On the Modular Isomorphism Problem

Speaker(s)

Leo Margolis

Affiliation

Universidad Autónoma de Madrid

Language of the talk

English

Date

June 12, 2025, 2:15 p.m.

Room

room 3220

Information about the event

Cykl dwóch wykładów (12 i 13 czerwca)

Seminar

Seminar Algebra

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A series of two lectures (90 minutes each one)

1. Thursday, June 12, 14:15, room 3220
2. Friday, June 13, 14:15, room 3240

Abstract. 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 FG as a ring determine the isomorphism type of G as a group, if G is a p-group and F a field of characteristic p?

In these lectures, after introducing the history of the problem, I will present the first negative answers to the problem and their generalizations. Also positive results and the methods underlying them will be presented. We also explain how the methods used here allow, at least on a conceptual level, also an interpretation in terms of algebraic geometry. 

This is joint work with Taro Sakurai.

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