Research Seminars @ Illinois

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Tailored for undergraduate researchers, this calendar is a curated list of research seminars at the University of Illinois. Explore the diverse world of research and expand your knowledge through engaging sessions designed to inspire and enlighten.

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GGT Seminar: Kaplansky's Conjecture: looking for zero divisors in group rings

Event Type
Seminar/Symposium
Sponsor
Groups, Geometry, and Topology Seminar
Location
347 Altgeld Hall
Date
Nov 7, 2024   11:00 am - 12:00 pm  
Speaker
Manisha Garg
Views
74
Originating Calendar
Groups, Geometry, and Topology Calendar

In 1956-1957 Kaplansky presented a list of several questions about group rings F[G], where F is a field and G is a group. Two of those questions are concerned with whether there are nontrivial units and zero-divisors in a torsion-free group ring. In 2021, G. Gardam found the first example of nontrivial units in a group ring over a virtually abelian group and a field of order 2.

In this talk we present a computational approach we employed in search of nontrivial units and zero divisors in a torsion-free group ring. We discuss an algorithm which aims to identify product structures satisfying various combinatorial conditions arising from the topology and geometry of the units and the zero-divisors. Identifying structures with such conditions could provide examples of groups with nontrivial units and zero-divisors.  This is joint work with Igor Mineyev.

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