Estimation of tail probabilities in systems that involve uncertain parameters or random forcing is important when these unlikely events have severe consequences. Examples of such events are hurricanes, energy grid blackouts, or failure of engineered systems. After explaining the challenges of estimating rare event probabilities, I will make a connection between extreme event probability estimation and constrained optimization that is established by large deviation theory. The approach leads to practical methods to estimate small probabilities, and a novel class of challenging, large-scale PDE-constrained optimization problems. I will show examples governed by the shallow water equation where one is interested in estimated the probability of large tsunamis on shore, and the randomly forced Navier Stokes equations, where one is interested in the probability of large point strains.

In this talk, I will give a brief overview of the scattering transform for the mKdV equation from the perspective of the Riemann-Hilbert method, and introduce the robust inverse scattering transform. I will describe how we use this perspective to produce rational solutions to the mKdV equation of arbitrary order, already present at low order in the literature. By examining a limiting Riemann Hilbert problem, we also produce rational solutions of infinite order. We realize these solutions as limits of the finite order solutions, and describe some of their asymptotics. This is ongoing and joint work with Deniz Bilman, Elliot Blackstone, and Peter Miller. 

The Boson-Fermion correspondence has found connection to symmetric functions through its application for deriving soliton solutions of the KP equations. In this framework, the space of Young diagrams is the Fermionic Fock space, while the ring of symmetric functions is the Bosonic Fock space. Then the (second part of) BF correspondence asserts that the map sending a partition to its Schur function forms an isomorphism as H-modules, with H being the Heisenberg algebra. In this talk, we give a generalization of this correspondence into the context of Schubert calculus, wherein the space of infinite permutations plays the role of the fermionic space, and the ring of back-stable symmetric functions represents the bosonic space.