It is believed that certain physical duality underlies various versions of Langlands duality in its geometric incarnation. By setting up a mathematical model for relevant physical theories, we suggest a program that enriches mathematical subjects such as geometric Langlands theory and symplectic duality. This talk is based on several works, main parts of which are joint with Chris Elliott and with Justin Hilburn.

Symplectic embedding problems are at the heart of the study of symplectic topology, and are closely related with Hamiltonian dynamics. In this talk we discuss how to compute the symplectic inner and outer radii of certain convex domains using the theory of integrable systems.
The talk is based on a joint work with Vinicius Ramos.

In the late 70s, Fathi showed that the group of compactly supported volume-preserving homeomorphisms of the ball is simple in dimensions greater than 2. We present our recent article which proves that the remaining group, that is area-preserving homeomorphisms of the disc, is not simple. This settles what is known as the simplicity conjecture in the affirmative. This is joint work with Dan Cristofaro-Gardiner and Vincent Humiliere.

I discuss some results and conjectures on contact aspects of non-equilibrium thermodynamics. Joint with Michael Entov, Shin-itiro Goto, Shai Lerer, and Lenya Ryzhik.

Let G be a compact Lie group and let M = G/H be a G-homogeneous space, equipped with an invariant metric. We prove that the spectral norm of any compact exact Lagrangian submanifold of the cotangent bundle T*M is bounded in terms of the diameter and dimension of G. Our proof is by sheaf theoretical methods; it recovers some results of Shelukhin and gives some other cases. This is a joint work in progress with Nicolas Vichery.