American Institute of Mathematics, Palo Alto, California
Robert Guralnick, Terrell Hodge, Brian Parshall, and Leonard Scott
This workshop, sponsored by AIM and the NSF, will be devoted to questions associated to the following 1984 conjecture of Guralnick's: There exists a ''universal constant'' $C$ which bounds 1-cohomology, in the sense that if $H$ is any finite group and $V$ is any faithful, absolutely irreducible H-module, then $$ \dim H^1(H, V ) \le C. $$
Over 25 years later, this conjecture remains open, but some recent developments have revealed new avenues for investigation. For example, relaxing the constraint on the term ''universal'' a bit, it has been recently shown that in the case of finite groups of Lie type $H = G(q)$, there are constants $C(\Phi)$, depending only on the root system $\Phi$ of the associated algebraic group $G$, which bound $\dim H^1(H, V )$, the 1-cohomology as above.
Significant topics envisioned for workshop investigations include:
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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