—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Fall 2017

Wednesdays, 3:45–4:35. Location TBA

Organizers:  Emily Peters, Peter Tingley

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.

October 4: Bin Gui. A unitary tensor product theory for unitary representations of unitary vertex operator algebras

Vertex operator algebra (VOA) is a mathematical formulation of chiral conformal field theory. Given a nice rational VOA V, Huang and Lepowsky constructed a braided tensor category Rep(V) of the representations of V, and showed later that Rep(V) is in fact a modular tensor category. In this talk, we study VOAs and their tensor categories from a unitary point of view. Assume that V is a unitary VOA, and that all the representations of V are unitarizable. We define a unitary structure on the tensor product of two unitary representations of V, and show that Rep(V) becomes a unitary modular tensor category.

October 11: Sahana Balasubramanya. Acylindrical group actions on quasi-trees

A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph is a (non-elementary) quasi-tree and the action of G on the Cayley graph is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As a by-product, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.

October 25: Nancy Scherich. An Application of Salem Numbers to Braid Group Representations

Many well known braid group representations have a parameter. I will show how to carefully choose evaluations of the parameter to force the representation to land in a lattice. It is surprising and exciting to see how careful algebraic constructions can lead to geometric results.

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.

November 8: Jason Gaddis. Auslander's Theorem for permutation actions on noncommutative algebras

Let G be a small finite group acting linearly on a polynomial ring A over an algebraically closed field of characteristic zero. A famous theorem of Auslander asserts that the skew group algebra A#G is isomorphic to the algebra of endomorphisms of A over the fixed ring of A by G. This result is intimately connected to the McKay correspondence and the study of graded isolated singularities. Two natural generalizations of this problem are to replace A with a noncommutative algebra or G with a Hopf algebra. Until recently there has been progress only when A has low (global) dimension. In this talk, I will discuss some of the history of this problem, the pertinency invariant developed by Bao, He, and Zhang, as well as joint work with Kirkman, Moore, and Won in extending Auslander's Theorem to permutation actions on certain noncommutative algebras.

November 15: Jackson Criswell. PBW bases and marginally large tableaux in types Bn and Cn

The original development of the infinity crystal involved an elaborate algebraic construction. Since the original papers by Kashiwara on the subject, these descriptions have been adapted to use various combinatorial objects. Since all of these constructions represent the same crystal, it is desirable to establish crystal-preserving bijections between these combinatorial objects. The two realizations discussed here are described in terms of marginally large tableaux in one case, and Kostant partitions in the other. In this work, for of type Bn or Cn, an explicit description of the isomorphism between the two realizations is given. We also present a stack notation for Kostant partitions which simplifies that realization.

November 29: Robert Muth. Schurifying superalgebras

I will describe how to "Schurify" a finite dimensional superalgebra. Some important algebras arise in this way; the classical Schur algebra is the Schurification of its ground field, and blocks of Hecke algebras were recently shown by Evseev and Kleshchev to be derived equivalent to Schurified zigzag algebras. Many nice properties of superalgebras are preserved under Schurification; in particular I will show that if a superalgebra is quasihereditary/cellular, then (under some restraints), its Schurification inherits that structure as well, and the decomposition numbers of standard modules can be conveniently described in terms of those of the original superalgebra and the classical Schur algebra. This is joint work with Alexander Kleshchev.

December 6: Chris Perez. The elementary theory of groups

The elementary theory of a group G is the set Th(G) of all boolean combinations of sentences in G using existential and universal quantifiers, i.e. the first-order theory of G. In 2006, Sela proved that any two finitely generated, non-abelian free groups have the same elementary theory, solving a problem first posed by Tarski in 1945.

The geometric approach used by Sela to solve Tarski's problem enabled him to apply the same techniques to study the elementary theory of torsion-free hyperbolic groups. He showed in 2009 that, given a fixed torsion-free hyperbolic group H, if G is a finitely generated group such that Th(G)=Th(H), then G is a torsion-free hyperbolic group (not necessarily isomorphic to H). The implication of such a result is that elementary theories are capable of detecting the geometric properties of groups.

In the past several years, geometric group theorists have been working to generalize the methods of Sela's program and apply them to other problems. In this talk, I will discuss some of these techniques and how I am applying them to study the elementary theory of toral relatively hyperbolic groups.