Abstract: We will discuss the combinatorial problem of counting the maximal number of tangent pairs in a collection of N spheres. First we will review what is known in the case of circles and of 3D spheres and then discuss what type of conditions on the collection should be imposed in higher dimensions in order to obtain a bound that beats the trivial O(N^2) one. We show that under such a condition (of "bounded complexity" type) and in any dimension the collection can only have far less than O(N^2) tangent pairs; this is achieved via a mix of polynomial method techniques. Time allowing, the proof will be sketched. 

Abstract: In this talk, we consider a fractional nonlinear wave equation with a general power-type nonlinearity (FNLW) on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for FNLW. We first construct the Gibbs measure associated with this equation. By introducing a suitable renormalisation, we then prove almost sure local well-posedness with respect to Gibbsian initial data. Finally, we extend solutions globally in time by applying Bourgain's invariant measure argument. This talk is based on a joint work with Luigi Forcella (University of Pisa). 

Abstract: The goal of this talk is to present a new point of view to study the Fourier restriction/extension problems. In a nutshell, it allows us to prove that the linear and multilinear L^2-based Fourier extension conjectures for the paraboloid are true (including the endpoint in the d-linear case) if one of the input functions is a full tensor. In the multilinear case, this result holds under the assumption that the domains of the inputs are weakly transversal, rather than transversal. The method extends to obtain “near-restriction” multilinear estimates (i.e., beyond the L^2-based theory) with and without transversality. We will also go over some applications of the tools used, which includes establishing an endpoint Restriction-Brascamp-Lieb conjecture for certain submanifolds by Bennett, Bez, Flock and Lee. If time allows, we will see another application of Mizohata-Takeuchi-type. This is joint work with Camil Muscalu.

Abstract: This talk is in two parts. In the first part we shall revisit the notion of visibility occuring in Guth's proof of the sharp multilinear Kakeya inequality in the light of a recent re-interpretation by Gressman, and we'll seek to convince the audience that visibility is a fundamental expression of the geometry of euclidean space. In the second part we shall discuss the Mizohata--Takeuchi conjecture and some recent results towards it obtained in joint work with Marina Iliopoulou and Hong Wang. Veracity of the Mizohata--Takeuchi conjecture further emphasises the perspective set out in the first part.