Polynomial maps between abelian groups

by Elias Schuster (University of Bonn)

Friday May 22, 2026, 2:15 PM → 3:45 PM Europe/Berlin

Endenicher Allee 60, Seminarraum 0.011 (Mathezentrum)

Abstract: 

Using discrete derivatives, one can define a notion of polynomials between arbitrary groups. Such polynomials arise naturally in inverse Gowers theory through a fundamental (and still only partially established) dichotomy: a bounded function $f \colon G \to \mathbb{C}$ either behaves pseudorandomly, or it correlates with a polynomial phase. This principle is crucial in establishing the existence of arithmetic patterns in subsets $A \subset G$.

Despite their importance, polynomial maps are only partially understood yet. To remedy this, it is valuable to develop algebraic characterizations of such functions. In this talk, we describe the construction of a universal group $\textup{Pol}_k^{ab}(G)$ which classifies all unital polynomials of degree at most $k$ from $G$ into an abelian group, building on work of Jamneshan and Thom. We then present classification results of $\textup{Pol}_k^{ab}(G)$ for a large class of abelian groups $G$.