COLLOQUIUM
3:00 p.m., Wednesday (April 9, 2008)
MATX 1100
Daniel K. Nakano
University of Georgia
Bridging Algebra and Geometry via Cohomology
Abstract: Cohomology theories were developed throughout the 20th century by topologists
to construct algebraic invariants for the investigation of manifolds
and topological spaces. During this time, cohomology was also defined
for algebraic structures like groups and Lie algebras to determine ways in
which their representations can be glued together.
The purpose of this talk will be to demonstrate how cohomology theories
for algebraic structures can be used to reintroduce the underlying geometry.
For finite groups, these ideas started with the work of D. Quillen and
J. Carlson. My talk with focus on the situation for the (small)
quantum group u_{q}({\mathfrak g}) where {\mathfrak g} is a complex
semisimple Lie algebra and q is a primitive lth root of unity.
In this setting the spectrum of the cohomology ring identifies with
certain subvarieties of the nilpotent cone {\mathcal N}. The nilpotent
cone is a wellstudied geometric object with beautiful combinatorial
properties related to the associated root system and Weyl group.
Results will be shown which illustrate new connections between the
(classical) orbit theory in {\mathcal N} and the cohomology theory
for quantum groups.
Refreshments will be served at 2:45 p.m. (MATX 1115, Math Lounge).
