Abstract: 

This largely survey talk is a prequel to the talk “Measure doubling of small sets in SO(3,ℝ)” (joint work with Jing Yifan and Zhang Ruixiang) given in Beijing Logic Meeting 2023. I will explain why the use of model theory is not accidental in such combinatorics flavored results and is to some degree the same reason we can show that large stable fields are separably closed (joint work with Will Johnson, Erik Walksberg, and Ye Jinhe).

More mathematically, we consider a group G equipped with a notion of size (cardinality, Haar measure, etc), and let  A be a subset of G. The following problems are of interest:

(i) Find inequalities relating the size of the product set A^k ={a_1  a_k: a_1, a_2 \in A} and the size of A. 

(ii)Understand when the size of  A^k is not too large compared to that of A. 

These are called the direct and inverse problems for growth in groups.

I will explain how these problems arise independently in many areas including number theory, additive combinatorics, convex geometry, analysis, geometric group theory, and model theory. Beside the result about SO(3,ℝ), I will also discuss some other recent progresses in addressing them including the solution of the Polynomial Freiman-Ruzsa Conjecture for (F_2)^n by Gowers, Green, Manners, Tao and sharp stability results for Brunn—Minkowski theorem by Figalli, van Hintum, and Tiba.