Posluchárna M1, 16:00, 27. března 2024

Speaker: Pablo Candela  (Autonomous University of Madrid), http://verso.mat.uam.es/~pablo.candela/

Title: Generalizing Fourier analysis using nilspace theory

Abstract:
Many applications of Fourier analysis in combinatorics rely on the following idea: the averages of a function over certain linear configurations in an abelian group can be usefully analyzed by approximating the function with its dominant Fourier components. A  far-reaching extension of this idea was initiated in the seminal work of Gowers on Szemerédi's theorem in the 1990s, leading to the theory known as higher-order Fourier analysis. A fundamental insight in this theory is that for many types of linear patterns, while the dominating Fourier components may not be helpful anymore, one can instead analyze the function effectively by approximating it with components that are based, not on the circle group (like classical Fourier characters), but rather on certain non-commutative extensions, such as nilmanifolds. This has led notably to the discovery of fascinating structures called nilspaces, which are a common generalization of abelian groups and nilmanifolds, and which have yielded further progress in higher-order Fourier analysis. I will give an introduction to this theory and discuss some recent results in this approach involving nilspaces, based on joint work with Balázs Szegedy.

Posluchárna M1, 16:00, 13. března 2024

Speaker: Matija Bucić  (Princeton University), https://sites.google.com/princeton.edu/matija-bucic

Title: Robust sublinear expanders

Abstract:
Expander graphs are perhaps one of the most widely useful classes of graphs ever considered. In this talk, we will focus on a fairly weak notion of expanders called sublinear expanders, first introduced by Komlós and Szemerédi around 30 years ago. They have found many remarkable applications ever since. In particular, we will focus on certain robustness conditions one may impose on sublinear expanders and some applications of this idea, which include:
- recent progress on the classical Erdős-Gallai cycle decomposition conjecture,
- essentially tight answer to the classical Erdős unit distance problem for "most" real normed spaces, and
- an asymptotic solution to the rainbow Turán problem for cycles, raised by Keevash, Mubayi, Sudakov and Verstraete, with an interesting corollary in additive number theory.