Colloquium 2009-10

March 11	Spring Break
March 18	David Blecher (University of Houston) Host: D. Sherman Title: Duality for Spaces of Hilbert Space Operators Functional analysis, as its name implies, is concerned in some large part with "functionals" on a space X. This amounts to understanding in detail (which can be very subtle in particular cases) the duality between X and its dual space, or predual if one exists. The study of vector spaces of Hilbert space operatorsis no exception to this principle. First I will introduce the structures we will be interested in, namely operator spaces, and important subclasses such as operator systems and operator algebras. Then we present the duality theory of these objects, mostly from recent work with Magajna.
March 25	Brian Boe (University of Georgia) Host: W. Wang Title: Lie Superalgebras and Varieties Lie superalgebras and their finite-dimensional representations play an important role in mathematical physics, where they arise in the context of "supersymmetry," as well as in several branches of mathematics. We study the category \mathcal F of finite-dimensional representations for a classical Lie superalgebra \mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1} over \mathbb C, which are completely reducible as \mathfrak g_{\bar 0}-modules. This category has a surprisingly rich structure---for instance, it is usually not semisimple. We discuss how cohomology can be used to understand the representation theory of \mathfrak g. Classical invariant theory plays an important role. The notion of support variety, borrowed from the setting of modular representations of finite groups, brings geometrical tools into the mix, and provides connections to the combinatorics of Lie superalgebras. This is joint work with Jonathan Kujawa and Daniel Nakano.
April 1
April 8	Michael Collins (Oxford University and UVA) Host: L. Scott Title: On Finite Subgroups of the Classical Groups
April 15	Alex Lubotzky (The Hebrew University of Jerusalem) Host: A. Rapinchuk Title: Counting Groups, Manifolds and Primes We will report of a series of works in the last decade around the following question: For a given simple Lie group G how many lattices (i.e., discrete subgroups of finite covolume) it has of covolume at most x. Equivalently, how many maniofolds (or volume at most x) are covered by the associated symmetric space. As many of these lattices are arithmetic, these questions often lead to deep number theoretic problems; counting primes etc. We will concentrate on recent works which give very sharp results for counting arithmetic lattices in SL(2). Here the representation theory of the symmetric groups comes also into the game.
April 22
April 29	Bruce Sagan (NSF and Michigan State University) Host: W. Wang Title: Combinatorial and colorful proofs of cyclic sieving phenomena

Colloquium 2009-10

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