A group of analysts (see the scientific committee below), working in the London or Paris area, have decided to organize a joint seminar. This Paris-London Analysis Seminar meets four times a year. Details are as follows.
11:00-12:00 Thierry Levy (Université Pierre et Marie Curie): The Douglas-Kazakov phase transition
Douglas and Kazakov predicted about twenty years ago that the pure Euclidean Yang-Mills theory on the two-dimensional sphere with structure group U(N) exhibits a phase transition, in the limit where N tends to infinity, when the area of the sphere crosses the critical value π 2 . In probabilistic language, this can be expressed as a phase transition for the Brownian bridge on the unitary group U(N) in the large N limit, when the length of the bridge crosses the same critical value π 2 . I will describe this phase transition from two points of view, on one hand by discussing the distribution of the eigenvalues of certain random unitary matrices, and on the other hand by looking for the dominant Fourier modes of the heat kernel on the unitary group. This is joint work with Mylne Mada (Lille).
13:00-14:00 Ivan Todorov (Queen’s University Belfast): Operator systems, non-signalling correlations and quantum graph parameters
Operator systems have played a central role in Operator Algebra Theory since their introduction in the 1970s. Capturing the features of noncommutative order, they and their morphisms completely positive maps have been prominent in C ∗ -algebra Theory, Operator Space Theory and, lately, Quantum Information Theory. The talk will be centred around their use in the description of classes of quantum correlations and in the introduction and study of quantum graph parameters, including quantum chromatic numbers and projective ranks. Their connection with Tsirelson’s Problem and Connes’ Embedding Problem will be discussed, and the link between these problems and the introduced graph parameters will be highlighted.
14:30-15:30 Christophe Lacave (Université Paris-Diderot): Incompressible fluids through a porous medium
In a perforated domain, the asymptotic behavior of the fluid motion depends on the rate (inter-hole distance)/(size of the holes). We will present the standard framework and explain how to find the critical rate where ”stange terms” appear. Next, we will compare the critical rate for the Laplace, Navier-Stokes and Euler equations
17:00-18:00 Gustav Holzegel (Imperial College): Linear Stability of the Schwarzschild solution under gravitational perturbations
The well-known Schwarzschild solution is a spherically symmetric static solution of the vacuum Einstein equations describing a black hole. In my talk, I will outline a recent proof, obtained in collaboration with Dafermos and Rodnianski, of the linear stability of the Schwarzschild solution under gravitational perturbations. The proof combines insights on the behaviour of linear waves on black hole backgrounds (proven in the last ten years) with a hierarchical structure in the system of linearised Einstein equations.
Lectures will take place in the Mathematical Building (Room 203) in the morning and in the Maths Lecture theatre in the afternoon.