Representation Theory Seminar -- Past seminars
Monday, January 29. Anna
Puskás, UMass. Demazure-Lusztig operators, Whittaker functions and crystals.
- Monday, February 12. Dongkwan Kim, MIT. On total Springer representations for classical types.
- Monday, March 5. Rahul Singh, Northeastern. The Conormal Variety of a Schubert Variety.
- Monday, March 19. Yiqiang Li, University at Buffalo. Quiver varieties and symmetric pairs.
- Monday, March 26. Makarand Sarnobat, IISER, Pune, India. Cohomology
of representations and Langlands functoriality.
- Monday, April 2. Cristian Lenart, SUNY Albany. Combinatorics of Lusztig's t-analogue of weight multiplicity.
- Monday, April 9. Justin Campbell, Harvard. Nearby cycles of Whittaker sheaves.
- Monday, April 23. Ana Balibanu, Harvard. The partial compactification of the universal centralizer.
- Monday, April 30. Tamar Friedmann, Smith. The action of the symmetric group on a generalization of the free Lie algebra: a CataLAnKe Theorem.
Monday 9/11. Organizational
Monday 9/18. Alejandro Morales,
polytopes and Kostant partition functions.
Monday 9/25. Tomoyuki Arakawa,
RIMS, Kyoto. Chiralization
of Moore-Tachikawa conjecture.
Monday 10/2. Spencer Leslie, BC.
Sheaves and Smith Theory.
Monday 10/16. Dinakar Muthiah,
results toward the geometry of double-affine flag varieties.
Monday 10/23. Zhijie Dong,
UMass. A relation between Mirković-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE.
Wednesday 10/25. Kostiantyn
Tolmachov, MIT. Towards
a categorification of a projection from an affine to a finite Hecke
algebra in type A.
Monday 10/30. Jacob Matherne,
polynomials of matroids.
Monday 11/13. Daniel Wong,
Cornell. Admissible modules and normality of classical nilpotent varieties.
Monday 11/27. Anna Puskás,
UMass. Postponed until Spring.
Monday 12/4. Neal Livesay, UC
spaces of irregular singular connections.
Monday 12/11. Robert Donley, Queensborough Community
College. Central Zeros of Clebsch-Gordan coefficients.
- Monday 2/6. Gufang Zhao, UMass. The elliptic affine Hecke algebra and the Hitchin system on an elliptic curve.
- Monday 2/27. Brian Hwang, Cornell. Degenerations of Grassmannians, the affine Grassmmannian, and Shimura varieties.
- Monday 3/6. Kyu-Hwan Lee, UConn. Description of real Schur roots of rank 3 acyclic quivers.
- Monday 3/20. Gus Lonergan, MIT. Fourier transform for the quantum Toda lattice.
- Monday 4/3. Ben Johnson, UMass. Defining Ideals of Nilpotent Orbit Closures
- Monday 4/10. [Canceled] José Simental Rodríguez, Northeastern. Harish-Chandra bimodules for rational Cherednik algebras.
- Monday 4/24. Mee Seong Im, West Point. On the representation theory of periplectic Lie superalgebras
- Monday 5/1. Nate Harman, MIT. Integral and quantum Deligne categories.
- Tuesday 5/16. Vinoth Nandakumar, Utah. The Exotic Robinson-Schensted Correspondence.
- Thursday 9/29. Ben Davison, IST Austria, The integrality conjecture and the Kac positivity conjecture
- Monday 10/3. Changlong Zhong, SUNY Albany,
Hecke actions and oriented cohomology of flag varieties.
- Monday 10/17. Jacob Matherne, UMass, Combinatorial Fourier transform for quiver representation varieties
in type A.
- Monday 11/7. Yaping Yang, UMass, Cohomology theories and affine quantum groups .
- Thursday 11/10. Greg Pearlstein, Texas A&M, Geometry of nilpotent cones in Hodge theory.
- Monday September 29. Eric Sommers, UMass, Exterior powers of the reflection representation in Springer theory.
- Monday October 6. Eric Sommers, UMass, Exterior powers of the reflection representation in Springer theory, II.
- Monday October 20. Pramod Achar, Louisiana State University, The affine Grassmannian and the Springer resolution in positive characteristic
- Monday October 27. William Slofstra, University of California, Davis, Free inversion arrangements and Peterson translation.
- Thursday October 30, Anna Bertiger, U. of Waterloo, The equivariant rim hook rule.
- Monday November 3. Martha Precup, Baylor University, The geometry of nilpotent Hessenberg varieties.
- Monday November 10. Alexei Oblomkov, UMass, Topology of the Affine Springer Fibers: examples and applications to the representation theory.
- Monday November 17. Paul Hacking, UMass, Canonical bases for cluster algebras.
- Monday October 24. Merrick Brown, UMass, The Saturated Tensor Cone for Symmetrizable Kac-Moody Algebras.
- Monday December 1. Elizabeth Drellich, UMass Equivariant Cohomology of Flags and Partial Flags.
This semester in addition to inivited talks by outside speakers we will have some talks by local speakers about affine Grassmannians in geometric representation theory.
- Monday, February 10. Ivan Mirković, UMass. Loop Grassmannians and Zastava spaces.
Thursday, February 14 Tuesday, February 18 (rescheduled due to snow), LGRT 1234. Roman Bezrukavnikov, MIT. Character sheaves and Drinfeld center.
- Monday, February 24. Ivan Mirković, UMass Loop Grassmannians and Zastava spaces II.
- Thursday, March 6. Roman Bezrukavnikov, MIT. Character sheaves and Springer theory.
- Monday, March 31. Myron Minn-Thu-Aye, UConn. Multiplicities for perverse coherent sheaves via the affine Grassmannian
- Monday, May 12. Roman Bezrukavnikov, MIT. Towards character sheaves on loop groups
This semester in addition to inivited talks by outside speakers we will have some more expository talks by local speakers, loosely organized around Springer Theory.
- Monday, September 16. Tom Braden, UMass. Constructing Springer's representations.
- Monday, September 23. Alexei Oblomkov, UMass. Springer's representations via convolution I.
- Monday, September 30. Alexei Oblomkov, UMass. Springer's representations via convolution II.
- Monday, October 7. Alexei Oblomkov and Ivan Mirkovic, UMass. Springer theory, continued.
- Tuesday, October 22. Laura Rider, MIT. Parity sheaves on the affine Grassmannian and the Mirkovic-Vilonen conjecture.
- Monday, November 4. Peter Samuelson, Toronto. On Cherednik's 2-variable Jones polynomials for torus knots.
- Monday, November 18. Dylan Rupel, Northeastern. Quantum Cluster Algebras and Feigin Homomorphisms.
This semester we will be running a reading seminar on days when there is no outside speaker. The goal will be to understand the recent proof of the positivity of coefficients of Kazhdan-Lustig polynomials by Elias and Williamson.
- Ben Elias, Geordie Williamson, The Hodge theory of Soergel
bimodules. preprint arXiv:1212.0791
- Wolfgang Soergel, "Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über
Polynomringen", available in English or German from
the author's webpage
- Some background reading on Hecke algebras and Kazhdan-Lusztig
polynomials, for instance the first section of Soergel's
"Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting
modules". Represent. Theory 1 (1997), 83-114.
- Section 1 from: Michel Brion, "Equivariant cohomology and
equivariant intersection theory", lecture notes available at
- Mark de Cataldo and Luca Migliorini, "The Decomposition Theorem and
the topology of algebraic maps", Bulletin of the A.M.S., Vol. 46, n.4,
(2009), 535-633, and "The Hard Lefschetz Theorem
and the topology of semismall maps", Ann.Scient.Ec.Norm.Sup., t.35. 2002, 759-772.
Available from the author's website
- Monday, February 4. Tom Braden, UMass. Organizational meeting for the reading seminar.
- Monday, February 11. Stephen Oloo, UMass.
Equivariant cohomology, flag varieties and Bott-Samelson varieties.
- Thursday, February 14. Sara Billey, University of Washington.
k-Vexillary permutations, Stanley symmetric functions, and generalized Specht modules.
- Monday, February 25. Tom Braden, UMass.
Hard Lefschetz, Hodge-Riemann bilinear relations, and the Decomposition Theorem
- Monday, March 4. Daniel Shenfeld, Princeton.
Quantum Cohomology of Hypertoric Varieties
- Monday, March 11. Tobias Wilson, UMass.
Soergel bimodules, part I
- Thursday, March 14. Eduardo Cattani, UMass.
The hard Lefschetz theorem
- Monday, March 25. Ivan Loseu, Northeastern University.
Uniqueness of tensor product categorifications.
Thursday, April 4 Thursday, April 18. Alexei Oblomkov, UMass. (rescheduled due to illness)
Soergel bimodules, part II. Special room: LGRT 1322
- Monday, April 22. Dinakar Muthiah, Brown University
Double Mirković-Vilonen Cycles and the Naito-Sagaki-Saito Crystal
- Monday, April 29. Mitya Boyarchenko, University of Michigan.
New geometric structures in the local Langlands program
This semester we will run an informal seminar on Double Affine Hecke
Algebras (DAHA's) led by Alexei Oblomkov. We will also have a few
outside speakers on other topics.
Here is some background material on DAHA's:
- Lecture notes
on Cherednik algebras by Pavel Etingof, Xiaoguang Ma.
- Calogero-Moser systems and representation theory by
This is a short book taken from a lecture course. It overlaps
substantially with the previous notes. Tom Braden has two copies and
can loan one of his copies out.
reflection algebras by Iain Gordon.
This is a nice survey article about the more general class of
symplectic reflection algebras, but it spends a lot of time
on the Cherednik/DAHA case.
- Tuesday, February 15. Alexei Oblomkov (UMass)
friends of the symmetric group: buzzwords introduction
- Tuesday, February 22. Alexei Oblomkov (UMass)
friends of the symmetric group, II
- Tuesday, March 1. Eric Sommers (UMass)
constructions of representations of S_n (after Springer)
- Tuesday, March 8. Tom Braden (UMass)
- March 15: no meeting, spring break.
- Tuesday, March 22. Suho Oh (MIT)
Title : Bruhat order and hyperplane arrangements
We link Schubert varieties in the generalized flag manifolds with
arrangements. For an element of a Weyl group, we construct a certain
graphical hyperplane arrangement. We show that the generating function
regions of this arrangement coincides with the Poincar\'e polynomial of
corresponding Schubert variety if and only if the Schubert variety is
- Tuesday, April 5. Ivan Mirkovic (UMass)
geometric theory of representations of affine Hecke algebras.
- Tuesday, April 12. Zhiwei Yun (MIT)
- Monday, February 22. Eric Sommers (UMass)
Title : Pieces of the exotic nilpotent cone
The exotic nilpotent cone was introduced by Kato. Besides establishing
many of its basic properties, Kato proved an exotic Springer
correspodence using the cone (involving the Weyl group of type C) and
he used the cone to study Hecke algebras with unequal parameters.
This talk will focus on the geometry of the cone, especially its
connection with nilpotent orbits in characteristic two. This is joint
work with Pramod Achar and Anthony Henderson.
- Tuesday, March 2. Daniel Juteau (CNRS and Caen, France)
Title : Parity Sheaves
- Monday, March 8. Peter Tingley (MIT)
Title : Quiver grassmannians, quiver varieties and the
Nakajima quiver varieties are certain moduli spaces that are
useful in geometric representation theory. For instance, they can be
used to give a geometric realization of the crystal for any irreducible
highest weight representation of a symmetric Kac-Moody algebra. It that
story, certain lagrangian sub-varieties that play a crucial role. We
present an alternative description of these lagrangian sub-varieties as
varieties of invariant subspaces of a fixed representation of a path
algebra. We find this ``grassmannian-type" description advantageous,
partly because it makes certain symmetries more evident. This talk is
based on joint work with Alistair Savage, and also incorporates work of
George Lusztig and of Ian Shipman.
- Wednesday, March 24. Ralf Schiffler (UConn) **special day, start
at 3:00 pm** in LGRT 1528 **special room**
Title : Cluster-tilted algebras
Cluster-tilted algebras are endomorphism algebras of tilting
objects in cluster categories. They can also be described as trivial
extensions of tilted algebras and as (a special case of) Jacobian
of quivers with potentials.
In this talk I will mainly explain the relation to the tilted algebras,
also touch the other interpretations.
- Monday, April 12. Ryan Kinser (UConn)
Title : Rank functions on quiver representations
A rank function assigns a nonnegative integer to any quiver
representation, generalizing the notion of the rank of a linear map.
They have algebraic properties similar to the classical rank function
(additive with respect to direct sum, multiplicative with respect to
tensor product, invariant under duality). We will discuss how to
construct a "global rank function" on any quiver (with explicit
examples) and how to derive more rank functions combinatorially.
These can be applied to study the structure of the representation ring
of a quiver. Geometrically, rank functions are not generally
semicontinuous on quiver representation spaces, but we will discuss a
proof that utilizes quiver Grassmannians to show that they are at
- Monday, May 3. Xinwen Zhu (Harvard)
Title: A categorical approach to Parshin reciprocity laws
on algebraic surfaces
Abstract: I will outline an intrinsic proof of Parshin
reciprocity laws for two-dimensional tame
symbols on an algebraic surface, which generalizes the proof of residue
formula on algebraic curves by
Tate and the proof of Weil reciprocity laws by Arbarello, De Concini
and Kac. The key ingredient is to
interpret the 2-dimensional tame symbol as the "commutator" of certain
central extension of a group by
a Picard groupoid. This is a joint work with D. Osipov.
- (**) Monday, September 21. Carl Mautner (Texas) **in LGRT 206**
Title : A Geometric Proof of a Modular Schur-Weyl
Modular representation theory is the study of representation theory
fields of positive characteristic. Generally, modular representation
theory is more complicated than its characteristic zero counterpart.
Recently, a number of theorems have appeared giving geometric
of categories of modular representations. A version of the geometric
Satake theorem due to Mirkovic-Vilonen implies that the modular
representation theory of the general linear group is encoded in a
of geometric objects called perverse sheaves on an affine Grassmannian.
the other hand, a version of Springer theory due to Juteau relates the
modular representation theory of the symmetric group with the geometry
the nilpotent cone in gl_n. This talk will explain how these two pieces
fit together to give a geometric explanation for connections between
modular representation theory of the general linear group and that of
- (**) Wednesday, September 30 at 4:30. Ivan Mirkovic (UMass) **
in LGRT 1634 ** Note special day, time, and room**
Title : Lusztig's conjectures on modular
This is the first of a series of related talks where certain
mechanisms will be applied to representation theory, Langlands
program, algebraic geometry and hopefully to knot invariants and
quantum field theory.
The representation theoretic aspect is a proof of Lusztig's
conjectures which describe
numerical structure of modular representation theory (with
The two key geometric ideas: (i) a construction of Azumaya
in positive characteristic as a tool for math/physics, (ii)
action of the affine braid groups on coherent sheaves on cotangent
bundles of flag varieties.
- Monday, October 5. Ivan Mirkovic (UMass) in LGRT 1322.
Title : Lusztig's conjectures on modular
representations (part II)
- Monday, November 9. Ting Xue (MIT) in LGRT 1322.
Title : Nilpotent orbits in characteristic 2 and
Abstract: Let k and F_q be an algebraically closed and
a finite field of characteristic 2 respectively. Let G be an
adjoint (resp. simply connected) algebraic group of type B,C or
D over k, g the Lie algebra of G and
g^* the dual vector space of g. We
construct the Springer correspondence for g (resp.
g^*) following Lusztig's method. The correspondence is
a bijective map from the set A_g (resp.
A_g^*) to the set of irreducible
characters of the Weyl group of G, where
A_g^*) is the set of all pairs
(c,F) with c a nilpotent G-orbit
in g (resp. g^*) and F an
irreducible G-equivariant local system on c (up to
isomorphism). In particular, we obtain classifications of nilpotent
orbits in orthogonal Lie algebras over F_q and in the
duals of classical Lie algebras over k and
F_q. Finally, we describe the explicit correspondence
using similar combinatorics that appears in the description of
generalized Springer correspondence (defined by Lusztig) for
classical groups in the case of characteristic not equal 2 and
unipotent case in characteristic 2.
- Monday, November 16. Apoorva Khare (Yale University) in LGRT
Title : Faces of polytopes and Koszul algebras
Abstract: Given a simple Lie algebra g and a
finite-dimensional simple g-module V, we study the category G of graded
finite-dimensional modules of the corresponding semidirect product Lie
algebra. This framework includes the truncated current Lie algebras as
well as those associated to folding of complex simple Lie algebras.
Given a face of the polytope formed by the weights of V, we introduce a
partial order on the simple objects in G. For certain finite subsets of
the affine weight lattice, we produce Koszul algebras of finite global
dimension equal to the number of weights of V which are on the face.
This is joint work with Vyjayanthi Chari and Tim Ridenour.
- Monday, November 23. Ryan Reich (Harvard).
Title : Twisted geometric Satake correspondence
- Monday, December 23. Tom Braden (UMass).
Title : Representation theory of hyperplane