—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Fall 2017

Wednesdays, 3:45–4:35. Location TBA

Webpages from previous semesters

Organizers:  Emily Peters, Peter Tingley

Peter Tingley (Loyola) When to hold 'em
Andrew Carroll (de Paul) Geometric techniques in representation theory of biserial algebras
Daniele Rosso (Indiana University Northwest)
Irreducible components of exotic Springer fibers
Bin Gui
Sahana Balasubramanya (Vandebilt)

Nancy Scherich (Santa Barbara)
1 Ben Salisbury (Central Michigan University) Rigged configurations and the infinity crystal
Jason Gaddis

No talk (thankgiving)
27 *Monday*
Robert Muth (Tarleton state university)
6 Maybe Christopher Perez


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

Parking is available on-campus 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.


September 13: Peter Tingley. When to hold 'em

I will discuss a fun project I did with Kaity Parsons and Emma Zajdela, two of our former undergraduates: We worked out optimal play in some interesting simplified poker games. This involves a lot of algebra is the pre-calculus sense of the word, although not much so much of the crazy algebra we usually talk about. But it was a lot of fun, and the results worked out really nicely. Here is the abstract of the resulting paper:
We consider the age old question: how do I win my fortune at poker? First a disclaimer: this is a poor career choice. But thinking about it involves some great math! Of course you need to know how good your hand is, which involves some super-fun counting and probability, but you are still left with questions: Should I hold 'em? Should I fold 'em? Should I bet all my money? It can be pretty hard to decide! Here we get some insight into these questions by thinking about simplified games. Along the way we introduce some ideas from game theory, including the idea of Nash equilibrium.

September 20: Andrew Carroll. Geometric techniques in representation theory of biserial algebras

A key technique in representation theory of associative algebras has been the study of associated geometric spaces, in particular, varieties whose points parameterize all representations of a common dimension (or dimension vector) over a given algebra. Isomorphism of representations can then be encoded as a group action under a product of general linear groups, which allows us to cast new questions in terms of invariant theory. One bold (and yet unsolved) question concerns the interaction between complexity of certain geometric quotients (known as moduli spaces) and the representation type of the algebra. (It has been shown, for example, that arbitrary projective varieties can arise as moduli spaces in general.) In this talk, I'll discuss some of the history of this point of view in the context of quivers with relations, and some satisfying results in the direction of answering the bold question above: Moduli spaces for many tame algebras appear to be products of projective spaces!

September 27: Daniele Rosso. Irreducible components of exotic Springer fibers

The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a reductive Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type C (Symplectic group). To make the symplectic case look more like the Type A case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux and obtain an exotic Robinson-Schensted correspondence. This is joint work with Vinoth Nandakumar and Neil Saunders.

November 1: Ben Salisbury. Rigged configurations and the infinity crystal

The crystal B(\infty) is a combinatorial skeleton of the negative half of the quantum group, and its importance in the theory of crystal bases has been highlighted since Kashiwara’s original papers on the subject. Since then, many combinatorial models for this crystal have been developed (i.e., tableaux, MV polytopes, quiver varieties, modified Nakajima monomials, etc). In this talk, we introduce yet another model; one that is uniform across all symmetrizable types. Our new model is a collection of rigged configurations, which are multipartitions whose parts are “rigged” with, or labeled by, integers. The connection between our model and the marginally large tableaux model will be discussed, as well as the calculation of the *-involution. This is joint work with Travis Scrimshaw.