- Pramod Achar
- David Ben-Zvi
- 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
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.
|9:00 am||Loseu (9:30)||Caraiani||Proudfoot||Caraiani||Achar (9:30)|
|10:30 am||Raskin (10:45)||Dodd||Hilburn||Toledano Laredo||Ben-Zvi (11:00)|
|11:30 am||Lunch (11:45)||Lunch||Lunch||Lunch|
|4:30 pm||Treumann||p-adic Q&A||Symplectic
|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 Finkelberg-Mirkovic 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.
|Ben-Zvi||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 3-space endows the corresponding moduli spaces with both canonical Poisson structures and canonical quantizations. In the four-dimensional setting our moduli spaces are noncommutative projective varieties exhibiting modular group symmetry.Betti Geometric Langlands
The Betti Geometric Langlands Conjecture (GLC) is a topological counterpart to the de Rham GLC of Beilinson-Drinfeld and Arinkin-Gaitsgory and the Dolbeault GLC of Donagi-Pantev, inspired by the work of Kapustin-Witten 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 p-adic analog of the affine Grassmannian is ind-projective (which improves on recent work of X. Zhu). This is based on joint work with P. Scholze.
|Caraiani||On p-divisible groups and the Fargues-Fontaine curve
The goal of this talk is to give some background on p-divisible 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 Fargues-Fontaine curve and will discuss the relationship between p-divisible groups and vector bundles on the curve. All these objects play an important role in understanding the local Langlands correspondence for p-adic fields.On the geometry of the Hodge-Tate period morphism
In this talk, I will introduce global versions of the objects discussed in the first talk: Shimura varieties and their associated Hodge-Tate period morphisms. I will describe the geometry of the Hodge-Tate 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
|Dodd||Higher Representation Theory, Coherent sheaves on Grassmannians, and Hodge Modules
In this talk, we will discuss some aspects of so-called 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'fand-Kapranov-Zelevinsky 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 Braverman-Finkelberg-Nakajima 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 p-adic Riemann-Hilbert
This is joint work in progress with Konstantin Ardakov, Oren Ben-Bassat 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 p-adic analog of a complex analytic result of Prosmans and Schneiders.Global analytic geometry
This is joint work in progress with Oren Ben-Bassat. 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
Iwahori-Hecke 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 marin-Pfeiffer 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|| W-algebras and Whittaker categories
Affine W-algebras 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 Feigin-Frenkel's duality theorem for them, which identifies W-algebras for a Lie algebra and for its Langlands dual by a somewhat complicated construction.
|Toledano Laredo||Quasi-Coxeter categories, the Casimir connection and quantum Weyl groups
A quasi-Coxeter 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 Kac-Moody 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 4-manifold, or rather a model for this moduli space coming from microlocal sheaf theory. When the 4-manifold 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. Kazhdan-Lusztig studied them first, and Goresky-Kottwitz-MacPherson 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 Braverman-Finkelberg-Nakajima 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 Braden-Licata-proudfoot 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.
Hand-written notes are available for all speakers who did not provide slides, courtesy of Richard Hughes. Slide talks link directly to the lecture capture file.
|Loseu||Caraiani 1||Proudfoot||Caraiani 2||Achar|
|Raskin||Dodd||Hilburn||Toledano Laredo (slides)||Ben-Zvi 2|
|Kamnizter||Kremnitzer 1||Webster (slides)||Ben-Zvi 1|
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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
"Beekeeping in Cambodia" lecture at Matthaei Botanical Gardens
Group run at Running Fit
Tuesday, May 10
Acoustic Tuesdays at Arbor Brewing Company
Wednesday, May 11
Trivia night at Arbor Brewing Company
Whiskey Wednesdays at The Last Word
Nature walk at Ford Heritage Park
Pete Seeger birthday tribute at The Ark
Thursday, May 12
Beer tasting at Arbor Brewing Company
"Irrational" (a musical about Pythagoras) at Theatre Nova
"Charley's Aunt" (play) at Riverside Arts Center
Weekly Go event at University of Michigan Go Club
Joe Pug at The Ark
Friday, May 13
Classical Baroque performance at Kerrytown Concert House