Fall 2018

- Date:
**8/31/18****Cain Edie-Michell, Vanderbilt University**- Title: Classifying small dimension bimodules of the hyperfinite type II_1 factor.
- Abstract: While the classification of modules over the hyperfinite type II_1 factor has long been known, the situation with bimodules is still far from being well understood. One of the few results in this direction is the classification of self-dual bimodules of Von Neumann-Murray dimension less than 2. Such bimodules are classified by the ADET Dynkin diagrams. In recent work I have been able to extend this classification, relaxing the condition that a bimodule M be self-dual, to just requiring that M commutes with its dual under the Connes tensor product. Along with the expected ADET bimodules, we find interesting and exotic new examples, such as a "twisted E_6" bimodule, along with many others.

- Date:
**9/7/18****Pieter Spaas, UC San Diego**- Title: II_1 factors with a unique McDuff decomposition.
- Abstract: We consider McDuff decompositions of II_1 factors. In particular, we will discuss a structural result for such a decomposition, when the central sequences of the non-McDuff part are captured by a Cartan subalgebra. This will allow us to deduce several new instances where the involved II_1 factor admits a unique McDuff decomposition. We will also comment on some related properties and results for equivalence relations.

- Date:
**9/14/18****Ben Hayes, University of Virginia**- Title: Pinsker algebras for 1-bounded entropy.
- Abstract: I will discuss the notion of a Pinkser algebra for 1-bounded entropy (a modification of free entropy dimension for strongly 1-bounded algebras in the sense of Jung). Given a tracial von Neumann algebra M, a Pinsker algebra in M is a subalgebra P of M which is maximal with respect to the property that the 1-bounded entropy of P in M is zero. Such algebras always exist. I will discuss properties of Pinkser algebras, as well as give at least one interesting example of such an algebra, and discuss the difficulties involved in producing more examples.

- Date:
**9/28/18****Noah Snyder, Indiana University**- Title: Module categories, graph planar algebra embeddings, and Extended Haagerup
- Abstract: A natural question that Pinhas Grossman and I have been studying is given a finite collection of finite index N-N bimodules what are all factors M containing N which can be built as a sum of these bimodules. This question is closely related to several "representation theoretic" questions about fusion categories, namely classifying module categories, finding Ocneanu's "maximal atlas", and finding Etingof-Nikshych-Ostrik's "Brauer-Picard groupoid." Unrelated to all of this, Vaughan Jones asked given a fixed subfactor planar algebra P, can you find all bipartite graphs \Gamma such that P embeds into the Graph Planar Algebra of \Gamma. Emily Peters gave partial evidence that for the Haagerup subfactor there were exactly three such graphs (the two principal graphs, and the broom). My main goal in this talk is to explain why these two questions are basically the same as each other. The key result is a GPA embedding theorem for module categories, which says that P embeds in the GPA(\Gamma) if and only if \Gamma is the fusion graph for some module category. In particular, I will show that it follows from my first paper with Pinhas that Emily's three graphs are the only graphs with Haagerup GPA embeddings. We are also able to use this approach to answer all these questions for the Extended Haagerup subfactor, showing that there are two new fusion categories EH3 and EH4 which still appear to be exceptional. This is joint work with Grossman, Morrison, Penneys, and Peters as part of our AIM Square.

- Date:
**10/5/18****Scott Atkinson, Vanderbilt University**- Title: A selective version of Lin's Theorem
- Abstract: Lin's Theorem says that a pair of nearly commuting self-adjoint matrices is near a pair of commuting self-adjoint matrices where "near" is with respect to the operator norm, and the estimates are independent of dimension. Such a statement is subject to many generalizations and variations?we will discuss some that fail and some that hold. To add to the list of affirmative generalizations and variations, we will show that in a finite factor von Neumann algebra, an n-tuple of self-adjoints for which pairs in a certain selection nearly commute is near an n-tuple of self-adjoints for which the pairs from the corresponding selection truly commute; in this case "near" is with respect to the Hilbert-Schmidt norm coming from the trace. To prove this theorem we obtain results on the tracial stability of certain graph products of abelian C*-algebras. Such results apply further to characterize the amenable traces of certain right-angled Artin groups.

- Date:
**10/12/18, 4:00-5:00pm****Corey Jones, Ohio State University**- Title: Vanishing categorical obstructions for permutation orbifolds.
- Abstract: It is an important open question whether all modular tensor categories arise as the DHR categories of completely rational conformal nets. Given a conformal net A and an action of a finite group G by global automorphisms, the fixed point net $A^{G}$ is a new net called the orbifold, whose DHR category is related to the original by a categorical construction called gauging. There is an obstruction to performing gauging at the categorical level which consequently must vanish if the category and its symmetry come from from conformal nets, yielding a potential obstruction to conformal net realizability. It has long been an open question whether these purely categorical obstructions vanish for permutation actions on tensor powers, since these symmetries always exist for tensor powers of nets. We will answer this question affirmatively, using stability properties of symmetric group cohomology. This constitutes a non-trivial test that all modular tensor categories arise from conformal field theories. Based on joint work with Terry Gannon.

- Date:
**10/19/18****No Meeting, Fall Break.**

- Date:
**10/26/18****Josh Edge, Indiana University**- Title: Classification of spin models on Yang-Baxter planar algebras
- Abstract: 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.

- Date:
**Monday 10/29/18, 4:10-5:30pm****Shamindra Ghosh, Indian Statistical Institute**- Title: Oriented extensions of a subfactor planar algebra
- Abstract: Given a finite index subfactor $N \subset M$, one may consider the associated bimodule category which is a rigid semisimple C*-2-category generated by the object $L^2(M)$ as an $N$-$M$-bimodule. If we know that $N$ is isomorphic to $M$, then using the isomorphism we can make the bimodule category sit inside a rigid semisimple C*-tensor category; we will see that this is possible in general (that is, without any isomorphism) using some "free" type construction and the associated subfactor planar algebra. Moreover, if we start with a hyperfinite subfactor, then our constructed C*-tensor category sits as a full subcategory of the $R$-$R$-bimodules. This is a joint work with Corey Jones and Madhav Reddy.

- Date:
**11/2/18, 4:10-5:00pm****Paramita Das, Indian Statistical Institute**- Title: Annular representations of group graded category
- Abstract: Given a diagonal subfactor $N \subset M$, the corresponding $N$-$N$ bimodule category turns out to be equivalent to the tensor category of $G$-graded finite dimensional Hilbert spaces for some group $G$. However, the tensor structure might not be not be strict and the associativity constraint is given by a scalar 3-cocycle of $G$. I will describe the annular representations of this category (which was explicitly computed in my joint work with Dietmar Bisch, Shamindra Ghosh & Narayan Rakshit) and also discuss the effect of cocycle twist on the annular representations in the more general set up of group graded categories (in the work of Bhowmick-Ghosh-Rakshit-Yamashita).

- Date:
**11/9/18****Yasu Kawahigashi, University of Tokyo**- Title: The relative Drinfeld commutants and the relative Verlinde formula
- Abstract: We study the relative Drinfeld commutant of a full subcategory of a unitary fusion category in terms of subfactor theory. We give the relative Verlinde formula for the fusion rules of the relative Drinfeld commutant which interpolates the fusion rules of the original fusion category and the standard Verlinde formula for its Drinfeld center.

- Date:
**11/23/18****No Meeting, Thanksgiving Break.**

- Date:
**11/30/18****Larry Rolen, Vanderbilt University**- Title: Locally harmonic Maass forms and central $L$-values
- Abstract: In this talk, we will discuss a relatively new modular-type object known as a locally harmonic Maass form. We will discuss recent joint work with Ehlen, Guerzhoy, and Kane with applications to the theory of $L$-functions. In particular, we find finite formulas for certain twisted central $L$-values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. Applications to the congruent number problem will be given.

- End of Fall Semester.

Some related conferences/workshops this semester: