abstracts

December 17, 2014

• M. Wong “Open de Rham Spaces for Irregular Connections on P^1.”

The main objects discussed will be the de Rham spaces of irregular connections over P^1, that is moduli space of meromorphic connections with higher order poles. Typically one takes the leading order term of the connection to be a regular diagonal matrix. In these cases, Boalch has given a construction of the open locus of the moduli space where the underlying bundle is trivial; this takes the form of a symplectic quotient of a product of simpler constituents.  Adapting the language of p-adic representation theory to that of meromorphic connections, Bremer and Sage have generalized this work to the “ wisted" case, that is, where the leading order term of the connection is not necessarily diagonal.  Here again, the moduli space is constructed as a symplectic quotient of a product.  I plan to give descriptions of the constituent spaces in such a way as to make them amenable to point-counting techniques to be applied to the de Rham spaces.

• M. Gualtieri "The exact WKB method"

December 10, 2014

• M. Wong “Open de Rham Spaces for Irregular Connections on P^1.”

The main objects discussed will be the de Rham spaces of irregular connections over P^1, that is moduli space of meromorphic connections with higher order poles. Typically one takes the leading order term of the connection to be a regular diagonal matrix. In these cases, Boalch has given a construction of the open locus of the moduli space where the underlying bundle is trivial; this takes the form of a symplectic quotient of a product of simpler constituents.  Adapting the language of p-adic representation theory to that of meromorphic connections, Bremer and Sage have generalized this work to the “ wisted" case, that is, where the leading order term of the connection is not necessarily diagonal.  Here again, the moduli space is constructed as a symplectic quotient of a product.  I plan to give descriptions of the constituent spaces in such a way as to make them amenable to point-counting techniques to be applied to the de Rham spaces.

November 6, 13, 27, December 4, 11,  2014

• J.Kamnitzer  "Geometric Satake, MV polytopes, and quiver varieties."

One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geometry of spaces called affine Grassmannians with the representation theory of reductive groups. This correspondence was originally developed from the viewpoint of the geometric Langlands program, but it has many other interesting
applications.  For example, this theory gives rise to a natural basis for irreducible representation.  This basis is labelled by certain varieties called Mirkovic-Vilonen cycles.

On the other hand, there exists another natural basis for irreducible representations, known as Lusztig's (semi)-canonical basis.  This basis is constructed through Lusztig's nilpotent varieties, which parametrize representations of preprojective algebras.  This leads to the question of the relationship between these two bases.

In my lectures, I will explain the construction of these two bases.  I will then describe a combinatorial link between them using the theory of MV polytopes.  Finally, I will explain ongoing work to give a more geometric foundation to this combinatorial link.
 

Literature:
- J. Kamnitzer, Lectures on geometric constructions of the irreducible representations of GL_n, 0912.0569.
- V. Ginzburg, Perverse sheaves on a loop group and Langlands duality, alg-geom/9511007.
- J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, math/0501365.
- P. Baumann, J. Kamnitzer, and P. Tingley, Affine Mirkovic-Vilonen polytopes, 1110.3661.
-C. Geiss, B. Leclerc, J. Schroer, Semicanonical bases and preprojective algebras, 0402.5448.

November 5, 2014

• A. Noll "The WKB approximation, harmonic maps to buildings and spectral networks (IV)"

We will continue investigating the complex WKB problem. In this talk we will focus on a particular example, and construct the universal building.

October 29, 2014

• A. Noll "The WKB approximation, harmonic maps to buildings and spectral networks (III)"

This is a continuation of the previous talks. We will focus on a particular building, namely the asymptotic cone of a symmetric space and construct a map from the Riemann surface to it.

October 22, 2014

• A.Noll "The WKB approximation, harmonic maps to buildings and spectral networks (II)"

We will continue the investigation of the complex WKB problem, again focussing on the case of G=SL_2. In this context, we will introduce the notion of an R-tree, which will serve as motivation for buildings. We will state a theorem characterizing the leaf space of the foliation as a “universal tree’’. Then, we start with an introduction to Euclidean buildings.

October 15, 2014

• A.Noll "The WKB approximation, harmonic maps to building and spectral network (I)"

In this first talk, I will state the WKB-problem. Then I will describe its solution for G=SL_2 in terms of maps from the universal cover of a Riemann surface to special kinds of metric spaces, called (Euclidean) trees. This is motivation for the WKB problem for more general groups, where trees will be replaced by Buildings.

• B.Davison "Cohomological hall Algebras, brane tilings and character varieties"

in this talk I will explain what a brane tiling is and how they give rise to Jacobi algebra presentations of fundamental group algebras of real 3-folds. I'll then explain how this gives rise to a cohomological Hall algebra structure on the compactly supported cohomology of character stacks associated to Riemann surfaces. I'll finish by saying what this (conjecturally) has to do with character varieties and Higgs bundles.

October 8, 2014

• M.McBreen “Mirror Symmetry for Hypertoric Varieties, Part II"

I will describe the multiplicative analogue of hypertoric varieties, and describe how one might prove that they are mirror to their non-multiplicative cousins. This is work in progress with Ben Webster. If time permits, I will also describe some results on the quantum cohomology of hypertoric varieties; this is joint work with Daniel Shenfeld, and (separately) Nicholas Proudfoot. 

• D.Boccalini “Symmetries for SL(n) and PGL(n) Higgs bundle on an elliptic curve."

I’ll try to convince you that, over an elliptic curve E, the moduli spaces of Higgs bundles with dual groups SL(n) and PGL(n) have related cohomologies and related derived categories of coherent sheafs, and why this was expected. The necessary calculations and equivalences are possible thanks to the identifications of these muduli spaces with well studied varieties built out of the Hilbert scheme of n points over T*E. This is joint project with Riccardo Grandi. 

October 1, 2014

• M. McBreen "Mirror Symmetry for Hypertoric Varieties"

I will introduce hypertoric varieties and describe their characteristic p quantization, in the spirit of my first talk. I will then propose a mirror space, the multiplicative hypertoric variety, and show how its Fukaya category (conjecturally) describes modules over the quantization. This is joint work with Ben Webster.

• T. Hausel "p-adic representations of quivers"

After reviewing the theory of representations of quivers over fields, I will recall results when the field is finite. Then we study the free representation theory over depth 2 PID's which we think of as the first approximation to the ring of p-adic integers. Our main result will relate this representation theory to the arithmetic of a Nakajima quiver stack. This is joint work with Letellier and Villegas.

September 24, 2014

• M. McBreen "Introduction to the Fukaya category and Mirror symmetry for Hypertoric Varieties"

 I will give a gentle intro to symplectic geometry, the Fukaya category and Mirror Symmetry: normally quite a mouthful, but I'll focus on some simple hypertoric examples relevant to my previous talk.

•    Michael Lennox Wong: “Hecke algebras and counting function on fission spaces"

Wild character varieties are the "Betti versions" of moduli spaces of meromorphic connections where one allows higher order poles.  These varieties are constructed via a quasi-Hamiltonian "fusion product" (a multiplicative version of a symplectic quotient) and the new fusion factors that are introduced are what P. Boalch calls "higher fission spaces".  We will look at how the traces of certain Hecke algebras, called Yokonuma--Hecke algebras, can be used to obtain point-counting functions on special cases of these fission spaces.

September 17, 2014

• T.Hausel "Arithmetic of wild character varieties"  slides

I explain some conjectures, originating both in arithmetic and physics, on the mixed Hodge polynomials and perverse Hodge polynomials of tame character varieties and moduli of parabolic Higgs bundles on Riemann surfaces. We will then study the arithmetic of one class of Boalch's wild character varieties using the character theory of Yokonuma-Hecke algebras, and point out the relationship of the point count, and a natural conjecture on their mixed Hodge polynomials to the tame case. Finally I will discuss the twisted case, where torus knots through their HOMFLY homology will appear naturally in our formulas. This is joint work with Martin Mereb and Michael Wong.

• M.McBreen "Representations of Lie Algebras over Characteristic p and Mirror Symmetry."

I'll explain what happens when you quantize a symplectic resolution over characteristic p (those who attended Joel Kamnitzer's talk may want to look over their notes, as the topic will be very close!). Time permitting, I'll say something about Mirror symmetry and how it can help understand the quantization.