Leo Margolis "Modular Isomorphism Problem - progress, solution and open challenges"

Say we are given only the R-algebra structure of a group ring RG of a finite group G over a commutative ring R. Can we then find the isomorphism type of G as a group? This so-called Isomorphism Problem has obvious negative answers, considering e.g. abelian groups over the complex numbers, but more specific formulations have led to many deep results and beautiful mathematics. The last classical open formulation was the so-called Modular Isomorphism Problem: Does the isomorphism type of kG as a ring determine the isomorphism type of G as a group, if G is a p-group and k a field of characteristic p?

Starting with an overview on the state of knowledge on general Isomorphism Problems and the modular one in particular, I will present a negative solution found in 2021 with D. García-Lucas and Á. del Río as well as a generalization obtained recently with T. Sakurai, but also positive structural results and several problems remaining open.