Speaker: Igor Mineyev (UIUC)

Title: The topology and geometry of units and zero-divisors: origami.

Since 1950’s and even before, Kaplansky conjectures are several famous and problems about group algebras. Two of those conjectures are the unit conjecture and the zero-divisor conjecture; they are asking whether in group algebras over torsion-free groups there exist nontrivial units or nontrivial zero-divisors. To address these question systematically, this project combines five areas of mathematics: using topology and geometry, algebraic problems are translated into combinatorial questions about graphs that can be verified by computational means. We define product structures, their corresponding cell complexes, associate to them the universal group G, and a pair of elements in a group ring RG that are either units or zero-divisors. The difficult part is to guarantee that the group is torsion-free and that the two units/zero-divisors are indeed nontrivial. This is where topology/geometry helps: we give lists of sufficient combinatorial conditions for a product structure that guarantee the existence of a metric of nonpositive curvature on the associated cell complex. If satisfied, these conditions imply both the “torsion-free” and “nontrivial” properties. These results allow using computer-based search to look for counterexamples to the Kaplansky unit and zero-divisor conjectures. The paper with the same title is available at https://mineyev.web.illinois.edu/ .

Also, on the same day with this talk, on Thursday, May 2, there is a poster session of the Illinois Mathematics Lab, from 2pm to 5pm in 314A Illini Union, where students will present a preliminary version of the “ColorTaiko!” computer game based on this project.