Let $G$ be a locally compact group with a left Haar measure $\mu$, and let $A,B \subseteq G$ be nonempty and compact. In 1964, Kemperman showed that if $G$ is unimodular (i.e., $\mu$ is also the right Haar measure, e.g., when $G$ is $\mathbb{R}/\mathbb{Z}$, $\mathrm{SL}_2(\mathbb{R})$, or $\mathrm{SO}_3(\mathbb{R})$), then $$\mu(AB) \geq \min \{\mu(A)+\mu(B), \mu(G)\} .$$ The inverse Kemperman problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem by Jinpeng An, Yifan Jing, Ruixiang Zhang, and myself highlighting some ideas from model-theoretic group theory.