## Future Talks

### Universal Traces

**Speaker:** Sean Sanford (OSU)

**Date:** Mar. 7, 2023

**Abstract:** In the category of vector spaces, the trace satisfies \(\operatorname{tr}(fg)=\operatorname{tr}(gf)\) and is universal with respect to this property. This characterization can be rephrased using the language of coends: traces live in the coend of the hom-functor. This interpretation of the trace as a universal invariant of endomorphisms can be generalized to any suitably small category by swapping Vec out for other categories. I will discuss the properties this invariant has in both additive and non-additive settings. This will give rise to a vast generalization of character theory, and we will discuss character tables valued in various categories. The talk will finish with a proposed generalization to 2-categories using 2-coends, and the connections to categorified traces for braided tensor categories *a la* Henriques-Penneys-Tener.

## Past Talks

### Cell systems for quantum \(\mathfrak{sl}_N\) module categories

**Speaker:** Cain Edie-Michell (University of New Hampshire)

**Date:** Mar. 2, 2023

**Abstract:** There has been a recent revival in the program to classify and construct module categories over the quantum group categories. This interest has mainly been fueled by applications to WZW conformal field theories. In this talk I will discuss a cell calculus which classifies these module categories in type A. Using this cell calculus, we are able to explicitly classify the module categories in the \(\mathfrak{sl}_4\) case. This settles an old claim made by Ocneanu regarding these modules.

### Bounding Quantum Chromatic Numbers for Quantum Graph Products

**Speaker:** A. Meenakshi McNamara (Purdue)

**Date:** Feb. 28, 2023

**Abstract:** We will provide a brief introduction to quantum graphs and quantum chromatic numbers, which are closely tied to quantum error-checking problems. Quantum graphs are a generalization of graphs using operator algebras, and quantum colorings are defined in terms of random strategies for non-local games using entanglement. We discuss existing bounds on quantum chromatic numbers and our work to expand upon these bounds. We define the lexicographic, cartesian, and categorical products of quantum graphs and investigate bounds on the resulting quantum chromatic number of these graph products. In particular, we define a quantum b-fold chromatic number which we use to derive bounds for the lexicographic product that are analogous to those in the classical case.

### Hyperbolic knots and torsion in Khovanov homology

**Speaker:** Micah Chrisman (OSU)

**Date:** Feb. 21, 2023

**Abstract:** Khovanov homology (KH) associates to each link in \(S^3\) a bi-graded chain complex whose homology is a link invariant. With integer coefficients, each bigrading \((i,j)\) is an abelian group. Applications of KH typically use only the free part of these groups. For example, the Jones polynomial is the graded Euler characteristic of KH. Another example is the Rasmussen invariant, which gives a lower bound on the smooth slice genus. The Rasmussen invariant is defined using rational coefficients in KH. Yet torsion in KH is (nearly) ubiquitous. Every non-split prime alternating link has either no torsion or only \(\mathbb{Z}_2\)-torsion. Odd torsion, however, is quite rare. The T(5,6) torus knot has both \(\mathbb{Z}_3\) and \(\mathbb{Z}_5\) torsion. Other examples have been found, but they are either torus links, twisted torus links, or connected sums of these. This leads to the natural question: are there any hyperbolic knots having odd torsion in their KH? We will answer this question affirmatively. It will be shown that if there is any knot having \(\mathbb{Z}_m\)-torsion in its KH, there are infinitely many hyperbolic knots having \(\mathbb{Z}_m\)-torsion in their KH. Similarly, the existence of a knot with \(\mathbb{Z}_m\)-torsion implies there are infinitely many prime satellite knots with \(\mathbb{Z}_m\)-torsion. This is joint work with Sujoy Mukherjee (2205.07747.pdf (arxiv.org)).

### A comparison between \(SL_n\) spider categories

**Speaker:** Anup Poudel (OSU)

**Date:** Feb. 14, 2023

**Abstract:** In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group \(U_q(SL_n)\). In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.

### Cone algebras for Kitaev's quantum double model are type \(\mathrm{II}_\infty\) factors

**Speaker:** Daniel Wallick (OSU)

**Date:** Feb. 7, 2023

**Abstract:** Fiedler and Naaijkens showed that the excitations for Kitaev's abelian quantum double on an infinite planar lattice can be described as superselection sectors localized in cone regions. Until recently, the type of the von Neumann algebras corresponding to these regions remained open. In a new paper, Ogata showed that these algebras are type \(\mathrm{II}_\infty\) factors. I will discuss the basics of the model in this talk and then describe Ogata's result, including a sketch of her argument.

### Characterization of Positive Definite, Radial Functions on Free Groups

**Speaker:** Chian Yeong Chuah (OSU)

**Date:** Jan. 24, 2023

**Abstract:** In this talk, we will give a brief account about the relationship between radial positive-definite functions on free groups and the moments of probability measures on the interval \([-1,1]\). The case for the commutative setting is proven by Bochner. Meanwhile, Haagerup and Knudby proved the case for \(\ell^1\) radial positive definite function. We explore the case for \(\ell^2\) radial positive definite functions (completely positive Fourier multipliers) on free groups.

### Bulk-boundary correspondence for symmetry protected topological phases

**Speaker:** Kyle Kawagoe (OSU)

**Date:** Jan. 18, 2023

**Abstract:** A universal property of symmetry protected topological (SPT) phases is that they have low energy edge modes which are protected under the symmetry. This fact inspires an important problem in the theory of SPT phases: How does one identify a bulk SPT phase given a low energy theory of its boundary degrees of freedom. This question is particularly challenging in the case of interacting SPT phases, where band theory approaches are inapplicable. In this talk, we will present a general method for solving this problem in the case of two-dimensional interacting bosonic systems with internal (non-spatial) symmetries.

### Identifying invertiblility of bimodule categories

**Speaker:** Jacob Bridgeman (Ghent University)

**Date:** Dec. 6, 2022

**Abstract:** Fusion categories, and their module categories, have many applications in both mathematics and physics.
One notion of equivalence between fusion categories is Morita equivalence, which is witnessed by an invertible bimodule category.
We provide a readily verifiable condition for deciding if a given bimodule category is invertible.
This condition makes use of the skeletal data, and utilizes orthogonality of characters of annular algebras.
We then extend this to generalized Schur orthogonality of matrix elements in this setting.
Generalized Schur orthogonality in this setting has application in physics.
We show that it is equivalent to the notion of MPO-injectivity, which is central to the study of topological orders with tensor networks.
This closes an open question concerning tensor network representations for string-net models and plays a central role in the study of generalized symmetries.
Based on arXiv:2211.01947
Work with Laurens Lootens and Frank Verstraete

### Kumjian-Pask Fibrations and Their C*-Algebras

**Speaker:** Lydia de Wolf (Kansas State University)

**Date:** Nov. 29, 2022

**Abstract:** In this talk, we will present discrete Conduche fibrations and Kumjian- Pask fibrations, which were studied as a generalization of k-graphs.
We will briefly summarize the basic results about their C*-algebras and path groupoids in general, and then demonstrate examples of KPfs in which specific choices allow for more exploration than the fully general case.

### Discrete extensions of simple C*-algebras

**Speaker:** Roberto Hernandez Palomares (U Waterloo)

**Date:** Nov. 15, 2022

**Abstract:** A subfactor is a unital inclusion of simple von Neumann algebras, and their 'size' is measured by the Jones index.
Vaughan Jones proved the striking Index Rigidity Theorem establishing the set of values the index takes is exactly \(\{4\cos^2(\pi/n)\}_{n\geq 3}\cup [4, \infty]\).
This discovery started the modern theory of subfactors and their classification program.
Discrete subfactors conform to an ample and well-behaved class resembling the properties of discrete groups.
These subfactors are studied through the Standard Invariant --which we view as an action of a unitary tensor category together with a chosen generator-- and can be reconstructed from it.
In this talk, I will define discreteness for inclusions of C*-algebras and establish a theorem stating that --similarly to their von Neumann counterparts,-- these can be reconstructed from a Standard Invariant.
This is work in progress, joint with Brent Nelson and Matthew Lorentz.

### Two non Gamma factors with non isomorphic ultrapowers

**Speaker:** Srivatsav Kunnawalkam Elayavalli (UCLA/IPAM)

**Date:** Oct. 25, 2022

**Abstract:** I will show you how to construct a non Gamma factor such that it and \(L(F_2)\) have non isomorphic ultrapowers.
This settles a fundamental open problem in the classification of \(\mathrm{II}_1\) factors.
This is joint work with Adrian Ioana and Ionut Chifan.

### Unitary anchored planar algebras II: the non-unitary correspondence

**Speaker:** David Penneys (OSU)

**Date:** Oct. 18, 2022

**Abstract:** We'll continue our series of lectures on unitary anchored planar algebras.
In this second talk, we'll discuss the correspondence between anchored planar algebras and pointed module tensor categories.
No knowledge from the first talk will be assumed.
Both talks I and II will be used for talk III.

### Unitary anchored planar algebras I: unitary adjunction

**Speaker:** David Penneys (OSU)

**Date:** Oct. 11, 2022

**Abstract:** I'll give the first talk in a series on a new result with Andre Henriques on unitarity for anchored planar algebras.
This first talk will focus on unitary adjunction between 2-Hilbert spaces.

### Anomalous symmetries on operator algebras

**Speaker:** Sergio Girón Pacheco (University of Oxford)

**Date:** Oct. 4, 2022

**Abstract:** An anomalous action on an operator algebra A is a mapping from a group G to the automorphism group of A, which is multiplicative up to inner automorphisms of A.
Anomalous symmetries can be rephrased as actions of pointed fusion categories on A.
Starting from the basics, I will introduce anomalous actions and discuss some history of their study in the literature.
I will then discuss their existence and classification on simple C*-algebras.

### An algebraic quantum field theoretic approach to toric code with boundary

**Speaker:** Daniel Wallick (OSU)

**Date:** Sep. 20, 2022

**Abstract:** Topologically ordered quantum spin systems have become an area of great interest, as they may provide a fault-tolerant means of quantum computation.
One of the simplest examples of such a spin system is Kitaev's toric code.
Naaijkens made mathematically rigorous the treatment of toric code on an infinite planar lattice (the thermodynamic limit), using an operator algebraic approach via algebraic quantum field theory.
We adapt his methods to study the case of toric code with boundary.
In particular, we recover the condensation results described in Kitaev and Kong and show that the boundary theory is a module tensor category over the bulk, as expected.

### Braided tensor categories from von Neumann algebras

**Speaker:** Quan Chen (OSU)

**Date:** Sep. 13, 2022

**Abstract:** Given a W*-category C, we construct a unitary braided tensor category End_loc(C) of local endofunctors on C, which is a new construction of a braided tensor category associated with an arbitrary W*-category.
For the W*-category of finitely generated projective modules over a von Neumann algebra M, this yields a unitary braiding on Popa's \(\widetilde{\chi}(M)\), which extends Connes' \(\chi(M)\) and Jone's kappa invariant.
Given a finite depth inclusion
\(M_0\subset M_1\) of non-Gamma \(\mathrm{II}_1\) factors, we show that \(\widetilde{\chi}(M)\) is equivalent to the Drinfeld center of the standard invariant, where \(M_\infty\) is the inductive limit of the Jones tower of basic construction.

### 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 \(\mathrm{II}_1\) factors settling an open problem from 1995; solid ergodicity for Gaussian actions without any mixing assumptions improving on results of Boutonnet, Chifan-Ioana.

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 \(A \subseteq B\), Vaughan Jones? basic construction gives a von Neumann algebra \(\langle B, e_A \rangle\) with \(B \subseteq \langle B, e_A \rangle\) a unital inclusion.
When \(A\) and \(B\) are \(\mathrm{II}_1\) factors and \(A \subseteq B\) is finite index, then \(\langle B, e_A \rangle\) is a \(\mathrm{II}_1\) 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 \(\langle B, e_A \rangle\) satisfying this Markov property, even if \(\langle B, e_A \rangle\) is finite.
We will give a necessary and sufficient condition for there to exist a Markov trace on \(\langle B, e_A \rangle\) 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 \(\mathrm{II}_1\) 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) \(\mathrm{II}_1\) 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 \(\mathrm{II}_1\)
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 \(\mathrm{II}_1\) 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 \(\mathrm{II}_1\) factor
of \(N\subset M\). This is joint work with Corey Jones
and Quan Chen.