- 10:00 Coffee in room 104 (Lounge)
- 11:00 Brian Parshall (University of Virginia):
*Reduced standard modules for reductive groups with applications to finite groups*

*Abstract*: Let G be a semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. Given a dominant weight, two representations for G can be constructed by "reduction mod p" from minimal and maximal lattices in the complex Lie algebra of the same type as G. These representations were broadly popularized by the famous Yale notes of Steinberg almost 40 years ago. The representations have other descriptions and remarkable homological properties, and they play an important role in the representation theory of G.

In this talk, we discuss an analogous family of representations of the algebraic group G -- this time, obtained by reduction mod p from quantum enveloping algebras at roots of unity. These modules also have remarkable homological properties and they can furthermore be applied to obtain new results on bounds for generic 1-cohomology for finite groups of Lie type. Such asymptotic results then provide positive evidence for a conjecture of R. Guralnick on finite group cohomology.

This talk is based on recent joint work with Ed Cline and Leonard Scott. - 12:15 Lunch in room 104 (Lounge)
- 14:00 Ragnar-Olaf Buchweitz (University of Toronto):
*Noncommutative Desingularisation of the Generic Determinant*: In this joint work with Graham Leuschke and Michel van den Bergh we show that the generic determinant admits a noncommutative crepant desingularization by a "Quiverized Clifford Algebra". The talk will explain these terms and show how this result relates to very concrete questions such as the following posed (and mainly answered using topoplogical methods in characteristic zero!)) by George Bergman: If X is an n-by-n matrix with indeterminate entries and adj(X) is its classical adjoint, can one factor adj(X) = UV with noninvertible n-by-n matrices U,V ?

Abstract - 15:30 Raman Parimala (Emory University):
*The symplectic discriminant*: The theory of Pfister forms revolutionised the algebraic theory of quadratic forms in the 60's. Pfister forms are simply a tensor product of binary forms. One looks for analogous notions for algebras with involution; the latter play a central role in the study of classical groups. One may call an algebra with an involution of first kind Pfister if it decomposes into a tensor product of quaternion algebras with involution. We explain an invariant associated to central simple algebras with a symplectic discriminant; this invariant defines the only obstruction to a degree 8 algebra with a symplectic involution to be Pfister. We derive some consequences concerning the Pfister Factor Conjecture. (Joint work with S. Garibaldi and J.-P. Tignol)

Abstract

* Abstract*:

Financial support for participating graduate students and postdoctoral fellows is available. If you are interested please contact one of the local organizers as soon as possible.