Speaker: Weiqiang Wang (University of Virginia)
Title: "Kostant $u$-homology for oscillator modules of Lie algebras and superalgebras"
Abstract: Formula for Kostant $u$-homology groups of the oscillator modules of Hermitian symmetric pairs was earlier obtained by Enright. In this talk we explain a conceptually simple and new approach to compute the Kostant $u$-homology groups and characters of oscillator modules, using Howe duality for infinite dimensional Lie algebras. Another advantage of our approach is that it works also for Lie superalgebras. This is a joint work with S.-J. Cheng and J.-H. Kwon.

Speaker: Christopher Bendel (University of Wisconsin, Stout)
Title: "Vanishing ranges for the cohomology of finite groups of Lie type"
Abstract: Let $G(\mathbb{F}_q)$ be a finite Chevalley group defined over the field of $q = p^r$ elements, and $k$ be an algebraically closed field of characteristic $p > 0$. A fundamental open and elusive problem has been the computation of the cohomology ring $H^{\bullet}(G(\mathbb{F}_q),k)$. In particular, the lowest positive degree in which cohomology occurs is not generally known. In this talk, we will discuss recent work on determining initial vanishing ranges when $p$ is larger than the Coxeter number. For certain root systems, the first non-trivial cohomology classes are determined. The determination makes use of homological calculations involving filtrations, line bundle cohomology for the flag variety G/B, and combinatorial data involving Kostant partition functions.

Speaker: Pramod Achar (Louisiana State University)
Title: "Green functions via hyperbolic localization"
Abstract: I will explain how to use hyperbolic localization to construct an exact functor from perverse sheaves on the nilpotent cone to sheaves on the flag variety. This functor leads to a new proof of the Lusztig-Shoji algorithm for computing stalks of simple perverse sheaves, and it points the way to a conjectural derived equivalence between mixed l-adic perverse sheaves on the nilpotent cone and the derived category of dg-modules over the Borel-Moore homology of the Steinberg variety.

Sunday May 23rd

Speaker: Christopher Drupieski (University of Georgia)
Title: "On injective modules and support varieties for the small quantum group"
Abstract: Let $u_\zeta$ be the small quantum group at a root of unity. It is a finite-dimensional Hopf-subalgebra of Lusztig's quantum group $U_\zeta$. In this talk I will discuss a new rank variety type result for the Borel subalgebra $u_\zeta(\mathfrak{b})$ of $u_\zeta$, analogous to the results of Friedlander and Parshall describing the support variety of a restricted Lie algebra. Specifically, given a $u_\zeta(\mathfrak{b})$-module $M$, a root vector is contained in the $u_\zeta(\mathfrak{b})$-support variety of $M$ if and only if $M$ is not projective over a corresponding cyclic subalgebra of $u_\zeta(\mathfrak{b})$. As an application, we can show that a finite-dimensional $U_\zeta$-module $M$ is injective for $u_\zeta$ if and only if $M$ is free over one root subalgebras of $u_\zeta$ corresponding to long roots.

Speaker: Amber Russell (Louisiana State University)
Title: "Graham's Variety and Perverse Sheaves on the Nilpotent Cone"
Abstract: In recent work, Graham has constructed a variety with a map to the nilpotent cone that is similar to the Springer resolution. However, Graham's map differs from the Springer resolution in that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone. In this talk, we will describe the summands of this perverse sheaf via the cohomology of the fibers of Graham's map.

Speaker: Benjamin Jones (University of Georgia)
Title: "Support Varieties for Demazure Modules"
Abstract: We report on joint work with Daniel Nakano on support varieties for Demazure modules. Specifically, if $\mathcal{L}(\lambda)$ is a line bundle on the flag variety $G/B$ corresponding to a dominant weight $\lambda$ and $X_w \subset G/B$ is a Schubert variety, consider the $B$-modules $M_{w, \lambda} = H0( X_w, \mathcal{L}(\lambda))$. We examine the support variety $\mathcal{V}_{B_1}(M_{w, \lambda})$ over the first Frobenius kernel $B_1$ and its dependence on $w$ and $\lambda$ using tools developed in the work of Nakano-Parshall-Vella. We present general results on the calculation of support varieties for certain $X_w$ as well as explicit calculations in rank 2.

Speaker: Xin Tang (Fayetteville State University)
Title: "Prime ideals and Derivations of a Down up Algebra"
Abstract: Let $U^{+}_{r,s}(sl_{3})$ denote the two-parameter quantized enveloping algebra of the maximal nilpotent subalgebra of the Lie algebra $sl_{3}$. Note that $U_{r,s}^{+}(sl_{3})$ features as special example of down up algebras. In this talk, we give a presentation of the determination of both the prime ideals and derivations of a down up algebra $U_{r,s}^{+}(sl_{3})$. As a demonstration of applications, we state some results on its prime ideal stratification and compute its first Hochschild cohomology group.

Speaker: Daniel Sage (Louisiana State University)
Title: "Moduli Spaces of Irregular Singular Connections"
Abstract: An important problem in the geometric Langlands correspondence is the construction of global meromorphic connections on the projective line with specified local behavior. Boalch has studied the moduli space of such connections in the case where the leading term of the connection is regular semisimple at each singular point. In this talk, I will describe joint work with Bremer in which we show how to construct moduli spaces of connections in much greater generality. I will define a more useful notion of the leading term of a connection in terms of fundamental strata, a concept adapted from the representation theory of p-adic groups. In particular, I will introduce the concept of a regular stratum; a formal connection containing a regular stratum generalizes the naive idea of a connection with regular semisimple leading term. I will then explain how to construct the moduli space of connections on the projective line with specified regular local formal isomorphism classes at a collection of singular points. This moduli space is a symplectic reduction of a direct product of manifolds encoding local data at the singularities. I will also show that this moduli space arises as a symplectic quotient of a smooth manifold by a torus action.

Speaker: Christopher Bremer (Louisiana State University)
Title: "A theory of fundamental strata for Higgs bundles"
Abstract: In the geometric version of the local Langlands correspondence, irregular singular connections play the role of wildly ramified Galois representations. One way to study irregular singular connections is to fix a basis and consider the corresponding Higgs bundle. Although this construction is highly non-canonical, a `fundamental stratum' gives a preferred affine flag for which the associated graded map induced by the Higgs field is non-nilpotent. Furthermore, fundamental strata illuminate the relationship between irregular singular connections and induction data for cuspidal automorphic representations. In this talk, I will show that every formal connection has a fundamental stratum, and then classify strata that are `regular semisimple' (in a graded sense). This talk is based on joint work with Sage.

Speaker: Anastasia Shabanskaya (University of Toledo)
Title: "Computational aspects of Lie algebras and Mubarakzyanov algebras"
Abstract: We investigate six-dimensional solvable indecomposable Lie Algebras that have a five-dimensional nilradical. Such algebras were classified by the Russian mathematician G. M. Mubarakzyanov in a paper published in 1963. According to the form of nilradical there are 9 paragraphs in this paper and 99 classes of algebras depending on up to four parameters. The paper contains errors because calculations were done by hand and the list of algebras is incomplete. Also there is very limited amount of details which makes it not clear how the algebras were derived. Some algebras are completely missing as in Paragraph 6 of the paper. I make extensive use of MAPLE and some new routines to help finesse Mubarakzyanov's list. In each paragraph I will give a change of basis that will take you directly to the answer without having to sort through very complicated arguments by hand.

Speaker: Garrett Johnson (University of California, Santa Barbara)
Title: "Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL_2"
Abstract: We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on certain subspaces of their polynomial representations. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.

Speaker: Jonathan Brundan (University of Oregon)
Title: "Cohomology of Spaltenstein varieties"
Abstract: I'll explain an extension to Spaltenstein varieties of the classical work of De Concini-Procesi-Tanisaki from the 1980s giving a presentation for the cohomology algebra of a Springer fiber of type A. I should also have time to discuss some of the applications.

Monday May 24th

Speaker: Georgia Benkart (University of Wisconsin, Madison)
Title: "Drinfeld Doubles - A New Twist"
Abstract: Radford gave a construction of certain modules for pointed Hopf algebras H as subspaces of the Hopf algebra H itself. We describe how this construction behaves under cocycle twists. Using a result of Majid, we obtain a category equivalence between simple modules for the Drinfeld double of H and the simple modules for the Drinfeld double of its twist. The motivation for this work came from looking at 2-parameter quantum groups atroots of unity. Under mild assumptions on the roots of unity r and s, the 2-parameter quantum group is a Drinfeld double. For certain choices of r and s, the simple modules for the quantum group are quite similar and for others, quite different. We give an explanation of this behavior using cocycle twists. This is joint work with Mariana Pereira and Sarah Witherspoon.

Speaker: David Hemmer (SUNY Buffalo)
Title: "Vanishing ranges and Young module cohomology for symmetric groups"
Abstract: In 1990 Benson, Carlson and Robinson proved that for each group G there is exists a natural number r(G) so that the vanishing of H^i(G,M) for r(G) consecutive values of i implies vanishing for all i. In recent joint work with Cohen and Nakano we used homological methods and algebraic topology to compute Young module cohomology H^i(G, Y^\lambda)$ for the symmetric group G and a Young module Y^\lambda. In this talk we discuss two related problems. The first is to compute the smallest degree i such that H^i(G, Y^\lambda) is nonzero. This computation leads one to discover some Young modules with large vanishing ranges. Thus the second problem is an attempt to use Young modules to realize the "maximal gap", in the sense of the BCR result. At this point the best upper bound known for r(\Sigma_n) is around n2. We will prove there are Young modules in all characteristics that vanish for about n^(3/2) consecutive degrees but are not identically zero. Related but not equivalent to this problem is the following: Given a module L(\lambda) for the general linear group, find the maximal \mu so that L(\lambda) is a constituent of the symmetric power S^\mu(V).

Speaker: Jonathan Kujawa (University of Oklahoma)
Title: "Generalized trace and modified dimension functions on ribbon categories"
Abstract: Inspired by topological techniques, we construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are non-zero. Such examples include categories of modules of certain Lie algebras and finite groups over a field of positive characteristic, representations of Lie superalgebras over the complex numbers, representations for quantum groups at a root of unity, and Deligne's category Rep(S_t). These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory. We will discuss both the basic setup and some examples and applications of this new point of view. This work is joint with Nathan Geer, Bertrand Patureau-Mirand.