Upcoming Seminars


Ongoing Seminars 
Past Seminars 
March 28, 2014
BA B025 45pm

Dave Penneys
Quantum doubles of fusion categories
I'll go over the construction of the quantum double (or
Drinfel'd center) of a fusion category. We'll then talk
about some applications.

Friday, March 14,
Stewart Library, 4pm 
ChiaCheng Liu
An introduction to Koszul duality
Roughly speaking, the Koszul duality is a derived equivalence
between (graded) module categories of a Koszul ring and
its "Koszul dual". We will
give an introduction to the theory, and then briefly discuss
its application in representation theory of Lie algebras.
The main reference is the 1996 paper Koszul duality patterns
in representation theory by Beilinson, Ginzburg, and Soergel.

Mar. 7 
Alex Weekes
Shift of Argument Algebras, Moduli Spaces, and the Cactus
Group
Shift of argument algebras are a certain family of maximal
commutative algebras of U(g), where g is a semisimple Lie
algebra. They yield some interesting examples as limits,
such as the GelfandTsetlin subalgebras in type A. The moduli
space of these subalgebras is also interesting and has the
cactus group of g as its fundamental group. This naturally
leads to two actions of the cactus group on crystals coming
from representations of g, which we conjecture to be the
same. We will describe these objects and the links between
them.

Feb. 28 
Iva Halacheva
Demazure and KR Crystals
After a general overview of crystals in representation
theory, we will talk about Demazure and KirillovReshetikhin
modules and their crystals. We will then discuss results
by FourierSchillingShimozono and SchillingTingley relating
the two.

Feb. 14
Room 230

Peter Crooks
An Introduction to Quantum Cohomology
Quantum cohomology is a deformation of the cup product
on the integral cohomology ring. I will describe the quantum
cohomology ring in the special case that the space in question
is a partial flag variety. This will require some consideration
of 3point GromovWitten invariants and enumeration problems
in geometry. We will conclude with a computation of the
quantum cohomology ring of complex projective space.
**Please note this semianr will take place in Room 230

Feb. 7 
Jamie Thind
Hall Algebras and Quantum Groups
I'll give an introduction to Hall Algebras. After introducing
the basics, I'll discuss how the Hall algebra of the category
of representations of an A,D,E quiver can be used to construct
(the positive half of) the corresponding quantum group.

Jan. 31 
Peter Samuelson
Rational Cherednik algebras and KZ equations
The rational Cherednik algebras H_k had their origins in
integrable systems and special functions. We give a very
brief historical introduction, and then define Dunkl operators
and the algebra H_k(g) associated to the root system of
a Lie algebra g. We then discuss category O, which shares
many nice properties with the category O for Lie algebras.
Finally, we'll define the KZ functor from category O to
the category of representations of the braid group of type
g. By the end of the talk it may or may not be clear how
(or if) the KZ functor is related to the KZ equations discussed
by Chia Cheng last semester.

Nov. 22 
ChiaCheng
KnizhnikZamolodchikov equations, part II
In this talk I will briefly review the theory of quantum
group associated to a Lie algebra g and its braided tensor
category. Then I will define the KnizhnikZamolodchikov
(KZ) equations and explain how to use the monodromy of KZ
equations to define a nontrivial tensor structure on the
category of finitedimensional representations of g. The
remarkable results of Drinfeld, KazhdanLusztig show that
the above two braided tensor categories are equivalent.

Nov. 15 
ChiaCheng
KnizhnikZamolodchikov equations
In this talk I will briefly review the theory of quantum
group associated to a Lie algebra g and its braided tensor
category. Then I will define the KnizhnikZamolodchikov
(KZ) equations and explain how to use the monodromy of KZ
equations to define a nontrivial tensor structure on the
category of finitedimensional representations of g. The
remarkable results of Drinfeld, KazhdanLusztig show that
the above two braided tensor categories are equivalent.

Nov. 1 
Dinakar Muthiah
On the Bernstein Presentation of the Affine Hecke Algebra
I will discuss two presentations of the affine Hecke algebra.
The first presentation, which is perhaps more familiar,
is the Coxeter (or KacMoody) presentation, which arises
by viewing a loop group as a KacMoody group. The second
presentation is the Bernstein presentation, which arises
by viewing the loop group over a finite field as a type
of padic group and studying its principal series representations.

Oct. 25 
Zsuzsanna Dancso
Bipartite algebras and a categorification of the flow lattice
of graphs
In this talk we will construct an algebra ("bipartite
algebra") associated to a bipartite quiver, and discuss
its representation theory and an application in which we
categorify the flow lattice of graphs. This is joint work
in progress with Anthony Licata.

Oct. 18 
Dave Penneys
TemperleyLieb
I will give a basic introduction to the TemperleyLieb
2category. We will then look at its idempotent completion
(Karoubi envelope), and we will discuss when this 2category
is unitary. Time permitting, we will discuss how some quotient
of TemperleyLieb appears in every rigid tensor category.

Oct. 11 
Iva Halacheva
QuasiHopf algebras
A quasiHopf algebra is a generalization of a Hopf algebra
with the coassociativity condition weakened. We will introduce
some of the theory behind them, including Drinfeld's motivation
for first defining them, and the twisted quantum double
as an example of particular interest.

Oct. 4 
Stephen Morgan
Walgebras and their applications in representation theory
Walgebras are algebraic structures defined from semisimple
Lie algebras along with a specified nilpotent element. Though
originally defined by and of interest to physicists, they
also have applications in representation theory. In particular,
the representation theory of Walgebras is closely related
to the block decomposition of category O. We'll discuss
Walgebras and how they arise through quantisation.

Sept. 27

Alex Weekes
Yangian and their applications
We will look at Yangians, which are quantizations Y(g)
of the universal enveloping algebra of polynomials g[t],
where g is a Lie algebra. We will define these, after reviewing
some background, and look at (some of) their (many) applications.

Sept. 20

Peter Samuelson
Double affine Hecke algebras and Macdonald polynomials
We'll discuss Cherednik's double affine Hecke algebra H(g),
which is a 2parameter quantum algebra associated to a (nice)
Lie algebra g. We'll give some basic properties and some
indications of how Cherednik used this algebra to prove
Macdonald's conjectures. (These involve several properties
of Macdonald polynomials, which are a family of symmetric
orthogonal polynomials associated to g). To keep things
concrete, most of the talk will focus on the case g=sl_2.
