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Tuesday, October 29, 2024
3:00 PM - 4:00 PM
Linde Hall 310

Mathematics Colloquium

Marton's Polynomial Freiman--Ruzsa conjecture
Frederick Manners, Department of Mathematics, UC San Diego,

A function $f : \mathbb{F}_2^n \to \mathbb{F}_2^n$ is linear if $f(x+y)=f(x)+f(y)$ for all pairs $(x,y)$. Now suppose $f$ is "a little bit linear" -- say, $f(x+y)=f(x)+f(y)$ for a 1% fraction of pairs $(x,y)$. What can you say about $f$? Must it be closely related to an actually linear function? If so, how closely?

This question turns out to be equivalent to asking for good quantitative bounds in the Freiman--Ruzsa theorem, a foundational result in additive combinatorics. Marton gave a formulation, equivalent to the statement above, which she conjectured should have polynomial bounds. I will discuss this conjecture, and its (relatively) recent proof (joint with Timothy Gowers, Ben Green and Terence Tao).

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].