L’institut Ottawa-Carleton de recherches en mathématiques et en statistique
et le Centre de Recherches Mathématiques présentent la
60e JOURNÉE D’ALGÈBRE
le samedi 31 mars 2007
Ottawa-Carleton Institute for Mathematics and Statistics, and the
Centre de Recherches Mathématiques
will host the
60th ALGEBRA DAY
Saturday, March 31, 2007
- 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
Abstract: 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 ?
- 15:30 Raman Parimala (Emory University): The symplectic discriminant
Abstract: 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)
Related event on Friday, March 30, 2007:
15:30 Karen V. H. Parshall, University of Virginia
4000 Years of Algebra: An Historical Tour from BM 13901 to Moderne Algebra
How is it that the high school analysis of polynomial equations and the modern algebra of the research mathematician---so seemingly different in their objectives, in their tools, and in their philosophical outlook---are both called "algebra"? Are they even related? The fact is that they are. This talk will sketch the long and complicated story of how they are related via a 4000-year-long history that stretches from Mesopotamia around 1800 B.C.E.---when mathematicians recorded an algorithm for solving quadratic equations on clay tablets like BM 13901, held today in the British Museum---to the publication in 1930 of Bartel van der Waerden's classic text, Moderne Algebra.
Location: All talks will be in the
Department of Mathematics and Statistics of the University of
Ottawa, 585 King Edward (KED on the campus map, between Laurier and Osgoode), room B5.
Parking: Participants will be given a parking permit for the parking lot T behind the department. Please talk to one of the local organizers as soon as possible after your arrival.
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.
Local organizers: Erhard Neher (firstname.lastname@example.org) and Michel