.
Grassmann embeddings and an analytic perspective on stability in the
sense of Gieseker and Mumford plays an important role.

We introduce a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space. Our study is motivated physical systems which consist of infinitely many particles in motion (This is a joint work with Andzrej Swiech).