Spring 2016

**Organizers:** Dietmar Bisch, Marcel Bischoff, Arnaud Brothier and Ben Hayes

**Fridays, 4:10-5:30pm in SC 1432**

**Ben Hayes, Vanderbilt University**- Title:
**A product formula for Pinsker Factors with application to completely positive entropy** - Abstract:

I will discuss part of ongoing joint work with Lewis Bowen and Brandon Seward. In it, we investigate structural properties of Pinsker factors for sofic entropy, as well as Rohklin entropy. The Pinsker factor is the largest zero entropy. In the nonamenable case, there is a modification of the Pinkser factor called the Outer Pinsker factor, the Outer Pinsker factor defined via extension entropy. Extension entropy turns out to fix many “pathological” monotonicity properties of entropy (namely increase of entropy under factors). We prove useful structural properties for Outer Pinsker factors. These properties imply that certain algebraic actions have completely positive entropy (which means that every factor has positive entropy). In the sofic case, our methods rely on a new property of actions called “the independent microstate lifting property” and in the Rokhlin entropy case they rely on the actions being weakly contained in Bernoulli shifts.

**No Meeting, due to the possibility of inclement weather.**

**Darren Creutz, Vanderbilt University**- Title:
**The Information Theory of Joinings** - Abstract: I will present ongoing research into an area I am developing based on the idea of treating joinings of quasi-invariant actions of groups on probability spaces along similar lines are treating random variables as representing information, in particular I consider the “mutual information” of two spaces in terms of their joinings.
Furstenberg entropy is a numerical measure of how far a quasi-invariant action of a group on a probability space is from measure-preserving. The main new tool I introduce is a relative version of this entropy measuring how far a homomorphism between such spaces is from being relatively measure-preserving.

I show that it enjoys the properties one would expect such as additivity over compositions and apply this notion to develop an Òinformation theoryÓ of joinings proving analogues of many of the key theorems in the information theory of random variables.

**Rudy Rodsphon, Vanderbilt University**- Title:
**Interactions between Operator Algebras and Differential Topology** - Abstract:

TBD

**No Meeting, Colloquium.**

**Yunxiang Ren , Vanderbilt University**- Title:
**TBD** - Abstract: TBD

**Noah Snyder , Indiana University Bloomington**- Title:
**The exceptional knot polynomial** - Abstract: Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding planar algebras fit into continuous familes GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These planar algebras are the ones attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don’t fit into any of the classical families: G_2, F_4, E_6, E_7, and E_8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-paramter family of planar algebras which introduces a variable q, and yields a new exceptional knot polynomial. In joint work with Scott Morrison and Dylan Thurston, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations.

**No Meeting, Spring Break.**

**Thomas Sinclair, Purdue University**- Title:
- Abstract:

- Title:
**Recent work updates** - Abstract:

A handful of speakers in the Subfactor research group will give short presentations on their recent work.

**Marcel Bischoff, Vanderbilt University**- Title:
**Commutative Algebras in Braided Tensor Categories and Quantum Symmetries** - Abstract:

Is there a commutative algebra of dimension 3+\sqrt3? The answer turns out to be yes, if we replace the category of vector spaces by a braided monoidal category. Commutative algebras in braided monoidal categories describe many phenomena including quantum subgroups, braided subfactors of type I, module tensor categories, extensions in chiral conformal field theory and gapped domain walls of topological phases.

I will review the structure of commutative algebras in (unitary) braided fusion categories, give examples, and discuss relations to the Longo-Rehren inclusions.

Ultimatively, I want to introduce a notion and give a classification of certain finite (quantum) actions by Markov maps on noncommutative probability spaces with applications to chiral conformal nets. This gives a generalization of fixed points by group symmetries to what might be called quantum symmetries.

**Joint VUUC Mini-Workshop, 2:10-4 pm SC 1312****2:10-2:40****James Tener, University of California, Santa Barbara**- Title:
**Construction of conformal field theories** - Abstract:

TBD

After defining a canonical “free graph algebra” associated to a weighted graph, I will explain how certain loop algebras from compact quantum metric spaces in the sense of Rieffel. I will then explain which convergence properties hold, and which properties “should” hold. This is joint work with Dave Penneys.

I will talk about the results on unique maximal extensions of certain subalgebras in a free group factor.

**Dave Penneys , University of California, Los Angeles**- Title:
**Bicommutant categories from (multi)fusion categories** - Abstract: I’ll discuss an ongoing joint project with Andr\’e

Henriques. Just as a tensor category is a categorification of a ring,

and its Drinfel’d center is a categorification of the center of a

ring, a bicommutant category is a categorification of a von Neumann

algebra. I’ll define the notion of the commutant C’ of a tensor

category C inside an ambient tensor category B. Since there is a

functor C’ to B, we can then take the bicommutant C” inside B, and C

naturally sits inside C”. Note, however, that C” is not always

equivalent to its bicommutant!Because we are interested in von Neumann algebras, we work in the

ambient category B=Bim(R), the tensor category of bimodules over a

hyperfinite von Neumann factor R, which can be thought of as a

categorification of B(H). Given a unitary fusion category C inside

Bim(R), we identify its bicommutant C”, and we show that C” is a

bicommutant category. This categorifies the theorem by which a finite

dimensional *-algebra that can be faithfully represented on a Hilbert

space is actually a von Neumann algebra.We will also discuss applications, including machinery to construct

elements of C’, the quantum double subfactor, and commutants of

multifusion categories.

**Corey Jones, Vanderbilt University**- Title:
**Quantum isometry groups** - Abstract:

We will discuss quantum isometry groups of various algebraic structures arising in quantum algebra. We then describe a general class of new examples of algebraic structures which admit a quantum isometry groups.