Marton's Conjecture, aka the Polynomial Freiman--Ruzsa conjecture

A function f:𝔽n2→𝔽n2 is linear if f(x+y)=f(x)+f(y) for all pairs (x,y).

Suppose f is "a bit linear" -- say, f(x+y)=f(x)+f(y) for 1% of pairs(x,y).  Must f agree with a truly linear function a positive proportion of the time?  How large a proportion?

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 outline a recent proof of this conjecture.

Joint work with Timothy Gowers, Ben Green and Terence Tao.