Planned webinars

Europe/Lisbon
Room P3.10, Mathematics Building Instituto Superior Técnicohttps://tecnico.ulisboa.pt

Mariusz Mirek

, Rutgers University

This will be a survey talk about recent progress on norm and pointwise convergence problems for classical and multiple ergodic averages along polynomial orbits. A celebrated theorem of Szemeredi asserts that every subset of integers with nonvanishing upper Banach density contains arbitrarily long arithmetic progressions. We will discuss the significance of using ergodic theory and Fourier analysis in solving this problem. We will also explain how this problem led to the conjecture of Furstenberg-Bergelson-Leibman, which is a major open problem in pointwise ergodic theory. Relations with number theory and additive combinatorics will be also discussed.

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa

Guy Bouchitté

Guy Bouchitté, Université de Toulon

The classical Kantorovich-Rubinstein duality theorem established a significant connection between Monge optimal transport and the maximization of a linear form on 1-Lipschitz functions. This result has been widely used in various research areas, particularly to demonstrate a bridge between Monge transport theory and some class of optimal design problems in mechanics.

The aim of this talk is to present a similar theory when the linear form is maximized over all real $C^{1,1}$ functions with a Hessian matrix spectral norm not exceeding one. It turns out that this new maximization problem can be viewed as the dual of a specific optimal transport problem. The task is to find a minimal three-point plan with given first two marginals, where the third is assigned to be larger than both in the sense of convex order. The existence of optimal plans allows to express solutions of the underlying Beckman problem as a combination of rank-one tensor measures supported by a graph. In the context of two-dimensional mechanics, this graph encodes the optimal location of a grillage to support a given bending load.