Organizers: Emily Peters, Peter Tingley
September  
5 
Ian Le (Perimeter institute)  From Dimers to Webs 
12 
No talk  
19 
Stephen Doty (Loyola) 
The partition algebra and Deligne's category \(\underline{\text{Re}}\text{p}(S_t)\) where \(S_t\) is a symmetric group for \(t \in \mathbb{C}\) 
26 
Josh Edge (Indiana University Bloomington) 

October  
3 
No talk 

10 
Emily Peters 
New categories from extended Haagerup 
17 
Mihnea Popa (Northwestern) 
Families of varieties and Hodge theory 
24 
No talk (conflict with colloquium) 

31 

November  
7  
14 


21 


28 


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 

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 higherrank version of the boundary measurement map which sends a network to a linear combination of SL_r webs, and is defined using the rfold 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 nonexistent 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 YangBaxter 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 socalled 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 socalled YangBaxter 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 YangBaxter 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 BischJones 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 AsaedaHaagerup 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.