Kontsevich's characteristic classes for framed smooth homology sphere bundles were defined by Kontsevich as a higher dimensional analogue of Chern-Simons perturbation theory in 3-dimension, developed by himself. In this talk, I will present an application of Kontsevich's characteristic class to a disproof of the 4-dimensional Smale conjecture, which says that the group of self-diffeomorphisms of the 4-sphere has the same homotopy type as the orthogonal group O(5). This leads, for example, to a negative answer to Eliashberg's problem, which asks if the space of compact-support symplectic structures on R^4 is contractible. In proving our result, we give and use a formula for the characteristic numbers for some bundles which counts gradient flow-graphs in the bundles. This is an analogue of K. Fukaya's Morse homotopy for families.

I will describe an approach to studying the Kakeya maximal function in high dimensions via the Guth--Katz polynomial partitioning method. Although the approach does not currently produce better bounds than the record set by Katz--Tao, it is rather flexible, provides a lot of interesting structural information and gives rise to some interesting algebraic/geometric problems.

Motivated by some classical results of Meyer, Pichorides and Zygmund, we present a variant of Yano's extrapolation theorem for analytic Hardy spaces over the torus. Some related questions will also be discussed.

Determinantal point processes are used to model the repulsion in certain sets of points. They capture negative correlations: the more similar two points are, the less likely they are to be sampled simultaneously. Therefore, these processes tend to generate sets of diverse or distant points. Unlike other repulsive processes, these have the advantage of being entirely determined by their kernel and there are exact algorithms to sample them. During this presentation, I will present the determinantal point processes in a general discrete framework and then in the one of the images: a 2D framework, stationary and periodic. We have studied the repulsion properties of such processes, in particular by using shot noise models, properties that are interesting for synthesizing microtextures. I will also present how the determinantal processes can be applied to the sub-sampling of an image in the patch space.

Quasistatic dynamical systems (QDS), introduced by Dobbs and Stenlund around 2015, model dynamics that transform slowly over time due to external influences. They are generalizations of conventional dynamical systems and belong to the realm of deterministic non-equilibrium processes.

I will first define QDSs and then give an ergodic theorem, which is needed since the usual theorem of Birkhoff does not apply in the absence of invariant measures. After briefly explaining some applications of the ergodic theorem, I will give results on the statistical properties of a particular QDS in which the evolution of states is described by intermittent Pomeau-Manneville type maps. One of these results is a functional central limit theorem, obtained by solving a well-posed martingale problem, which (in a certain parameter range) describes statistical behavior as a stochastic diffusion process.

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This prove a conjecture of Zakharov-Dyachenko based on the formal Birkhoff integrability of the waver waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order $\epsilon^{−3}$.