Cardiff University

In this talk, I will review an inverse scattering construction of interacting integrable quantum field theories on two-dimensional Minkowski space and its ramifications. The construction starts from a given two-body S-matrix instead of a classical Lagrangean, and defines corresponding quantum field theories in a non-perturbative manner in two steps: First certain semi-local fields are constructed explicitly, and then the analysis of the local observable content is carried out with operator-algebraic methods (Tomita-Takesaki modular theory, split subfactor inclusions). I will explain how this construction solves the inverse scattering problem for a large family of interactions, and also discuss perspectives on extensions of this program to higher dimensions and/or non-integrable theories.

Abstract: In the talk we first introduce the level set equation for the evolution by mean curvature flow, explaining the main difference between the standard Euclidean case and the horizontal evolution.

Then we will introduce a stochastic representation formula for the viscosity solution of the level set equation related to the value function of suitable associated stochastic controlled ODEs which are motivated by a concept of intrinsic Brownian motion in Carnot-Caratheodory spaces.

I will describe a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic

two-material composites with high contrast in a finite-strain setting. I will start by introducing the nonlinear

description of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions. I shall then focus

on the loading regimes when the applied load is small or of order one in terms of the period of the composite structure.

I will show that this corresponds to the situation when the displacements on the stiff component are situated in the vicinity

of a rigid-body motion. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component

by its linearised version. As a main result, I derive (rigorously in the spirit of $\Gamma$-convergence) a limit functional

that allows to establish a precise two-scale expansion for minimising sequences. This is joint work with M. Cherdantsev and

S. Neukamm.

In the first part of the talk I will introduce a notion of convexity ($\mathcal{X}$-convexity) which applies to any given family of vector fields: the main model which we have in mind is the case of vector fields satisfying the H\"ormander condition.

Then I will give a PDE-characterization for $\mathcal{X}$-convex functions using a viscosity inequality for the intrinsic Hessian and I will derive bounds for the intrinsic gradient and intrinsic local Lipschitz-continuity for this class of functions.\\

In the second part of the talk I will introduce a notion of subdifferential for any given family of vector fields (namely $\mathcal{X}$-subdifferential) and show that a non empty $\mathcal{X}$-subdifferential at any point characterizes the class of $\mathcal{X}$-convex functions.

As application I will prove a Jensen-type inequality for $\mathcal{X}$-convex functions in the case of Carnot-type vector fields. {\em (Joint work with Martino Bardi)}.

Using the spectral multiplicities of the standard torus, we
endow the Laplace eigenspaces with Gaussian probability measures.
This induces a notion of random Gaussian eigenfunctions
on the torus ("arithmetic random waves''.)  We study the
distribution of the nodal length of random Laplace eigenfunctions for high
eigenvalues,and our primary result is that the asymptotics for the variance is
non-universal, and is intimately related to the arithmetic of
lattice points lying on a circle with radius corresponding to the
energy. This work is joint with Manjunath Krishnapur and Par Kurlberg

Dirichlet to Neumann maps and their generalizations are exceptionally useful tools in the study of eigenvalue problems for ODEs and PDEs. They also have real physical significance through their occurrence in electrical impedance tomography, with applications to medical imagine, landmine detection and non-destructive testing. This talk will review some of the basic properties of Dirichlet to Neumann maps, some new abstract results which make it easier to use them for a wide variety of models, and some analytical/numerical results which depend on them, including detection and elimination of spectral pollution.