—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Fall 2018

Wednesdays, 3:45–4:35. IES 111

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 5: Ian Le. From Dimers to Webs

Weighted planar bipartite networks in a disc give a way of parameterizing positroid strata. The key to this is Postnikov's boundary measurement map, which matches Plucker coordinates on the Grassmannian with weighted sums over perfect matchings. We give a higher-rank version of the boundary measurement map which sends a network to a linear combination of SL_r webs, and is defined using the r-fold dimer model on the network. The main result is that the higher rank map factors through Postnikov's map. An interesting consequence is that Postnikov networks and webs are in some sense "dual" to each other, so that the moves relating equivalent networks can be translated into relations on the space of SL_r webs. We will explain all the necessary background and give examples along the way. This is joint work with Chris Fraser and Thomas Lam.

September 19: Stephen Doty. The partition algebra and Deligne's category \(\underline{\text{Re}}\text{p}(S_t)\) where \(S_t\) is a symmetric group for \(t \in \mathbb{C}\)

A recent trend in mathematics is a renewed focus on the idea of "categorification", in which some useful mathematical structure is replaced by a catogory that models the original structure in some way, such that the original structure is recovered by taking isomorphism calsses of objects. (This is NOT a precise definition.) For example, the category of Sets categorifies the natural numbers N. The notion of monoidal category (also known as tensor category) is a categorification of monoid. Tensor categories have been studied since MacLane and others in the 1960s, but there is renewed interest in them. Indeed, there is a recent book on the subject.

Deligne (2007) constructed a tensor category \(\underline{\text{Re}}\text{p}(S_t)\), analogous to the category \(\text{Rep}(S_n)\) of complex representations of the symmetric group \(S_n\), except that \(t\) is allowed to be any complex number. Of course, there is no such thing as a symmetric group on \(1.75\), \(\pi\), \(\sqrt{2}\), or \(1+i\) elements! So how can a non-existent thing have representations? I will try to answer this question, using combinatorial gadgets called partition diagrams (which are related to the partition algebra independently discovered by V. Jones and P.P. Martin in the 1990s). This talk is expository and I will say nothing original. I will mainly follow a 2011 paper of Comes and Ostrik.

REFERENCES
1. MIT Seminar on Deligne Categories with loads of links to resources.
2. Blog posts one and two by Akhil Mathew.
3. This post by David Speyer in the Secret Blogging Seminar.

September 26: Josh Edge. Classification of spin models on Yang-Baxter planar algebras

After the discovery of the Jones polynomial in the 1980s, many mathematicians were interested in finding sources for more invariants of knots and links. One promising method pursued by Kauffman, Jaeger, de la Harpe, and Kuperberg among others was via so-called spin models, whose original purpose was to explain magnetism in certain physical models. The classification of such models for the Jones polynomial was first noted by Kauffman in 1986, which Jaeger then generalized to the classification of spin models for the Kauffman polynomial (or BMW algebra) in 1995 by connecting the existence of such a model to the existence of graphs satisfying certain properties. In 2015, Liu finished the classification began by Bisch and Jones of so-called Yang-Baxter planar algebras (YBPAs), planar algebras that satisfy a generalization of the Reidermeister moves. In this talk, we will use the classification of YBPAs to generalize Jaeger's result about spin models of the Kauffman polynomial (which itself is a YBPA) to classify all spin models of Yang-Baxter planar algebras by making a connection to graphs similar to Jaeger. In particular, we will demonstrate that aside from the spin models arising from BMW classified by Jaeger, the only other YBPAs giving spin models are the Bisch-Jones algebra and the Jones polynomial at a discrete sets of values.

October 10: Emily Peters. New categories from extended Haagerup

Very few exotic fusion categories remain exotic for long: the Haagerup fusion categories are now known to be part of a probably infinite family, and the Asaeda-Haagerup fusion categories have recently been shown to be related to the same family. The extended Haagerup categories are among the few certifiably exotic fusion categories that remain. I will talk about the Morita equivalence class of extended Haagerup, and a new tool for calculating Morita equivalences for diagrammatic categories. This is joint work with Grossman, Morrison, Penneys, and Snyder.

October 17: Mihnea Popa (Northwestern). Families of varieties and Hodge theory

Allowing the isomorphism class of algebraic varieties to vary in a family usually imposes strong conditions on the space parametrizing that family. This has been thoroughly studied for families of varieties whose canonical bundle is positive, leading to what is called the hyperbolicity of the moduli stack of such varieties. I will explain how recent advances in Hodge theory and in the study of holomorphic forms, facilitated especially by M. Saito's theory of Hodge modules, allow us to answer questions regarding the base spaces of more general (and conjecturally arbitrary) families of varieties.

November 14: Alexander Stolin (Gotenburg). In Search of New Quantum Groups

A new class of quantum groups will be discussed, including a new quantum group structure on sl(2), (after the Coll-Gerstenhaber-Giaquinto formula).

November 28: Aaron Lauve (Loyola). Rational Series over the Free Group

We discuss the "word problem" for the free skew field. That is, given some rational expression R built from polynomials over a set X of noncommuting variables, determine (effectively) whether or not R = 0. If time permits, we also discuss: (1) a characterization of rational series over the free group, conjectured by Alain Connes; (2) a connection to noncommutative flag varieties; (3) a connection to recent work on computational complexity. This is joint work with Christophe Reutenauer (UQAM), arXiv: 1806.03540.

December 5: Emily Peters (Loyola). New categories from extended Haagerup, part II

This is a continuation of the talk from Oct 10. I will of course assume that you remember everything I said then in complete detail.