Survey of some problems on group rings

Operator Algebra Seminar by Andreas Naes Aaserud, UCLA

I will cover the history and present state of some problems on group rings that were raised by Graham Higman (1939) and Irving Kaplansky (1956).  The main problem, which is usually called the zero-divisor conjecture or the Kaplansky conjecture, asks whether the (complex) group ring of a torsion-free group always lacks zero-divisors. I will also talk about the Kadison-Kaplansky conjecture, which states that the reduced group C*-algebra of a torsion-free group always lacks non-trivial idempotents, and the connection between the zero-divisor conjecture and the Atiyah conjecture.