Organizers: Stephen Doty, Tony Giaquinto, Aaron Lauve, Peter Tingley
February  
6 
Aaron Lauve (Loyola University Chicago) 
Convolution powers of the identity in graded Hopf algebras 
13 
No talk 

20 
No talk 

27 
Jenny Wilson (Chicago)

FImodules and stability phenomena of representations of the classical Weyl groups 
March  
6 
Spring break 

13 
Amy Pang (Stanford) 
Card Shuffling and other Hopfpower Markov Chains 
20 
Alexander Yong (UIUC) 
Varieties in flag manifolds and their patch ideals 
27 
Alexander Ellis (Columbia) 
Odd symmetric functions and odd categorified quantum sl(2) 
April  
3 
Tom Halverson (Macalester College) 
Groups, Representation Graphs, and Centralizer Algebras 
10 
Canceled 

17 
Dave Penneys (Toronto)  Planar algebras 
24 
Tony Giaquinto 
On the cohomology of the Weyl algebra, the quantum plane, and the qWeyl algebra 
Directions 

Loyola Hall is located at 1110 W. Loyola Avenue, Chicago, IL (map)
Public parking available oncampus in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.
Abstracts 


Feb 6, Aaron Lauve (Loyola) 4:00PM: Convolution powers of the identity in graded Hopf algebras
We answer the question, "what are the eigenvalues of the kth convolution power
of the identity operator acting on a graded Hopf algebra?"... well, the answer
is (+1/1). So I'll spend most of the time saying things about multiplicity
and implications. Additionally, I will connect it to the classical "indicators"
of Frobenius and Schur, which help classify representations of finite groups.
(These were reformulated in the language of Hopf algebras by Susan Montgomery
and coauthors.) This is joint work with Marcelo Aguiar.

Feb 27, Jenny Wilson (U. of Chicago) 4:00pm: FImodules and stability phenomena of representations of the classical Weyl groups Last year, Church, Ellenberg, and Farb developed a new framework for studying sequences of representations of the symmetric groups, using a concept they call an FImodule. I will give an overview of this theory, and describe how it generalizes to sequences of representations of the classical Weyl groups in type B/C and D. The theory of FImodules has provided a wealth of new results by numerous authors working in algebra, geometry, and topology. I will outline some of these results, including applications to coinvariant algebras, and the cohomology of configurations spaces and hyperplane complements. 
A Hopfpower Markov chain is a random walk where the transition probabilities are given by the coproductthenproduct operator on a combinatorial Hopf algebra. Key examples include the GilbertShannonReeds model of riffleshuffling of a deck of cards, a model of rockbreaking, and the restrictiontheninduction of representations of the symmetric group. I'll give the general definition of these processes, and explain how Hopf algebra structure theory, such as the CartierMilnorMoore and PoincareBirkhoffWitt theorems, give the stationary distributions and other information about their long term behaviour. This is a generalisation of joint work with Persi Diaconis and Arun Ram 
This talk addresses the problem of how to analyze and discuss singularities of a variety X that ``naturally'' sits inside a flag manifold. Our three main examples are Schubert varieties, Richardson varieties and Peterson varieties. The overarching theme is to use combinatorics and commutative algebra to study the *patch ideals*, which encode local coordinates and equations of X. Thereby, we obtain formulas and conjectures about X's invariants. We will report on projects with (subsets of) Erik Insko (Florida Gulf Coast U.), Allen Knutson (Cornell), Li Li (Oakland University) and Alexander Woo (U. Idaho). 
We introduce odd analogues of the symmetric functions, the nilHecke algebra, and the cohomology of Grassmannians. These algebras are used in constructing an odd categorification of quantum sl(2) and, conjecturally, odd Khovanov homology. By work of Hill and Wang, there is a relation to the categorification of KacMoody superalgebras as well. Joint with Mikhail Khovanov and Aaron Lauda. 
For a group G and a Gmodule V over the complex numbers, we construct a representation graph \Gamma with vertices labeled by the irreducible Gmodules and edges given by the rule for tensoring by V. We show that many of the graphtheoretic properties of \Gamma come from the representationtheoretic properties of G (or vice versa). We then construct the tensor power centralizer algebra Z of endomorphisms that commute with G on the kfold tensor product of V. We see that the representation theory of Z is also controlled by the graph \Gamma. We will observe the examples that lead to Z being the symmetric group, the Brauer algebra, the partition algebra, and the TemperleyLieb algebra. We then study the special case where G is a finite subgroup of SU(2), and via the McKay correspondence the graph is one of the extended affine Dynkin diagrams. A beautiful combinatorial representation theory emerges.
I will give an overview of Jones' planar algebras, with attention to specific examples. I will not assume any familiarity with the subject beyond some basic linear algebra. I will then explain how planar algebras are useful in various areas of mathematics, including knot theory, graph theory, and tensor categories.
Deformation theory can be used to compute the cohomology of a deformed algebra with coefficients in itself from that of the original. Using the invariance of the EulerPoincare characteristic under deformation, it is applied here to compute the cohomology of the Weyl algebra, the algebra of the quantum plane, and the qWeyl algebra. The behavior of the cohomology when q is a root of unity may encode some number theoretic information.
