Abstract:
Additive Combinatorics is a mathematical field intermediate between Combinatorics and Number Theory studying combinatorial questions involving the group operation.
In our talk we discuss an important characteristic of the area which can be formulated non rigorously as a principle of the existence of a rigid structure provided some “critical” relation takes place. We mean the following abstract setting. Suppose that there is a family of objects endowed with a real functional. Let us assume that the maximal value of the functional equals $M$. Take a parameter $K>1$ (not really large) and ask the question: what can be said about the objects of our family with the value of the functional greater than $M/K$? Is it true that the objects have some structure? Ñonversely, does the structure guarantee that the functional will has the large value automatically? Remarkably for the wide class of additive combinatorial questions the answer is affirmative and the “extremal” subobjects are highly structured.
In the talk we observe recent results on polynomial Freiman-Ruzsa conjecture, inverse theorems for Gowers norms and structural results on sets having extremal relations between different additive energies.