This is a joint work with Cheng JingRui.

]]>In this talk, I will try to address the following two questions:

- How machine learning will impact computational mathematics and computational science?
- How computational mathematics, particularly numerical analysis, can impact machine learning? We describe some of the most important progresses that have been made on these issues so far. Our hope is to put things into a perspective that will help to integrate machine learning with computational science.

- The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
- The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me to associate to each partition of N a unique monster potential with N roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer N, the number of monster potentials with N roots coincide with the number of integer partitions of N, which is the dimension of the level N subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

This is joint work with Riccardo Conti (Group of Mathematical Physics of Lisbon University).

]]>The purpose of the talk is to describe a framework for studying highly anisotropic singularities. In particular, for analysing the asymptotics of solutions to linear systems of wave equations on the corresponding backgrounds and deducing information concerning the geometry.

The talk will begin with an overview of existing results. This will serve as a background and motivation for the problem considered, but also as a justification for the assumptions defining the framework we develop.

Following this overview, the talk will conclude with a rough description of the results.

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This Riemann-Hilbert representation can be used to derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal, and it reveals a remarkable connection with a family of unbounded solutions to the Korteweg-de Vries (KdV) equation and with an integro-differential version of the Painlevé II equation.

]]>Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary $\Sigma$, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and $\Sigma$ is a closed surface (i.e. $\partial \Sigma = \emptyset$), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require $\partial \Sigma \neq \emptyset$ for these representations to be defined. We will try to explain what goes wrong when $\Sigma$ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.

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