Advances in Geometric Representation Theory
Organizers: S. Kitchen,
T. Nevins.
Speakers to include:
 Pramod Achar
 David BenZvi
 Bhargav Bhatt
 Ana Caraiani
 Tudor Dimofte
 Chris Dodd
 Justin Hilburn
 Joel Kamnitzer
 Kobi Kremnitzer
 Invisible placeholder
 Ivan Loseu
 Ivan Mirkovic
 Nicholas Proudfoot
 Sam Raskin
 Valerio Toledano Laredo
 David Treumann
 Julianna Tymoczko
 Ben Webster
 Invisible placeholder
Program
All activities will take place in USB 1230. The entrance to the Undergraduate Science Building (USB) that is closest to this room is on the plaza by the Fletcher Street parking structure. Due to construction, the best way to access the plaza is by the path between the parking deck and performing arts center on Fletcher Street.
Schedule
Monday  Tuesday  Wednesday  Thursday  Friday  

8:00 am  Refreshments  Refreshments  Refreshments  Refreshments  Refreshments 
9:00 am  Loseu (9:30)  Caraiani  Proudfoot  Caraiani  Achar (9:30) 
10:30 am  Raskin (10:45)  Dodd  Hilburn  Toledano Laredo  BenZvi (11:00) 
11:30 am  Lunch (11:45)  Lunch  Lunch  Lunch  
1:30 pm  Tymoczko  Bhatt  Dimofte  Kremnitzer  
3:00 pm  Kamnitzer  Kremnitzer  Webster  BenZvi  
4:00 pm  Refreshments  Refreshments  Refreshments  Refreshments  
4:30 pm  Treumann  padic Q&A  Symplectic Duality Panel 
Mirkovic 
Titles & Abstracts
Achar  Reductive groups, the loop Grassmannian, and Springer resolution
Let G be a reductive group over an algebraically closed field k of characteristic p > 0. In the late 1990s, Finkelberg and Mirkovic conjectured that the principal block of Rep(G) should be able to be realized in terms of a certain category of perverse sheaves on the dual loop Grassmannian. Moreover, this relationship should be compatible with (i.e., be a refinement of) the famous geometric Satake equivalence. In my talk, I will explain these statements, and try to give some context and motivation. The FinkelbergMirkovic conjecture remains open, but a kind of ``graded'' analogue of it is now a theorem. If time permits, I will discuss some of the ingredients in its proof. This is joint work with Simon Riche.

BenZvi  Algebraic Geometry of Topological Field Theories
I will describe joint work with Andy Neitzke, concerned with attaching moduli spaces to topological field theories in three and four dimensions. The algebra underlying the topology of configurations of points in 3space endows the corresponding moduli spaces with both canonical Poisson structures and canonical quantizations. In the fourdimensional setting our moduli spaces are noncommutative projective varieties exhibiting modular group symmetry. Betti Geometric LanglandsThe Betti Geometric Langlands Conjecture (GLC) is a topological counterpart to the de Rham GLC of BeilinsonDrinfeld and ArinkinGaitsgory and the Dolbeault GLC of DonagiPantev, inspired by the work of KapustinWitten in supersymmetric gauge theory and introduced recently with David Nadler. On the one hand, eigensheaves in the de Rham and Betti sense coincide, so that the Betti conjecture provides a different ``integration measure'' on the same fundamental objects. On the other hand, the Betti spectral categories are more explicit than their de Rham counterparts and the conjecture is expected to be easier. The Betti program also enjoys greatly enhanced symmetry coming from topological field theory. 
Bhatt  The Witt vector affine Grassmannian
I will describe some surprisingly nice features of algebraic geometry in the world of perfect schemes of characteristic p (i.e, schemes where Frobenius is an isomorphism). As an application, we will see why the padic analog of the affine Grassmannian is indprojective (which improves on recent work of X. Zhu). This is based on joint work with P. Scholze. 
Caraiani  On pdivisible groups and the FarguesFontaine curve The goal of this talk is to give some background on pdivisible groups, their moduli spaces and associated period morphisms. In particular, I will describe a classification of abelian varieties over the complex nubmers. In the second half of the talk, I will introduce a more geometric perspective via the FarguesFontaine curve and will discuss the relationship between pdivisible groups and vector bundles on the curve. All these objects play an important role in understanding the local Langlands correspondence for padic fields. On the geometry of the HodgeTate period morphismIn this talk, I will introduce global versions of the objects discussed in the first talk: Shimura varieties and their associated HodgeTate period morphisms. I will describe the geometry of the HodgeTate morphism: defining a Newton stratification and computing the fibers of the morphism above a given Newton stratum. I will mention an application to the global Langlands program, more precisely to understanding torsion in the cohomology of certain compact Shimura varieties. This is joint work with Peter Scholze. 
Dimofte  Vortices, monopoles, and finite AGT
TBA 
Dodd  Higher Representation Theory, Coherent sheaves on Grassmannians, and Hodge Modules
In this talk, we will discuss some aspects of socalled higher representation theory and their connection to geometry. We will recall how Grassmannians and their cohomology relate to the representation theory of sl_2, and we will discuss the connection with Hodge theory and some natural constructions in algebraic geometry. 
Hilburn  GKZ Hypergeometric Systems and Projective Modules in Hypertoric Category O
I will show that indecomposable projective and tilting modules in hypertoric category O are obtained by applying a variant of the geometric Jacquet functor of Emerton, Nadler, and Vilonen to certain Gel'fandKapranovZelevinsky hypergeometric systems. This proves the abelian case of a conjecture of Bullimore, Gaiotto, Dimofte, and Hilburn on the behavior of generic Dirichlet boundary conditions in 3d N=4 SUSY gauge theories. 
Kamnitzer  Generalized slices in the affine Grassmannian and symplectic duality
The recent work of BravermanFinkelbergNakajima gives a construction of the symplectic dual of a symplectic reduction of the cotangent bundle of a representation. In particular, their construction shows that generalized slices in the affine Grassmannian are the symplectic duals of Nakajima quiver varieties. I will introduce these generalized slices (also known as moduli spaces of singular monopoles) and their quantizations, called truncated shifted Yangians. I will explain many recent results and open questions, especially concerning their representation theory. 
Kremnitzer  Towards a padic RiemannHilbert
This is joint work in progress with Konstantin Ardakov, Oren BenBassat and Thomas Bitoun. I will define a sheaf of infinite order differential operators on a Berkovich space and show that a certain category of modules over it embeds into sheaves of bornological spaces. This is a padic analog of a complex analytic result of Prosmans and Schneiders. Global analytic geometryThis is joint work in progress with Oren BenBassat. I will explain an approach to (derived) analytic geometry over the integers and over the "field with one element". 
Loseu  Hecke algebras for complex reflection groups
IwahoriHecke algebras are classical objects in Representation theory. An important basic property is that these algebras are flat deformations of the group algebra of the corresponding real reflection group. In 1998 Broue, Malle and Rouquier have extended the definition of a Hecke algebra to the case of complex reflection groups. They conjectured that the Hecke algebras are still flat deformations of the group algebras. Recently, the proof of this conjecture was completed by myself and marinPfeiffer in the case when the base field has characteristic 0. In my talk I will introduce Hecke algebras for complex reflection groups and explain some ideas of the proof of the BMR. conjecture. 
Mirkovic  Remarks on the form of the geometric Class Field Theory
The talk covers some speculations on the nature of class field theory and its geometric meaning. 
Proudfoot  An introduction to symplectic duality
I will define category O for a bionic symplectic resolution and give an introduction to the symplectic duality program. I will also briefly discuss the hope of using symplectic duality to relate Nakajima’s construction of weight spaces of simple representations of simply laced Lie algebras to the construction arising from the geometric Satake equivalence. 
Raskin  Walgebras and Whittaker categories
Affine Walgebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of attention because of FeiginFrenkel's duality theorem for them, which identifies Walgebras for a Lie algebra and for its Langlands dual by a somewhat complicated construction. 
Toledano Laredo  QuasiCoxeter categories, the Casimir connection and quantum Weyl groups
A quasiCoxeter category is a braided tensor category which carries an action of a generalised braid group B_W on the tensor powers of its objects. The data which defines the action of B_W is similar in flavour to the associativity contraints in a monoidal category, but is related to the coherence of a family of fiber functors on C. I will outline how to construct such a structure on integrable, category O representations of a symmetrizable KacMoody algebra g, in a way that incorporates the monodromy of the KZ and Casimir connections of g. The rigidity of this structure implies in particular that the monodromy of the latter connection is given by the quantum Weyl group operators of the quantum group U_h(g). This is joint work with Andrea Appel. 
Treumann  Moduli spaces of Lagrangian surfaces
I will discuss work with Shende, H. Williams, and Zaslow on the moduli space of Lagrangian surfaces in a symplectic 4manifold, or rather a model for this moduli space coming from microlocal sheaf theory. When the 4manifold is a cotangent bundle, these spaces generalize the moduli of locally constant sheaves on the base, and one might think of the Lagrangian surfaces as being Betti analogs of spectral curves. Exact Lagrangians play a special role, so that analyzing our moduli space gives a lower bound, often infinite, for the number of them. 
Tymoczko  Generating the equivariant cohomology of affine Springer fibers
The finite Springer fiber is the collection of fixed flags under a given linear operator. Affine Springer fibers generalize this notion to the affine Grassmannian. KazhdanLusztig studied them first, and GoreskyKottwitzMacPherson used them to prove a special case of the Fundamental Lemma. We will discuss some recent results that construct generating sets for the equivariant cohomology of affine Springer fibers. 
Webster  The discreet charm of the Coulomb branch
For many years, my collaborators and I tried to understand the Coulomb branches of certain field theories from physics and failed miserably. Luckily, recent work of BravermanFinkelbergNakajima gives a mathematical construction of these spaces, and algebras quantizing them. I'll discuss an approach to the representation theory of these algebras (building on joint work with BradenLicataproudfoot and mny other authors). Applications include a version of the Koszul duality between the Higgs and Coulomb branches of such a theory, a new perspective on category O for Cherednik algebras, and a new understanding of coherent sheaves on Coulomb branches. 
Downloadable Program
Audio Recordings
Notes
Handwritten notes are available for all speakers who did not provide slides, courtesy of Richard Hughes. Slide talks link directly to the lecture capture file.
Information
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Sunday, May 8
Jazz performance at Kerrytown Concert House
Piano dissertation recital at Umich School of Music
"Extreme Time" exhibit at Ruthven Museum
Monday, May 9
Live music at Arbor Brewing Company
Classical choir performance at Kerrytown Concert House
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Group run at Running Fit
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"Irrational" (a musical about Pythagoras) at Theatre Nova
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Joe Pug at The Ark
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