Edinburgh Winter 2025 

Abstract: The Hilbert transform $H$ is a basic example of Fourier multipliers, which plays an essential role in modern harmonic analysis.  Riesz and Cotlar proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.  In this talk, we will first introduce a  noncommutative analogue for Cotlar identity.  Then we will explain a new connection between group actions on real-trees and 

Hilbert transforms on groups.    These groups include all the free products and Higman-Neumann-Neumann (HNN) extensions.  Very recently,  we have  generalised the previous results into a broader setting --groups which admit actions on right-angled buildings-- to deal with new examples of groups which can not be covered in the previous tree models. 

Based on a joint work with A. Gonz\'alez-P\'erez and J. Parcet,  and a joint work with Xiaoqi Lu.

Abstract: In the first part of the seminar we recall basic definition and classical results about scattering theory for solutions to NLS. In the second part we present a recent result showing that the scattering operator can be written as identity plus a smoother term. In order to simplify the presentation we focus on the quintic 1-d NLS. The strategy is quite flexible and can be extended in more general situations.

This is a joint work with N. Burq, H. Koch, N. Tzvetkov

Abstract: We consider a general family of multilinear, polynomial, ergodic averages associated to a family of commuting, measure-preserving transformations and establish pointwise almost everywhere convergence for these averages under the assumption that the polynomials have distinct degrees. Besides the linear case, only two previous results were known, both in the bilinear setting and both in the single transformation case where one polynomial is linear. This is joint work with D. Kosz, M. Mirek and S. Peluse.

Abstract:  We establish a deterministic analogue of Beck's local-to-global principle for Kronecker sequences. This gives rise to a novel reformulation of Littlewood's conjecture in diophantine approximation.