# Activities for Summer 2020

#1: April 22nd
Speaker: David Green (The Ohio State University)
Title: Computer assisted calculations in the classification program for MTC’s.

#2: June 19th
Speaker: Matthew Harper (The Ohio State University)
Title: Presentations of 3-manifolds by Heegaard splittings and surgery links, and their related quantum invariants.

#3: June 26th
Speaker: Peter Huston (The Ohio State University)
Title: Lagrangian algebra objects and gapped boundaries of (2+1)-D topological phases of matter.
Abstract: The idea will be to absorb some of the basic ideas presented in section 3 of this paper (https://arxiv.org/pdf/1109.5558.pdf) by Davydov, Nikshych, and Ostrik, as well as the physical interpretation given in this paper (https://arxiv.org/pdf/1307.8244.pdf).

#4: July 3rd
Speaker: Adrián Franco Rubio (Max Plank Institute)
Title: How (some) physicists think about Conformal Field Theory.
Abstract: Conformal Field Theory (CFT) has become a subject of interest for a certain set of mathematicians (with nonempty intersection with operator algebraists and category theorists). In this informal talk I will give a brief introduction to what a CFT is for many physicists, what language we use to describe it, some easy examples, and why it is interesting to us. The goal is to give the more mathematically oriented audience the chance to talk about CFT from a possibly different perspective than they might be used to.

#5: July 10th
Speaker: Peter Huston (The Ohio State University)
Title: Lagrangian algebra objects and gapped boundaries of (2+1)-D topological phases of matter. II
Abstract: Second part of talk on June 26th.

#6: July 17th
Speaker: Christoph Weis (Oxford University)
Title: What is a tensor category with a compact manifold of simple objects?
Abstract: Fusion categories are tensor categories with finitely many simple objects. There is motivation coming from Conformal Field Theory and Chern-Simons-Theory to study generalizations thereof – tensor categories with a compact manifold of simple objects. Just as fusion categories sometimes behave like finite groups, these “Manifold Tensor Categories” behave like Lie Groups. I will show how one might define these objects, discuss a few fun examples and state a result or two.

#7: July 24th
Speaker: Ramona Wolf (Leibniz Universität Hannover)
Title: Towards a Haagerup CFT
Abstract: It is an open question whether every subfactor has something to do with a conformal field theory. While it is possible to always reconstruct a subfactor from a conformal net, the other direction is more mysterious. So far, only specific examples have been worked out, which do not include the most fascinating ones: exotic subfactors (i.e., subfactors not coming from quantum groups), the simplest one (in some sense) being the Haagerup subfactor. Since I am a physicist, I try to approach the problem with methods from physics: Lattice models and numerical investigations. Since every finite-depth subfactor gives rise to two fusion categories, we can use them to build so-called anyon chains and investigate whether these give rise to a CFT. In this talk, I will present how to construct an anyon chain from a fusion category and tell you what we have learned about a possible Haagerup CFT from numerical studying the Haagerup anyon chain.

#8: July 31st
Speaker: Sergio Girón Pacheco (Oxford University)
Title: Realizations of graded vector space categories over cyclic groups as bimodules over Z.
Abstract: The standard invariant of a subfactor works as an invariant for amenable subfactors of the hyperfinite II_1 factor R, one of its formulations is as a C* tensor category of bimodules over R, it is true that any unitary fusion category can be realized as a full subcategory of bimodules over R. This theory motivates what I will be looking at in this talk, I will give a brief introduction to the classification programme for C* algebras and the Jiang-Su algebra (Z), trying to assume as little knowledge as possible on C*-algebras. Afterwards, I will talk about realizing categories of  graded vector spaces over cyclic groups twisted by a cocycle as bimodules of the Jiang-Su algebra, or also time permitting of UHF-algebras, these algebras can be seen as C* analogues of R. Time permitting I will arrive at the result, clashing with the case of R, that there are no fully faithful functors from these categories when twisted by non-trivial cocycles into Bimodules of Z.

#9: August 7th
Speaker: Quan Chen (The Ohio State University)
Title: standard $\lambda$-lattice, tensor category, module and bimodule
Abstract:

#10: August 14th
Speaker: Qing Zhang (Purdue University)
Title: Modular categories with transitive Galois actions
Abstract: The absolute Galois group acts on the set of isomorphism classes of simple objects of any modular category via the characters of the fusion ring.  In this talk, we consider modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of SL2(Z) associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category C(sl2, p − 2) for some prime p > 3. As a consequence, we completely classify transitive modular categories. This talk is based on joint work with S.-H. Ng and Y. Wang. The paper can be found at https://arxiv.org/abs/2007.01366.
Notes: Modular Categories with Transitive Galois Actions

#11: August 21st
Speaker: Sachin J. Valera (Bergen University)
Title: Skein-theoretic methods for unitary fusion categories
Abstract:  Given a fusion rule $q\otimes q \cong 1\oplus\bigoplus^k_{i=1}x_{i}$ in a unitary fusion category $\C$, we extract information using skein-theoretic methods and a rotation operator. For instance, one can classify all associated framed link invariants when k=1,2 and $\C$ is ribbon. In particular, we will also consider the instances where $q$ is antisymmetrically self-dual. The main exposition will follow from considering the action of the rotation operator on a “canonical basis”. Assuming self-duality of the summands $x_{i}$, we will explore some properties of $F$-symbols. If time permits, we will explore the ramifications of some of these results in a physical setting.
Background reading: our paper in arxiv: https://arxiv.org/abs/2008.07129 and a paper by Morrison-Peters-Snyder (Thms 3.1 & 3.2): https://arxiv.org/abs/1003.0022

#12: August 28th
Speaker: Roberto Hernández Palomares (The Ohio State University)
Title: Representaions of unitary categories over GJS C*-algebras
Abstract: We will construct a representation of an arbitrary countably generated RC*TC onto a subcategory of finitely generated projective Hilbert C*-bimodules over a simple separable unital monotracial C*-algebra, using the diagrammatic methods from Guionnet, Jones and Shlyakhtenko. Out of this category we will construct another functor into the bifinite and spherical bimodules over a type II_1 factor, turning C*-bimodules into honest Hilbert spaces. The composite of these two functors recovers the representation constructed by Brothier, Hartglass and Penneys.
Background reading:  My joint article with Harglass https://arxiv.org/abs/2005.09821