Upcoming
Seminars 2013

Tuesday
June 25
3:00 p.m.
Room 332 
Peter Crooks
Weyl Group Representations on the Homology of Springer Fibres
Using the convolution algebra framework, we will construct Weyl group
representations on the topdegree BorelMoore homology of each Springer
fibre. Some emphasis will be placed on examples arising in the case
of sln. We will also consider these Weyl group representations in
the context of the Springer Correspondence.

Past Seminars
201213 
Tuesday
June 18
3:00 p.m.
Room 332

Peter Samuelson
In this talk we'll define BorelMoore homology and give some basic
properties. We'll also describe the basic setup of convolution algebras
and give some simple examples. Then we'll discuss the convolution
algebra associated to the Steinberg variety (at least when g = sl_n).

Tuesday
June 11
3:00 p.m.
*Revised location Fields Institute, Room 332

Peter Crooks
An Introduction to Springer Theory
Let G be a connected, simplyconnected complex semisimple linear
algebraic group with Lie algebra g. One may construct a natural resolution
of the (singular) nilpotent cone of g, called the Springer resolution.
This has a realization in symplectic geometry as a moment map for
a Hamiltonian action of G on the cotangent bundle of the flag variety
of g. The fibres of the Springer resolution, called Springer fibres,
are of particular interest. The topdegree BorelMoore homology of
each fibre carries a representation of the Weyl group, W. The construction
of this representation is somewhat geometric in nature, as it involves
identifying the group algebra of W with a subalgebra of the BorelMoore
homology of the Steinberg variety. It turns out that one can obtain
every complex irreducible Wmodule as a certain summand of the top
BorelMoore homology of a Springer fibre. This is part of the Springer
correspondence.
This talk will begin with a survey of a few ideas that will be fundamental
to our studying Springer Theory. Also, we will consider several examples
arising in the case G=SLn.

Thursday
June 6, 4 p.m.
Room 230

Chris Dodd 
Tuesday
June 4
3 p.m.
Room 230

Chris Dodd
Localization for Modular Representations and Lusztig's conjecture
This is an expository talk about the work of Bezrukavnikov, Mirkovic,
and Ruminyin about localization for a semisimple lie algebra in positive
characteristic and its application to understanding the category of
representations. I'll start from scratch with modular reprsentations,
talk about localization (in general), and explain how the affine hecke
algebra works its way into the picture. Probably much of this will
spill into the second week.

Tuesday
May 28
11 a.m.
Room 230

Chris Dodd
Localization for Modular Representations and Lusztig's conjecture
This is an expository talk about the work of Bezrukavnikov, Mirkovic,
and Ruminyin about localization for a semisimple lie algebra in positive
characteristic and its application to understanding the category of
representations. I'll start from scratch with modular reprsentations,
talk about localization (in general), and explain how the affine hecke
algebra works its way into the picture. Probably much of this will
spill into the second week.

Tuesday
May 21
11 a.m.
Stewart Library

Chris Dodd
Localization for Modular Representations and Lusztig's conjecture
This is an expository talk about the work of Bezrukavnikov, Mirkovic,
and Ruminyin about localization for a semisimple lie algebra in positive
characteristic and its application to understanding the category of
representations. I'll start from scratch with modular reprsentations,
talk about localization (in general), and explain how the affine hecke
algebra works its way into the picture. Probably much of this will
spill into the second week.

Tuesday
May 14
3 p.m.
Stewart Library

Chris Dodd
Localization for Modular Representations and Lusztig's conjecture
This is an expository talk about the work of Bezrukavnikov, Mirkovic,
and Ruminyin about localization for a semisimple lie algebra in positive
characteristic and its application to understanding the category of
representations. I'll start from scratch with modular reprsentations,
talk about localization (in general), and explain how the affine hecke
algebra works its way into the picture. Probably much of this will
spill into the second week.

Tuesday
May 7
3 p.m.
Stewart Library

Joel Kamnitzer
Categorification of Lie algebras [d'apres Rouquier, KhovanovLauda,
...]
Given a vector space with an action of a semisimple Lie algebra,
we can try to "categorify" this representation, which means
finding a category where the generators of the Lie algebra act by
functors. Such categorical representations arise naturally in geometric
representation theory. A framework for studying these categorical
representations was introduced by Rouquier and KhovanovLauda. Their
definitions are algebraic/combinatorial, but are connected to the
topology of quiver varieties by the work of VaragnoloVasserot.

Thursday
May 2
3 p.m.
Stewart Library

Philippe Humbert, University of Strasbourg
Beak diagrams and surface tangles
The fact that the category of tangles can be "algebraicized"
via its braided monoidal structure has played a fundamental role in
the theory of quantum and finitetype invariants.
How can this algebraization be generalized to tangles lying in a cylinder
over an arbitrary surface? In this talk I will present one of the
possible answers. Even though the approach may not be the most natural
one, the good point is that it is based on a rather explicit description
of surface tangles (by some kind of planar diagrams that I call "beak
diagrams").

Thursday
Apr. 11
3 p.m.
Stewart Library

Kiumars Kaveh, University of Pittsburgh
Asymptotic behavior of multiplicities of reductive group actions
We consider the action of a connected reductive algebraic group G
on the graded algebra A of sections of a line bundle on a projective
variety X. The asymptotic of multiplicities of irreducible representations
appearing in A is related to the DuistermaatHeckman function and
RiemannRoch theorem for multiplicities due to GuilleminSternberg,
Meinrenken and others.
For a given representation $\lambda$ let $m_{k, \lambda}$ denote the
multiplicity of $\lambda$ appearing in the kth degree piece A_k .
We describe the asymptotic behavior of $m_{k, \lambda}$ as k goes
to infinity. Our methods have elementary convex geometric nature and
use the theory of NewtonOkounkov bodies. This work recovers and extends
some previous results and of Brion and Paoletti who obtain similar
results using ReimannRoch theorem for multiplicities.This is a joint
work with Takuya Murata.

Thursday
Apr. 4
3 p.m.
Stewart Library

Stephen Morgan, University of Toronto
Quantum Hamiltonian reduction of Walgebras
A Walgebra is an algebraic structure constructed from a universal
enveloping algebra and a nilpotent element of the underlying Lie algebra;
more precisely it can be constructed by reducing the universal enveloping
algebra in a manner analogous to Hamiltonian reduction of Poisson
varieties, known as quantum Hamiltonian reduction. In fact, Walgebras
form a quantisation of the ring of functions on the appropriate Slodowy
slices corresponding to the chosen nilpotent elements. More generally,
we will show that Walgebras corresponding to more regular nilpotent
elements can be obtained from more singular Walgebras using quantum
Hamiltonian reduction, and mention some applications this has to categorification
of tensor products of simple representations of sl2.

Thursday
Mar. 21
3 p.m.
Room 332

Lucy Zhang, Perimeter Institute/ University of Toronto
A classification of SET phases by Gextensions of spherical fusion
categories
We introduce the concept of topological phases and symmetry enriched
topological (SET) phases. We augment the physical concepts with the
study of Ggraded fusion categories. In this talk, we will explore
my work in progress with XiaoGang Wen on classifying the SET

Thursday
Mar. 14
3 p.m.
Stewart Library

Kirill Zaynullin, University of Ottawa
Formal group laws, Hecke algebras and Oriented cohomology theories
We generalize the construction of the nil Hecke ring of KostantKumar
to the context of an arbitrary algebraic oriented cohomology theory
of LevineMorel (e.g. Chow groups, Grothendieck's K_0, connective
Ktheory and algebraic cobordism). The resulting object, called a
formal (affine) Demazure algebra, is parameterized by a onedimensional
commutative formal group law and has the following important property:
specialization to the additive and multiplicative periodic formal
group laws yields completions of the nil Hecke and the 0Hecke rings
respectively. We also introduce a deformed version of the formal (affine)
Demazure algebra, which we call a formal (affine) Hecke algebra. We
show that the specialization of the formal (affine) Hecke algebra
to the additive and multiplicative periodic formal group laws gives
completions of the degenerate (affine) Hecke algebra and the usual
(affine) Hecke algebra respectively. We apply it to construct an algebraic
model of the Tequivariant algebraic oriented cohomology of the variety
of complete flags. The talk is based on two recent preprints arXiv:1208.4114,
arXiv:1209.1676 and the paper [Invariants, torsion indices and cohomology
of complete flags. Ann. Sci. Ecole Norm. Sup. (4) 46 (2013), no.3.].

Thursday
Feb. 28
3 p.m.
Stewart Library

XiaoGang Wen, Perimeter Institute
The mathematical languages for patterns of quantum entanglement
Some quantum phases of matter are described different patterns of
quantum entanglement. The patterns of quantum entanglement are new
phenomena that happen in nature. But what kind of mathematical language
do we use to describe quantum entanglement. Here I would like to explain
that fusion category theory and group cohomology theory are nature
languages to describe various patterns of quantum entanglement. As
a result, we can use fusion category theory and group cohomology theory
to classify new quantum states of matter.

Thursday,
Feb. 7
3 p.m.
Stewart Library

Michael McBreen, Princeton
Quantum cohomology of hypertoric spaces, Bethe equations and mirror
formulae
A hypertoric variety is a hyperkahler analogue of a toric variety.
I will discuss joint work with Daniel Shenfeld which computes the
equivariant quantum cohomology ring of any smooth hypertoric variety,
and provides integral formulas for the associated quantum differential
equation. One could optimistically view these formulas as a form of
mirror symmetry for the reduced genus 0 GromovWitten theory of hypertoric
spaces.
Time permitting, I will also sketch possible applications to the quantum
cohomology of Nakajima quiver varieties and the Bethe equations of
various finite dimensional quantum integrable systems, or alternatively
the representation theory of the Yangian.

Thursday,
Jan 31, 3pm
Stewart Library 
TBA

Thursday,
Jan 24, 3pm
Stewart Library 
Andre Henriques (Utrecht)
Extended conformal field theories
A conformal field theory is a functor from the cobordism category
of Riemann surfaces with boundary to the category of Hilbert spaces.
Generalizing that idea, an extended conformal field theory is a 2functor
from a 2category of Riemann surfaces with corners to an appropriate
target 2category that I'll describe. I'll explain how to
(partially) construct extended conformal field theories.

Thursday,
Jan 17, 3pm
Stewart Library 
Bhairav Singh (MIT)
Twisted Geometric Satake Equivalence
One of the fundamental results in geometric representation theory
is the geometric Satake equivalence due to Lusztig, Ginzburg, MirkovicVilonen,
and BeilinsonDrinfeld, betweenequivariant perverse sheaves on the
affine Grassmannian of a reductive group G and representations of
of its Langlands dual group GV . Recently, FinkelbergLysenko gave
a similar description for categories of monodromic perverse sheaves
on the determinant line bundle of the affine Grassmannian. We will
give a motivated introduction to the work of FinkelbergLysenko, and
explain how some results in geometric representation theory generalize
to themonodromic setting.

January 10, 2013
Room 230 
Chris Dodd, University of Toronto
Modules over Algebraic Quantizations and representation theory
Recently, there has been a great deal of interest in the theory of
modules over algebraic quantizations of socalled symplectic resolutions.
In this talk I'll discuss some new joint work that open the door to
giving a geometric description to certain categories of such modules;
generalizing classical theorems of Kashiwara and Bernstein in the
case of Dmodules on an algebraic variety.

Nov. 30*
11:00 a.m.
Room 210
*nonstandard time and place

Stavros Garoufalidis (Georgia Tech)
The colored HOMFLY polynomial is qholonomic
We will prove that the colored HOMFLY polynomial of links colored
by symmetric powers representation is qholonomic. This fulfils some
of the wishes of physics, has ramifications on the the SL(2,C) character
variety of knots and motivates questions on the "web" approach
of representation theory.

Nov. 29

Joel Kamnitzer (Toronto)
Crystals, geometric Satake, and the nonnegative tropical flag variety
IV

Nov. 22

Joel Kamnitzer (Toronto)
Crystals, geometric Satake, and the nonnegative tropical flag variety
III

Nov.8

Adam Sikora (University of Buffalo)
Character varieties as completely integrable systems
We will give an introduction to character varieties. It is known
that SL2character varieties of surfaces are completely integrable.
We prove an analogous statement for rank 2 Lie groups. Time permitting,
we will discuss the relevance of this result to quantum invariants
of WittenReshetikhinTuraev.

Nov.15

Oded Yacobi (Toronto)
Yangians and quantizations of slices in the affine grassmannian
We study slices to Schubert varieties in the affine Grassmannian,
which arise naturally in the context of geometric representation theory.
These slices carry a natural Poisson structure, and our main result
is a quantization of these slices using subquotients of quantum groups
called Yangians. We discuss also conjectural applications of these
results to categorical representation theory. This is based on joint
work with Joel Kamnitzer, Ben Webster, and Alex Weekes.

Nov.1

Joel Kamnizter (Toronto)
Crystals, geometric Satake, and the nonnegative tropical flag variety
II

Friday Oct. 26, 2012
Room 210
3:30 p.m.
*Please note nonstandard date and time

Allen Knutson (Cornell University)
Combinatorial rules for branching to symmetric subgroups
Given a pair G>K of compact connected Lie groups, and a dominant
Gweightlambda, it is easy to use character theory to say how the
irrep V_lambda decomposes as a Krepresentation. If G = K x K, this
is tensor product decomposition, for which we have an enumerative
formula: the constituents can be counted as a number of Littelmann
paths or MV polytopes.
I'll give a positive formula in the more general case that K is a
symmetric subgroup of G, i.e., the (identity component of) the fixedpoint
set of an involution. The combinatorics is controlled by the poset
of Korbits on the flag manifold G/B, which reduces to the Bruhat
order in the case G = K x K. I can prove this formula in the asymptotic
(or, symplectic) situation replacing lambda by a large multiple, and
nonasymptotically for certain pairs (G,K).
This is a joint meeting with the Algebraic Combinatorics Seminar.

Oct. 18

No Seminar
David Nadler (Northwestern University) will be giving a talk 'Traces
and loops' as part of the Fields
Symposium at 2:45 p.m. in Room 230 of the Fields Institute.

Oct. 19
Bahen Centre, Room 4010
4:00 p.m.
*Please note change in time

Masoud Kamgar (MPI, Bonn)
Ramified Satake Isomorphisms
I will explain how to associate a Sataketype isomorphism to certain
characters of the compact torus of a split reductive group over a local
field. I will then discuss the geometric analogue of this isomorphism
and its possible applications. (Joint work with T. Schedler).

Oct. 11

Joel Kamnitzer (Toronto)
Crystals, geometric Satake, and nonnegative tropical flag variety
Crystals are a combinatorial model for studying the representation
theory of reductive groups. In 2004, A. Henriques and I proved that
the octahedron recurrence controls the category of GL_ncrystals.
I will explain how this result can be generalized to any reductive
group G, via configurations of points in the affine Grassmannian and
the nonnegative tropical flag variety.

Oct.4

Josh Grochow (Toronto)
Introduction to Geometric Complexity Theory
The Geometric Complexity Theory (GCT) program was introduced by Mulmuley
and Sohoni to attack fundamental lower bound problems in computational
complexity theory—such as P vs NP—using algebraic geometry
and representation theory. In addition to presenting the basic structure
of the GCT program, I will discuss some of the intuition behind the
use of representation theory in complexity, as well as how GCT relates
to classical questions in representation theory such as the LittlewoodRichardson
rule for the decomposition of tensor products of representations of
GL_n into irreducibles.

Sept. 27

Dave Penneys (Toronto)
Classifying subfactors
A subfactor is an inclusion of von Neumann algebras with trivial
centers. We use several invariants to classify subfactors, including
the index, the principal graphs, and the standard invariant. The standard
invariant forms a unitary 2category, so the typical planar calculus
gives us the structure of a planar algebra. I will discuss
all these notions, and I will give a brief overview of the classification
program of small index subfactors.

Sept. 20

Peter Samuelson (UofT)
Quantizations of character varieties
If \pi is a finitely generated group, the set $Hom(\pi, SL_2(C))$
has a natural scheme structure. We recall a description of its algebra
of functions $O(\pi)$ and explain that if M is a 3manifold, the Kauffman
bracket skein module $K_q(M)$ gives a quantization of $O(\pi_1(M))$.
If $M = S^3 \ K$ for a knot K, then $K_q(M)$ is a module over (a subalgebra
of) the quantum torus $A_q$ which encodes the colored Jones polynomials
of K. We give an indication of the types of modules that arise from
this construction, and if time permits we'll discuss deformations
of $K_q(M)$ to a module over the double affine Hecke algebra $H_{q,t}$.
(In the talk I intend to define all unfamiliar words in this abstract.
This is work in progress with Yuri Berest.)
