2017
University of North Carolina at Chapel Hill
Fridays 4pm, PH-367 or PH-385
The aim of this seminar is to bring speakers from this area and outside to speak on topics related to Representation Theory (specially geometric and topological methods employed in Representation Theory). The speakers are expected to give their talks at a level suitable for graduate students. The seminar is organized by Shrawan Kumar.
2017 Fall
| V.Balaji Degenerations of the moduli space of Higgs bundles on curves We work over the field of complex numbers. In this this talk I will prove certain basic results on the construction of a degeneration of the moduli space of Higgs bundles (and more generally Hitchin pairs) on smooth curves, as the smooth curve Xk degenerates to an nodal curve with a single node. As an application we get new compactifications of the Picard variety for smooth curves degenerating to nodal curves as toric blow-ups of the Oda-Seshadri compactification. This is joint work with P. Barik and D.S.Nagaraj. On semi-simplicity of tensor products in positive characteristics Abstract: We work over an algebraically closed field k of characteristic p > 0. In 1994, Serre showed that if semi-simple representations Vi of a group Γ are such that (dimVi − 1) < p, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where Γ is a subgroup of G(k), for G a reductive group, and answered the question of “complete reducibility” of Γ in G, (Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the Vi are semi-simple representations of a group scheme G. In my talk I present the recent work of mine with Deligne and Parameswaran where we consider the case when G is a subgroup scheme of a reductive group G and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes G of reductive groups G. As an application, we obtain an analogue of classical Luna’s ́etale slice theorem in positive characteristics. |
| Larry Rolen Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems Abstract: In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has been proved for degrees $d\leq3$. We obtain an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$, which allows us to prove thehyperbolicity of $100\%$ of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang. |
| Pramod Achar Koszul duality and characters of tilting modules Abstract: This talk is about the “Hecke category,” a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right. It turns out that there is also a second, “hidden” action of the Hecke category on the left. The symmetry between the “hidden” left action and the “obvious” right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson. |
| Xiping Zhang Local Euler Obstruction, Characteristic Complex and (Equivariant) Characteristic Classes of Determinantal Varieties Abstract: The (generic) determinantal variety is the projective variety consisting of m by n matrices with kernel dimension \geq k, which arises naturally from many aspects. In this talk I will give formulas for the local Euler obstructions on them. With the formulas I will prove that characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle, and hence is irreducible. This is an interesting and rare phenomenon, and has been studied for many spaces. I will also give formulas for the (equivariant) Chern-Mather/Chern-Schwartz-MacPherson classes of determinantal varieties. The formulas are based on calculations of degrees of certain Chern classes of the universal bundles over Grassmannians. For low dimensions we use Macaulay2 to exhibit some examples, and formulate conjectures concerning the positivity of the classes and the vanishing of specific terms in the Chern-Schwartz-MacPherson classes of the largest strata. |
| Daniel Thompson Representation theory of global Cherednik algebras Abstract: We will introduce the representation theory of sheaves of rational Cherednik algebras associated to a complex algebraic variety with the action of a finite group. Viewing this theory as a deformation of the theory of equivariant $\mathscr{D}$-modules, we can characterize holonomic modules in this context. The main technical result we obtain is that the pushforward functor under certain equivariant maps preserves holonomicity. This allows us to give a filtration by support on the category of (holonomic) modules for the global Cherednik algebra, and to describe its Serre subquotients in terms of representation theory of the usual rational Cherednik algebra. This is partially based on joint work with P. Etingof and G. Bellamy. |
| Leonardo Mihalcea Chern-Schwartz-MacPherson classes for Schubert cells: geometry and representation theory Abstract: A compact manifold has a tangent bundle, and a natural question is to find a replacement for the Chern classes of the tangent bundle, in the case when the space is singular. The Chern-Schwartz-MacPherson (CSM) classes are homology classes which “behave like” the Chern classes of the tangent bundle, and are determined by a functoriality property. The existence of these classes was conjectured by Grothendieck and Deligne, and proved by MacPherson in 1970’s. The calculation of the CSM classes for Schubert cells and Schubert varieties in flag manifolds was obtained only recently, and it exhibited some unexpected features. For instance, these classes are determined by a Demazure-Lusztig operator, and they are essentially equivalent to certain Lagrangian cycles in the cotangent bundle of the flag manifold, showing up in the proof of Kazhdan-Lusztig conjectures. They are also equivalent to the stable envelopes of Maulik and Okounkov. In this talk I will survey some of these developments. No prior knowledge about the CSM classes will be assumed. This is joint work with P. Aluffi, and ongoing joint work with P. Aluffi, J. Schurmann and C. Su. |
| Xinwen Zhu Hilbert’s twenty-first problem for p-adic varieties Abstract: Hilbert’s twenty-first problem, formulated to generalize Riemann’s work on hypergeometric equations, concerns the existence of linear differential equations of Fuchsian type on the complex plane with specified singular points and monodromic group. Its modern solution, due to Deligne and known as the Riemann-Hilbert correspondence, establishes an equivalence between two different types of data on a complex algebraic manifold X: the representations of the fundamental group of X (topological data) and the linear systems of algebraic differential equations on X with regular singularieties (algebraic data). I’ll review this classical theory, and discuss some recent progress to solve similar problems for p-adic manifolds. Spectral operators as a trace Abstract: I will first give a new formalism of (some parts of) the Springer theory and Deligne-Lusztig theory. Then I will use the new formalism to study the affine case, and construct the spectral operators acting on the local Hitchin moduli and the moduli of local Shtukas. In the former case, the construction generalizes the action of the affine Weyl group on the cohomology of affine Springer fibers constructed by Lusztig. In the latter case, the construction generalizes V. Lafforgue’s S-operators. |
| Weiqiang Wang Canonical bases arising from quantum symmetric pairs and Kazhdan-Lusztig theory Abstract: A quantum symmetric pairs consists of $(U, U^i)$, where U is a quantum group and $U^i$ is a coideal subalgebra (corresponding to a fixed point subalgebra by an involution). We focus on one particular example of $(U, U^i)$ of type AIII, explain new i-canonical bases for $U^i$ and for the tensor product U-modules, which admit favorable properties such as positivity and geometric interpretation. As an application of the i-canonical bases we establish a Kazhdan-Lusztig theory for Lie superalgebras of type BCD (which amounts to a new formulation for the not super KL theory). This is joint work with Huanchen Bao (Maryland). |
| Gufang Zhao Quiver varieties and elliptic quantum groups Abstract: In this talk I will define a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. This definition is naturally obtained from the elliptic cohomological Hall algebra of a preprojective algebra. The sheafified elliptic quantum group is an algebra object in a certain monoidal category of coherent sheaves on the colored Hilbert scheme of an elliptic curve. This monoidal structure is related to the factorisation structure of Beilinson-Drinfeld. I will show that the elliptic quantum group acts on the equivariant elliptic cohomology of Nakajima quiver varieties. Taking suitable rational sections provides Drinfeld currents, which satisfy the commutation relations of the dynamical elliptic quantum group studied by Felder and Gautam-Toledano Laredo. This talk is based on my joint work with Yaping Yang. |
| Catharina Stroppel Quiver Schur algebras and connections to number theory Abstract:I will start with recalling the notion of affine Schur algebras and briefly indicate how it appears in number theory (in p-adic Langlands). Then I will sketch the geometric construction of the quiver Schur algebra attached to an affine quiver of type A (involving Borel-Moore homology on a generalized Steinberg variety) and give an algebraic definition. Since the definition is not quite direct I will try to illustrate this by examples. The main result will then be a connection between the two algebras. As a consequence we obtain a grading on affine Schur algebras (at least after some completion).symmetric pairs and their appearance in branching rules Abstract: In this talk I will define examples of quantized symmetric spaces. These are certain coideals in quantum groups quantizing the fixed point Lie algebras under an involution. I will explain categorial actions of these algebras and connect them to branching rules in the representation theory of e.g. semisimple Lie algebras, of Lie superalgebras and of Brauer algebras. |
2017 Spring
| Date | Speaker | Affiliation | Title |
|---|---|---|---|
| May 05 | Andrea Appel | University of South California | Monodromy of the Casimir connection and Coxeter categories |
| April 21 | Reuven Hodges | Northeastern | Levi subgroup actions on Schubert varieties in the Grassmannian |
| April 18 | Arik Wilbert | University of Bonn | Two-block Springer fibers and Springer representations in type D |
| April 7 | Curtis Porter | NCSU | Straightening out degeneracy in CR Geometry: When can it be done? |
| Mar 31 4:40pm | Gabor Pataki | UNC | Combinatorial characterizations in semidefinite programming duality |
| Mar 24 | Xuhua He | Maryland | Cocenters and representations of p-adic groups |
| Mar 10 | Olivier Debarre | University Paris 7 and EMS | UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS |
| Feb 10 | Goncalo Oliveira | Duke | Gerbes on G2-manifolds |
| Andrea Appel Monodromy of the Casimir connection and Coxeter categories Abstract: A Coxeter category is a braided tensor category which carries an action of a generalised braid group B_w on its objects. The axioms of a Coxeter category and the data defining the action of B_W are similar in flavor to the associativity and commutativity constraints in a monoidal category, but are related to the coherence of a family of fiber functors. We will show how to construct two examples of such structure on the integrable category O representations of a symmetrisable Kac–Moody algebra g, the first one arising from the associated quantum group, and the second one encoding the monodromy of the KZ and Casimir connections of g. The rigidity of this structure implies in particular that the monodromy of the Casimir connection is given by the quantum Weyl group operators. This is a joint work with Valerio Toledano Laredo. |
| Reuven Hodges Levi subgroup actions on Schubert varieties in the Grassmannian Abstract: Let L be the Levi part of the stabilizer in GL_N(C) (for left multiplication) of a Schubert variety X(w) in the Grassmannian. For the induced action of L on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plucker embedding), I will give a combinatorial description of the decomposition of C[X(w)] into irreducible L-modules. Using this combinatorial description, I give a classification of all Schubert varieties X(w) in the Grassmannian for which C[X(w)] has a decomposition into irreducible L-modules that is multiplicity free. This classification is then used to show that certain classes of Schubert varieties are spherical L-varieties. Also, I will describe interesting related results on the singular locus of X(w) and multiplicities at points in X(w). |
| Arik Wilbert Two-block Springer fibers and Springer representations in type D Abstract: We explain how to construct an explicit topological model for every two-block Springer fiber of type D. These so-called topological Springer fibers are homeomorphic to their corresponding algebro-geometric Springer fiber. They are defined combinatorially using cup diagrams which appear in the context of finding closed formulas for parabolic Kazhdan-Lusztig polynomials of type D with respect to a maximal parabolic of type A. As an application it is discussed how the topological Springer fibers can be used to reconstruct the famous Springer representation in an elementary and combinatorial way. |
| Curtis Porter Straightening out degeneracy in CR Geometry: When can it be done? Abstract: CR geometry studies boundaries of domains in C^n and their generalizations. A central role is played by the Levi form L of a CR manifold M, which measures the failure of the CR bundle to be integrable, so that when L has a nontrivial kernel of constant rank, M is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold N, then we say M is CR-straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to N. It remains to classify those M for which L is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, and Medori-Spiro. I will discuss their results as well as my recent progress on the problem in dimensions 7 and beyond. |
| Gabor Pataki Combinatorial characterizations in semidefinite programming duality Abstract: I will discuss optimization problems over affine slices of positive definite symmetric matrices. These problems, called semidefinite programs (SDPs), have numerous applications. Several basic properties of SDP’s are rooted in the fact that the linear image of the cone of symmetric positive semidefinite matrices may not be closed. Examples of these properties include the non-triviality of the infeasibility problem (when the set of positive semidefinite matrices has an empty intersection with an affine subspace) and pathological properties of dual SDP’s. In this talk I survey recent, somewhat surprising combinatorial type characterizations for several fundamental problems in SDP duality. The main tool is very simple: we use elementary row operations – inherited from Gaussian elimination – to bring an SDP to a format in which properties – such as infeasibility – are trivial to recognize. Part of this work is joint with Minghui Liu. |
| Xuhua He Cocenters and representations of p-adic groups Abstract: It is known that the number of conjugacy classes of a finite group equals the number of irreducible representations (over complex numbers). The conjugacy classes of a finite group give a natural basis of the cocenter of its group algebra. Thus the above equality can be reformulated as a duality between the cocenter (i.e. the group algebra modulo its commutator) and the finite dimensional representations.Now let us move from the finite groups to the $p$-adic groups. In this case, one needs to replace the group algebra by the Hecke algebra. The work of Bernstein, Deligne and Kazhdan in the 80’s establish the duality between the cocenter of the Hecke algebra and the complex representations. It is an interesting, yet challenging problem to fully understand the structure of the cocenter of the Hecke algebra.In this talk, I will discuss a new discovery on the structure of the cocenter and then some applications to the complex and modular representations of $p$-adic groups, including: a generalization of Howe’s conjecture on twisted invariant distributions, trace Paley-Wiener theorem for smooth admissible representations, and the abstract Selberg Principle for projective representations. It is partially joint with D. Ciubotaru. |
| Olivier Debarre UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS Abstract: In 1985, Beauville and Donagi showed by an explicit geometric construction that the variety of lines contained in a Pfaffian cubic hypersurface in $P^5$ is isomorphic to a canonical desingularization of the symmetric self-product of a K3 surface (called its Hilbert square). Both of these projective fourfolds are hyperkähler (or symplectic): they carry a symplectic 2-form. In 1998, Hassett showed by a deformation argument that this phenomenon occurs for countably many families of cubic hypersurfaces in $P^5$. Using the Verbitsky-Markman Torelli theorem and results of Bayer-Macri, we show these unexpected isomorphisms (or automorphisms) occur for many other families of hyperkähler fourfolds. This involves playing around with Pell-type diophantine equations. This is joint work with Emanuele Macrí. |
| Goncalo Oliveira Gerbes on G2-manifolds Abstract: On a projective complex manifold, the Abelian group of Divisors (Div) maps onto that of holomorphic line bundles (the Picard group). I shall explain a similar construction for G2-manifolds. This uses coassociative submanifolds to define an analogue of Div, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the Picard group. |