The Kemperman inverse problem

Let G be a connected locally compact group with a left Haar measure μ, and let A,B ⊆ G be nonempty and compact. Assume further that G is unimodular, i.e., μ is also the right Haar measure; this holds, e.g., when G is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that

μ(AB) ≥ min {μ(A)+μ(B), μ(G)} .
The Kemperman inverse problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem, highlighting the connections to model theory. (Joint with Jinpeng An, Yifan Jing, and Ruixiang Zhang).