Olof Sisask: Approximate groups and almost-periodicity

- Lunch is served at 12:00 noon (register at doodle  by Sunday Dec 15).
- The presentation starts at 12:10 pm and ends at 1 pm.
- Those of us who wish reconvene after a short break for ca two hours of more technical discussions.

ABSTRACT
One can characterise the finite subgroups of an abelian group -- or rather cosets of subgroups -- as those non-empty subsets A of the group for which the sumset A+A = {a+b : a,b in A} has the same size as A itself. What if one relaxes this condition to |A+A| < K|A| for some fixed number K -- must such sets look a bit like subgroups?

This kind of question forms the backdrop for a fundamental result of additive combinatorics, known as Freiman's theorem, that gives a partial classification of such 'approximate subgroups'. There has been great progress in the past few years on obtaining stronger descriptions of such sets, but there are still important unsolved problems in this direction, the most famous probably being the Polynomial Freiman-Ruzsa conjecture. A positive resolution of this conjecture would provide very interesting and strong classifications of some fundamental approximate algebraic objects, such as approximate subgroups and approximate homomorphisms, and has been shown to have nice applications in other areas -- among them theoretical computer science. The plan for this talk is to describe some of the background of the Polynomial Freiman-Ruzsa conjecture and some of the underlying theory, and possibly to mention some of the application areas in TCS.