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Project name: Approximate Algebraic Structures of Higher Order: Theory, Quantitative Aspects and Applications, A-PLUS (2024–2025)

Project Reference No: 11143, supported by the Science Fund of the Republic of Serbia

Project leader: Luka Milićević

Project description

The theory of Gowers uniformity norms, stemming from Gowers's celebrated work on Szemerédi's theorem on arithmetic progression, has stimulated a tremendous amount of exciting mathematical developments, giving rise to the field of higher order Fourier analysis. Over time, it became apparent that approximate algebraic objects play a vital role in this theory. The goal of this Project is to study such objects with aim of improving the understanding of Gowers uniformity norms and related topics, and also with a view towards applications.
Despite significant effort of many researchers, the question of optimal bounds in the inverse theorems for Gowers uniformity norms is still far from resolved even in the most studied cases of finite vector spaces and cyclic groups. Furthermore, there are no inverse theorems that work in arbitrary finite abelian groups, just to name some of the most salient of questions. This Project’s main goal is to resolve such questions by studying approximate algebraic structures of higher order. Namely, we aim to develop theory of a particular class of such structures (Freiman multihomomorphisms) in general finite abelian groups and to apply it to prove general inverse theorem for Gowers uniformity norms. Another aim is to prove such results with quasipolynomial bounds. Given the lack of understanding of these norms in such a context, we expect the results of this Project to be rather significant and novel.
We will rely heavily on algebraic regularity method, developed by Gowers and Milićević, the principal investigator, and one of the main goals of the Project is to generalize this method from finite vector spaces to finite abelian groups. Additionally, we will use a number of tools from additive combinatorics, including Croot-Sisask-Sanders theory, dependant random choice, symmetrization arguments, discrete Fourier analysis, etc.
Finally, during the project, we shall work on applications of this theory in number theory and combinatorics.

List of researchers

Luka Milićević		Research assistant professor		Mathematical Institute SANU		luka.milicevic@mi.sanu.ac.rs
Danijela Popović		research assistant		Mathematical Institute SANU		danijela.mitrovic@mi.sanu.ac.rs
Marija Jelić Milutinović		assistant professor		Faculty of Mathematics, University of Belgrade		marija.jelic@matf.bg.ac.rs