## Plateau’s problem as a capillarity problem

## On minimizers and critical points for anisotropic isoperimetric problems

Anisotropic surface energies are a natural generalization of the perimeter functional that arise in models in crystallography and in scaling limits for certain probabilistic models on lattices. This talk focuses on two results concerning isoperimetric problems with anisotropic surface energies. In the first part of the talk, we will discuss a weak characterization of critical points in the anisotropic isoperimetric problem (joint work with Delgadino, Maggi, and Mihaila).

## Lorentzian polynomials

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The class of Lorentzian polynomials contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. I will give several combinatorial applications. No specific background will be needed to enjoy the talk. Joint work with Petter Brändén (https://arxiv.org/abs/1902.03719).

## Elliptic measures and the geometry of domains

Given a bounded domain $\Omega$, the harmonic measure $\omega$ is a probability measure on $\partial \Omega$ and it characterizes where a Brownian traveller moving in $\Omega$ is likely to exit the domain from. The elliptic measure is a non-homogenous variant of harmonic measure.

## An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by

Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called *refined* $\mathscr{L}$*invariant*, which happens to be a Hecke operator. We apply some results of the theory of Mazur's

Eisenstein ideal to study in which power of the Eisenstein ideal $\mathscr{L}$ belongs. One corollary of our