# Schedule of Talks, Abstracts

All talks will be held in the math department, SC 1308 (Stevenson Center, Building 1, room 1308). Talks will start at 9:30am on Saturday, 9/30, and end around 1pm on Sunday, 10/1.

## Schedule of Talks

**Friday, 9/29/17**

**Ralph Kaufmann**will speak in the Subfactor Seminar (4:10pm in SC 1432).

**Saturday, 9/30/17**

Talks start at 9:30am and are 50 minutes long.

**Tobias Osborne**

*Coffee break*

**Zhengwei Liu**

*Lunch break*

Talks start at 2pm and are 50 minutes long.

**Terry Gannon**

**Birgit Kaufmann**

*Coffee break*

**Dylan Thurston**

*Beer & Pizza in SC 1425 (starts after Dylan’s talk)*

**Sunday, 10/1/17**

Talks start at 9:30am and are 50 minutes long.

**James Tener**

*Coffee break*

**Feng Xu**

**Simon Wood**

*End of workshop at around 1pm.*

## Titles and Abstracts

**Terry Gannon, University of Alberta**

*Significant sharpening of the Ocneanu-Schopieray bound for modular*

invariants

invariants

- Abstract: Modular invariant partition functions are the basic combinatorial

data describing bulk conformal field theories, module categories, extensions

of vertex operator algebras, etc. The first and most famous result in this

direction was the 1986 A-D-E classification of affine sl(2) modular

invariants; they are also known for affine sl(3) (mid 1990s). In the early

2000s, Ocneanu gave a bound for affine sl(2) modular invariants and

suggested that it generalizes to all other affine algebras. Recently,

Schopieray (following Ostrik) fleshed out the details for the three rank 2

algebras, and it is clear that his argument indeed generalizes to all affine

algebras. His bound grows exponentially with the number of roots, and is 3

million for sl(3) and around a google for E8. In this paper we use a

different method, obtaining a much sharper bound, which grows with the cube

of the rank. E.g. sl(3) has 38 bad levels, sl(9) has 1202, and E8 has 12476

bad levels. Using the new bounds, the modular invariant classification can

be completed for all classical algebras up to rank 6 as well as G2.

**Birgit Kaufmann, Purdue University**

*Bethe-Ansatz for an SU(3) Hecke quotient*

- Abstract: The Temperly-Lieb (TL) algebra is ubiquitous in mathematical

physics appearing for instance in statistical mechanics in the form of

$U_q(su(2))$ invariant spin chains and the theory of planar algebras.

The TL-algebra can be realized as a quotient of the Hecke algebra for

the A-Coxeter systems. There are other quotients of this Hecke algebra

which are also of interest in many situations. We will discuss a

particular integrable model that arises in physics in the context of

reaction-diffusion systems with several species of particles. This

model can also be viewed a $U_q(su(3))$ invariant spin chain. We

derive Bethe Ansatz equations and use them to find the dynamical

critical exponent in this integrable model. Furthermore there is an

interesting connection to a surface growth model governed by the KPZ

equation, which we will discuss if time permits.

**Zhengwei Liu, Harvard University**

*From subfactors to quantum information and back*

- Abstract: We first talk about the applications of subfactor theory in

quantum information. Inspired by a question in quantum information, we

introduce surface algebras as an extension of planar algebras from the plane

to surfaces. As an application, we give a new proof of the Verlinde formula

for any modular tensor category (MTC) and any genus surface. We also prove

new identities for MTCs including the 6j symbol self-duality. The proof is

based on a recent result joint with Feng Xu, which identifies two different

Fourier dualities in subfactors and MTCs.

**Tobias Osborne, University of Hannover**

*Tensor network representations of spacetime symmetries*

- Abstract: Tensor network models for spacetime have enjoyed considerable recent interest in physics. However, what exactly is a tensor network model for a given spacetime? Certainly we know one when we see one, e.g., the multiscale

entanglement renormalisation ansatz is understood to be a good model for a

(time slice of) anti de Sitter space. But there are still ambiguities if you

want to make things precise. In this talk I attempt to supply a

mathematician-friendly definition of a “tensor network representation of

spacetime”. I’ll argue that a tensor network model is a (projective) unitary

representation of the symmetry group of spacetime via tensor-network unitary

operators. Properties of these representations will be discussed and

connections to Jones’ unitary representations of Thompson’s group T will be

sketched.

**James Tener, UC Santa Barbara**

*Geometry of conformal nets*

- Abstract: The category of representations of a two-dimensional chiral

conformal field theory is thought of as a braided tensor category, but in

fact it has quite a bit more structure. According to Graeme Segal, there

should be a tensor product operation for every complex pair of pants, and

indeed every complex cobordism should produce some operation on this

category. I will describe a construction of Segal CFTs from conformal nets

and a geometric picture underlying certain subfactors constructed from the

nets. This is joint work in progress with Andre Henriques.

**Dylan Thurston, Indiana University**

*Quantum exceptional series*

**Simon Wood, Cardiff University**

*Classifying positive energy modules in conformal field theory*

- Abstract: When attempting to construct a conformal field theory from some

vertex operator algebra, a major first obstacle that needs to be overcome is

the classification of modules. In this talk I will show how free field

algebras and symmetric functions can be used to prove module classification

theorems for some important families of vertex operator algebras.

**Feng Xu, UC Riverside**

*Triality, golden ratio and subfactors*

- Abstract: I will describe some interesting examples of subfactors and their symmetries motivated by reconstruction program.

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