Organizers: Aaron Lauve, Peter Tingley
January  
14 
No talk 

21 
No talk 

28 
No talk 

February  
4 
Alexander Stolin (Gothenburg) 
Classification of quantum groups and Lie bialgebras 
11 ***IES 123*** 
Peter Tingley 
Crystals, PBW bases, and related combinatorics. 
18 
No talk 

25 ***IES 123*** 
Josh Hallam (Michigan State) 
Increasing Forests 
March  
4  No talk (spring break)  
11 
Shamil Shakirov (Berkeley) 
TBA 
18 


25 
Tony Giaquinto (Loyola) 

April  
1 (near Easter) 

8 
Andrew Carroll (DePaul) 

15 

22 
Stefan Catoiu (DePaul University) 
TBA 
29 
Christophe Hohlweg (Universite du Quebec a Montreal) 

Directions 

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)
Parking is available oncampus for $7 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 

Given an arbitrary field F of characteristic 0, we study Lie bialgebra structures on sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure, the classical double is isomorphic to the tensor product of sl(n,F) with either F[e]/e^2, F+F, or a quadratic field extension of F. In the first case, the classification leads to quasiFrobenius Lie subalgebras of sl(n,F). In the second and third cases, a BelavinDrinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The BelavinDrinfeld untwisted and twisted cohomology sets associated to an rmatrix are computed. For the CremmerGervais rmatrix in sl(3), we also construct a natural map of sets between the total BelavinDrinfeld twisted cohomology set and the Brauer group of the field F.
Feb 11: Peter Tingley (Loyola), Crystals, PBW bases, and related combinatorics.
Kashiwara's crystals are combinatorial objects used to study complex simple Lie groups and their highest weight representations. A crystal consists of an underlying set, which roughly indexes a basis for a representation, along with combinatorial operations related to the Chevalley generators. Here we are interested in types A and D, where the crystals are realized using Young tableaux and KashiwaraNakashima tableaux respectively. However the combinatorics is not always enough, and one would like to understand an actual basis corresponding to the crystal. This can be done using Lusztig's theory of PBW bases, and so there is a unique structurepreserving bijection between the PBW basis and the combinatorially constructed crystal. We describe that bijection explicitly. In type A the answer was to some extent known to experts, but in type D we believe it is new. I will explain as much of this as I can without assuming any prior knowledge of crystals or PBW bases, mainly through examples. This is joint work with two Loyola students (John Claxton and Adam Shultze), and also with Ben Salisbury.
Feb 25: Joshua Hallam (Michigan State), Increasing Forests
Let G be a graph with vertex set {1,2,...,n}.
We say a subtree of G is increasing if the labels along any path
starting at the minimum vertex increase. Moreover, we call a
subforest of G increasing if each of its components is an increasing tree.
We will show that the generating function for increasing forests of graph
always factors with nonnegative integer roots. We also give a
characterization of when the generating function is equal to the
graph's chromatic polynomial. We finish by generalizing this work to
simplicial complexes.
This joint work with Jeremy Martin and Bruce Sagan.