Seminar: Frederick Manners (University of California, San Diego): "Inverse theorems and approximate structure"

Zoom link: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09
Speaker: Frederick Manners (University of California, San Diego)
Title: Inverse theorems and approximate structure
Abstract: We call a function f linear if f(x+y) = f(x) + f(y) holds for all x,y.  It is natural to call f "99% linear" if instead this identity holds for most pairs (x,y); say, 99% of pairs. Similarly, we could say f is "1% linear" if this identity holds 1% of the time.  A natural question is then: what can we say about the structure of "99% linear" or "1% linear" functions?  Are they always just perturbations of true 100% linear functions, or are there other examples?
 
Given almost any algebraic definition, you can similarly ask about its approximate variants, and if you can prove a strong positive statement, it tends to have applications.  In particular, I will discuss how 1% linear functions relate to the Polynomial Freiman-Ruzsa conjecture, and how 1% polynomial functions relate to the Inverse Theorem for the Gowers norms.