## Future Talks

### K-theoretical classification of inductive limit actions of unitary fusion categories on AF-algebras

**Speaker:** Quan Chen (OSU)

**Date:** Jul. 26, 2022

**Abstract:** We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras.
We show that this is a complete invariant for inductive limits of finite dimensional actions of fusion categories on unital AF-algebras.
In particular, this gives a complete invariant for inductive limit actions of finite groups on AF-algebras.
We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF-algebras.
This is joint work with Roberto Hernandez Palomares and Corey Jones.

### The sheaf theory of classical and virtual knots

**Speaker:** Micah Chrisman (OSU)

**Date:** Jul. 19, 2022

**Abstract:**
Virtual knots are typically defined combinatorially.
They are collection of planar diagrams that are considered equivalent up to a finite sequence of moves.
By contrast, knots in the 3-sphere can be defined geometrically. They are the points of a space of knots.
The space has a topology so that equivalent knots lie in the same path component.
Here we will give geometric construction of virtual knots using sheaf theory.
We define a site $(VK,J_{VK})$ so that the category $Sh(VK,J_{VK})$ of sheaves on this site can be naturally interpreted as the “space of virtual knots”.
The points of this Grothendieck topos correspond exactly to virtual knots.
An equivalence of virtual knots corresponds to a path in this space, or more precisely, a geometric morphism $Sh([0,1]) \to Sh(VK,J_{VK})$.
Many other combinatorial concepts in virtual knot theory can likewise be given a geometric reformulation using the language of sheaves.

### From C*-categories to W*-categories

**Speaker:** Giovanni Ferrer (OSU)

**Date:** Jun. 28, 2022

**Abstract:**
In this talk, we will construct the free W*-category generated by a C*-category.
This leads us to prove the Sherman-Takeda theorem along the way, which states that the double-dual of a C*-category agrees with the bicommutant of its universal representation.

**Notes:** PDF file available

### Tanaka Duality (Part II)

**Speaker:** David Green (OSU)

**Date:** Jun. 21, 2022

### Composing topological domain walls and anyon mobility

**Speaker:** David Penneys (OSU)

**Date:** Jun. 14, 2022

**Abstract:**
We study the concatenations of topological domain walls and their decompositions into superselection sectors.
Our approach uses a description of particle mobility across domain walls in terms of tunneling operators.
These are formalized in a 3-category of (2+1)D topological orders with a fixed anomaly described by a unitary modular tensor category A, algebraically characterized by the 3-category of A-enriched unitary fusion categories.
This is joint work with Fiona Burnell, Peter Huston, and Corey Jones.

### Gray-categories model algebraic tricategories (Part II)

**Speaker:** Giovanni Ferrer (OSU)

**Date:** May 4, 2022 (Wednesday, OUTSIDE!)

**Abstract:**
In this talk, we will adapt the technique of Gurski, Johnson, and Osorno to show the localization of GrayCat at the weak equivalences is equivalent to the category of algebraic tricategories and pseudo-natural equivalence classes of weak 3-functors.

### Small at infinity compactification of a von Neumann algebra

**Speaker:** Srivatsav Kunnawalkam Elayavalli (Vanderbilt)

**Date:** May 10, 2022

**Abstract:** We will discuss recent work with C. Ding and J. Peterson where we develop the theory of the small at infinity compactification of a von Neumann algebra.
An important feature of this theory is that it can be seen as an application of the theory of operator bimodules to the classification of von Neumann algebras, thereby building on the program initiated by Connes, Haagerup and various other mathematicians in the era of 1970's to 2010's wherein ideas from operator system/space theory such as injectivity and so on were applied to the classification of von Neumann algebras with great success.
We use this to define the notion of proper proximality for von Neumann algebras, and find several applications including the solution of a question of Popa asking if L(G) where G is an inner amenable group can embed into L(F_2); the equivalence between the Haagerup property and the compact approximation property for II_1 factors settling an open problem from 1995; solid ergodicity for Gaussian actions without any mixing assumptions improving on results of Boutonnet, Chifan-Ioana.

## Past Talks

Some of our previous talks can be found on the OSU Math youtube channel.

### Gray-categories model algebraic tricategories (Part I)

**Speaker:** Giovanni Ferrer (OSU)

**Date:** Apr. 26, 2022 (OUTSIDE!)

**Abstract:** Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences.
In this talk, we will go over basic facts about 3-dimensional category theory and model structures.

### Some basics on the basic construction

**Speaker:** Daniel Wallick (OSU)

**Date:** Apr. 19, 2022

**Abstract:** Given a unital inclusion of tracial von Neumann algebras A ⊆ B, Vaughan Jones’ basic construction gives a von Neumann algebra ⟨B,e_A⟩ with B⊆⟨B,e_A⟩ a unital inclusion.
When A and B are II_1 factors and A ⊆ B is finite index, then ⟨B,e_A⟩ is a II_1 factor, with the unique trace satisfying a Markov property.
However, if B is an arbitrary tracial von Neumann algebra, then there may not exist a trace on ⟨B,e_A⟩ satisfying this Markov property, even if ⟨B,e_A⟩ is finite.
We will give a necessary and sufficient condition for there to exist a Markov trace on ⟨B,e_A⟩ when B is finite-dimensional.
The results in this talk are from Jones' seminal 1983 paper "Index for Subfactors."

### Q-system realization and applications

**Speaker:** Quan Chen (OSU)

**Date:** Apr. 12, 2022

**Abstract:** Q-systems are unitary versions of Frobenius algebra objects which appeared in the theory of subfactors.
I will discuss higher unitary idempotent completion for C*/W* 2-categories called Q-system completion and its inverse 2-functor called realization.
We will explain several applications of Q-system realization, including the coend construction to describe the inductive limit II_1 factor of a Jones tower and induced actions of unitary tensor categories on C*-algebras.
Time permitting, we will explain the relationship between the equivariant K-theory and the K-theory of the realization.

### Graded extensions of generalized Haagerup categories

**Speaker:** Pinhas Grossman (UNSW)

**Date:** Mar. 29, 2022

**Abstract:** There is an obstruction theory for graded extensions of fusion categories due to Etingof, Nikshych, and Ostrik.
However, computing the obstructions can be difficult in concrete examples.
In this talk we will discuss a direct construction of graded extensions of generalized Haagerup categories using operator algebras, which leads to a number of new examples of fusion categories.
This is joint work with Masaki Izumi and Noah Snyder.

### Filtered Frobenius algebras in monoidal categories

**Speaker:** Harshit Yadav (Rice University)

**Date:** Mar. 22, 2022

**Abstract:** We develop filtered-graded techniques for algebras in monoidal categories with the goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well.
Towards the goal, we construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov.
We then produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form.
These two results of independent interest are used to achieve our goal.
As an application of our main result, we show that any exact module category over a symmetric finite tensor category is represented by a Frobenius algebra in it.
This is joint work with Dr. Chelsea Walton (Rice University).

### Braided quantum symmetries of graph C*-algebras

**Speaker: Sutanu Roy ** (National Institute of Science Education and Research (NISER) Bhubaneswar)

**Date:** Mar. 8, 2022

**Abstract:** In this talk, we will explain the existence of a universal braided compact quantum group acting on a graph C*-algebra in the twisted monoidal category of C*-algebras equipped with an action of the circle group.
To achieve this we construct a braided version of the free unitary quantum group.
Finally, we will compute this universal braided compact quantum group for the Cuntz algebra.
This is a joint work in progress with Suvrajit Bhattacharjee and Soumalya Joardar.

### From 3-manifolds to modular data

**Speaker:** Yang Qiu (UC Santa Barbara)

**Date:** Jan. 18, 2022

**Abstract:** The progress of TQFT has revealed connections between the algebraic world of tensor categories and the topological world of 3-manifolds, such as Reshetikhin-Turaev and Turaev-Viro theories.
Motivated by M-theory in physics, Cho-Gang-Kim recently proposed another relation by outlining a program to construct modular data from certain classes of closed oriented 3-manifolds.
In this talk, I will talk about our mathematical exploration of this program.
This talk is based on the joint works: [Cui-Qiu-Wang, arXiv: 2101.01674], [Cui-Gustafson-Qiu-Zhang, arXiv: 2106.01959].

### Microscopic definitions of anyon data

**Speaker:** Kyle Kawagoe (University of Chicago)

**Date:** Jan. 11, 2022

**Abstract:** The theory of anyons represents an enormously successful collaboration between the worlds of mathematics and physics.
Although unified by topological quantum field theory, this topic is discussed very differently in these two fields.
In math, we describe these theories by unitary modular braided fusion categories.
In physics, these theories are described by Hilbert spaces with gapped Hamiltonians and ground states with long range entanglement.
These differences raise a fundamental question: How can we understand the mathematical data in an anyon theory in a physical context?
In this talk, we will give precise definitions of these data for any given microscopic model.
These definitions are also operational in that they come with a method for calculating these data.
We will also give a brief overview of how this formalism can be extended to other problems in the mathematical physics of topological phases of matter.

### Fusion Categories over Non–Algebraically Closed Fields

**Speaker:** Sean Sanford (Indiana University)

**Date:** Dec. 7, 2021

**Abstract:** Much of the early work on Fusion Categories was inspired by physicists desire for rigorous foundations of topological quantum field theory.
One effect of this was that base fields other than the complex numbers were rarely considered, if at all.
The relevant features of $\mathbb{C}$ that make the theory work are the fact that it is characteristic zero, and algebraically closed.

This talk will focus on the interesting things that can be found when the algebraically closed requirement is removed.
The content will start with lots of examples, and slowly accelerate into higher categorical implications.

### Milnor's $\bar{\mu}$-invariants for knots in thickened surfaces, virtual links, and welded links.

**Speaker:** Micah Chrisman (OSU)

**Date:** Nov. 9, 2021

**Abstract:** Milnor's $\bar{\mu}$-invariants are concordance invariants
of links in $S^3$. For knots in $S^3$, the invariants are always
vanishing. Here we construct a non-trivial extension of Milnor's
$\bar{\mu}$-invariants to knots (and links) in thickened surfaces
$\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. The
extended Milnor invariants are invariant under concordance and vanish
on homologically trivial trivial knots in $\Sigma \times [0,1]$. They
are always stronger as slice obstructions than the generalized
Alexander polynomial. There are examples in which the extended Milnor
invariants provide stronger slice obstructions than the graded genus,
writhe polynomial, Rasmussen invariant, and parity projection. As an
application, we show that the concordance group of long virtual knots
is not abelian. This answers a question posed in 2008 by Turaev.

### Spin model subfactors

**Speaker:** Michael Montgomery (Vanderbilt)

**Date:** Nov. 2, 2021

**Abstract:** Complex Hadamard matrices generate a class of irreducible
hyperfinite subfactors with integer Jones index coming from spin model
commuting squares. I will prove a theorem that establishes a criterion
implying that these subfactors have infinite depth. I then show that
Paley type II and Petrescu's continuous family of Hadamard matrices
yield infinite depth subfactors. Furthermore, infinite depth
subfactors are a generic feature of continuous families of complex
Hadamard matrices. In this talk I will present an outline for the
proof of these results.

### Strong 1-boundedness for von Neumann algebras and Property T

**Speaker:** Srivatsav Kunnawalkam Elayavalli (Vanderbilt)

**Date:** Oct. 26, 2021

**Abstract:** Strong 1-boundedness is a notion introduced by Kenley Jung which captures (among Connes-embeddable von Neumann algebras) the property of having "a small amount of matrix approximations". Some examples are diffuse hyperfinite von Neumann algebras, vNa’s with Property Gamma, vNa’s that are non prime, vNa’s that have a diffuse hyperfinite regular subalgebra and so on. This is a von Neumann algebra invariant, and was used to prove in a unified approach remarkable rigidity theorems. One such is the following: A non trivial free product of von Neumann algebras can never be generated by two strongly 1-bounded von Neumann subalgebras with diffuse intersection. This result cannot be recovered by any other methods for even hyperfinite subalgebras. In this talk, I will present joint work with David Jekel and Ben Hayes, settling an open question from 2005, whether Property (T) II_1 factors are strongly 1-bounded, in the affirmative. Time permitting, I will discuss some insights and future directions.

### Spectral bounds for chromatic number of quantum graphs

**Speaker:** Priyanga Ganesan (Texas A&M)

**Date:** Oct. 5, 2021

**Abstract:** Quantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator systems theory, non-commutative topology and quantum information theory. In this talk, I will review the different perspectives to quantum graphs and introduce a chromatic number for quantum graphs using a non-local game with quantum inputs and classical outputs. I will then show that many spectral lower bounds for chromatic numbers in the classical case (such as Hoffman’s bound) also hold in the setting of quantum graphs. This is achieved using an algebraic formulation of quantum graph coloring and tools from linear algebra.

### A categorical Connes' $\chi(M)$

**Speaker:** David Penneys (OSU)

**Date:** Sep. 21, 2021

**Abstract:** We consider the W* tensor category $\widetilde{\chi}(M)$ of
approximately inner and centrally trivial bimodules over a $\rm II_1$
factor $M$, which generalizes the usual notions for automorphisms in
Connes' definition of $\chi(M)$. We construct a unitary braiding on
$\widetilde{\chi}(M)$ extending Jones' $\kappa$ invariant on
$\chi(M)$, solving a problem posed by Popa from 1994. For a non-Gamma
finite depth $\rm II_1$ subfactor $N\subset M$, we prove that the
unitary braided tensor category
$\widetilde{\chi}_{\operatorname{fus}}(M_\infty)\subset
\tilde{\chi}(M_\infty)$ of finite depth objects is the Drinfeld center
$Z(\mathcal{C})$, where $\mathcal{C}$ is the standard invariant of
$N\subset M$, and $M_\infty$ is the inductive limit $\rm II_1$ factor
of $N\subset M$. This is joint work with Corey Jones
and Quan Chen.