## Student Harmonic Analysis and PDE Seminar (HADES): Gowers norms in Euclidean spaces: Introduction, applications and stability

Seminar | April 4 | 3:40-5 p.m. | 740 Evans Hall

Anh Nguyen, UC Berkeley

Department of Mathematics

Gowers norms are useful tools in additive combinatorics in measuring additivity of a subset $A$ in an abelian group $Z$ - how close $A$ is to being a subgroup of $Z$. In this talk, I demonstrate that there are uses of Gowers norms in Euclidean spaces. In particular I show that a measurable subset $E \subset \mathbb R^n, |E| = 1$ must assume convexity or near convexity if its $k$th Gowers norm is maximized or nearly maximized, respectively. Similar statements can be made about a measurable function in $\mathbb R^n$. This talk is based on recent papers of Christ, Eisner and Tao, and my current work.

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