QSS is a small seminar on quantum symmetries organized by David Penneys and Daniel Wallick at the Ohio State University.

Topics of interest

  • Tensor and fusion categories
  • Higher categories
  • Subfactors and operator algebras
  • Topological phases of matter
  • Quantum information

Meeting Time

We usually meet on Tuesdays 1:50 pm - 2:50 pm, US Eastern Time Zone. However, sometimes we meet online at a possibly different hour. See future talks for the precise meeting time of a given talk.


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

Future Talks

Finite-dimensional quantum groups of type super A

Speaker: Guillermo Sanmarco (University of Washington)

Date: Feb. 20 (VIRTUAL)

Abstract: We construct a class of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of super type A, for a root of unity q. In the case that q has even order, we classify ribbon structures for these quantum groups; such structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the positive Borel Hopf subalgebra. As applications, we derive that the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. We finish by explicitly computing link invariants associated to a four-dimensional simple module of the rank-two quantum group, based on generalized traces. These knot invariants distinguish certain knots not distinguished by the Jones or HOMFLYPT polynomials. This is based on joint work with Robert Laugwitz (arxiv 2301.10685).

To be determined

Speaker: Kate Ponto (University of Kentucky)

Date: Mar. 5

To be determined

Speaker: Niles Johnson (OSU)

Date: Apr. 2

To be determined

Speaker: Kylan Schatz (NC State University)

Date: Apr. 9

An Absence of Quantifier Reduction for \(\textrm{II}_1\) Factors, using Quantum Expanders

Speaker: Jennifer Pi (UC Irvine)

Date: Apr. 16

Abstract: A basic question in model theory is whether a theory admits any kind of quantifier reduction. One form of quantifier reduction is called model completeness, and broadly refers to when arbitrary formulas can be "replaced" by existential formulas. Prior to the negative resolution of the Connes Embedding Problem (CEP), a result of Goldbring, Hart, and Sinclair showed that a positive solution to CEP would imply that there is no \(\textrm{II}_1\) factor with a theory which is model-complete. In this talk, we discuss work on the question of quantifier reduction for general tracial von Neumann algebras. In particular, we prove a complete classification for which tracial von Neumann algebras admit complete elimination of quantifiers. Furthermore, we show that no \(\textrm{II}_1\) factor (satisfying a weaker assumption than CEP) has a theory that is model complete by using Hastings' quantum expanders. This is joint work with Ilijas Farah and David Jekel.

To be determined

Speaker: Vijay Higgins (Michigan State University)

Date: Apr. 23

To be determined

Speaker: Nick Gurski (Case Western Reserve University)

Date: Apr. 30

Past Talks

An index for quantum cellular automata on fusion spin chains

Speaker: Junhwi Lim (Vanderbilt)

Date: Feb. 13

Abstract: The index for 1D quantum cellular automata (QCA) was introduced to measure the flow of the information by Gross, Nesme, Vogts, and Werner. Interpreting the index as the ratio of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators on 2D topologically ordered spin systems. We introduce our generalization of index and show that it is a complete invariant for the group of QCA modulo finite depth circuits for the fusion spin chains built from the fusion category Fib. This talk is based on a joint work with Corey Jones.

Classification of anyon sectors in Kitaev's quantum double models

Speaker: Siddharth Vadnerkar (UC Davis)

Date: Jan. 30

Abstract: Kitaev's quantum double models, which generalize the Toric Code for finite groups G, are a family of celebrated exactly solvable models that have allowed us to understand many properties of topological phases on a microscopic level. These models support the existence of 'anyons', 2d localised quasiparticles that exhibit non-trivial braiding and fusion. When one puts this model on an infinite 2d orientable lattice like the triangular lattice, a robust infinite volume definition of so-called global 'anyon types' emerges. In this talk, we will discuss this infinite volume definition of an 'anyon type', called an anyon sector. We will then sketch our strategy for showing the following result: in the quantum double models in infinite volume, there exist certain anyon sectors labelled by irreducible representations of the quantum double algebra \(D(G)\), and any anyon sector is unitarily equivalent to one of these sectors. This result is the content of our recent paper (https://arxiv.org/pdf/2310.19661.pdf) co-authored with Alex Bols.

Super Multiset RSK and the Mixed Multiset Partition Algebra

Speaker: Alexander Wilson (Oberlin)

Date: Jan. 16

Abstract: A main focus of my research in combinatorial representation theory is finding diagrammatic descriptions of certain categories of symmetric group representations. These descriptions can help us understand representations of corresponding endomorphism algebras which then give us information about the particular symmmetric group representations. In this talk, I will introduce the mixed multiset partition algebra as well as an RSK-like algorithm which helps to construct its irreducible representations.

Non-semisimple skein 4-TQFTs

Speaker: Benjamin Haïoun (University of Toulouse)

Date: Dec. 12 (VIRTUAL)

Abstract: I will present a recent construction of non-semisimple skein (3+1)-TQFTs with F. Costantino, N. Geer and B. Patureau-Mirand and explain how it is expected to extend all the way down. I will give the explicit handle attachment formulas in dimension 4, the admissible skein modules in dimension 3, the expected values on surfaces, and, if time permits, how to obtain the description on lower-dimensions using factorization homology.

The homotopy coherent classification of fusion 2-categories

Speaker: Thibault Décoppet (Harvard)

Date: Nov. 28

Abstract: I will explain how to describe the space of all fusion 2-categories, and monoidal equivalences. The starting point is the observation that every fusion 2-category is Morita connected. In particular, an important part of our proof consists in understanding the Witt groups of braided fusion 1-categories. More precisely, we prove that the functor sending a symmetric fusion 1-category to the associated Witt space preserves limits. This last fact can be used to show that fusion 2-categories are classified by a single non-degenerate braided fusion 1-category together with group-theoretic data. As consequences of our classification, we obtain Ocneanu rigidity and rank finiteness for fusion 2-categories, as well as strong constraints on the associated hypergroups. This is joint work in progress with Huston, Johnson-Freyd, Penneys, Plavnik, Nikshych, Reutter, and Yu.

Low rank symmetric fusion categories in positive characteristic

Speaker: Agustina Czenky (University of Oregon)

Date: Nov. 14

Abstract In this talk, we look at the classification problem for symmetric fusion categories in positive characteristic. We recall the second Adams operation on the Grothendieck ring and use its properties to obtain some classification results. In particular, we show that the Adams operation is not the identity for any non-trivial symmetric fusion category. We also give lower bounds for the rank of a (non-super-Tannakian) symmetric fusion category in terms of the characteristic of the field. As an application of these results, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects.

Categorical Quantum Groups, Braided Monoidal 2-Categories and the 4d Kitaev Model

Speaker: Hank Chen (University of Waterloo)

Date: Oct. 31

Abstract: It is well-known since the late 20th century that Hopf algebra quantum groups play a signification role in both physics and mathematics. In particular, the category of representations of quantum groups are braided, and hence captures invariants of knots. This talk is based on arXiv:2304.07398 & JHEP 2023 141, where we develop a categorification of the theory of quantum groups/bialgebras, including homotopy refinements, and prove that their 2-representations form a cohesive braided monoidal (tensor) 2-category. We will then apply our general theory to describe the 4D toric code and its spin variant, which unites the 2-categorical and 2-group gauge theory frameworks for topological orders.

Anomalies on von Neumann algebras

Speaker: Pradyut Karmakar (Ohio University)

Date: Oct. 17

Abstract: Connes showed that anomalies of finite cyclic groups can be realized in \(\mathrm{II}_1\) factors. Later, Jones showed that anomalies can be realized in \(\mathrm{II}_1\) factors for finite groups and Sutherland showed that for discrete groups. We have shown anomalies can be realized in type \(\mathrm{III}\) factors for certain compact groups.

Floquet codes, TQFTs, automorphisms, and quantum computation

Speaker: Nat Tantivasadakarn (Caltech)

Date: Oct. 3

Abstract: Topological quantum error-correcting (QEC) codes such as the toric code have a deep connection to topological quantum field theory (TQFT). The recently introduced Floquet codes are implemented by a sequence of anticommuting measurements, causing the encoded information to be toggled between various subspaces at different times. This unusual property raises the question of whether it is possible to understand Floquet codes using TQFTs.

I will review a recent interpretation that these anticommuting measurements are the lattice counterpart of anyon condensation. Using this understanding, I will propose a generalization of Floquet codes called dynamic automorphism (DA) codes, that not only encodes quantum information, but can also simultaneously perform quantum computation.

From the lens of TQFT, the quantum computation can be understood as a sequence of time-like domain walls that implements automorphisms in a TQFT. The preservation of encoded information corresponds to the invertibility of the domain wall, and I will describe how to efficiently compute automorphisms from a sequence of condensations. I will further show that these sequences of condensations in a particular 2+1D TQFT with boundaries is sufficient to implement the full Clifford group of logical gates in the corresponding code. A similar setup in a 3+1D TQFT allows us to implement a non-Clifford gate, making the first step towards a new measurement-based model of universal quantum computation.

Immersion in entanglement bootstrap

Speaker: Bowen Shi (UC San Diego)

Date: Sep. 12

Abstract: Immersion (i.e., locally embedding) of a manifold in another is a basic concept in topology. Its appearance in physics has been rare until recently. We explain why immersion is a valuable concept in entanglement bootstrap, an ongoing research program that starts with a many-body wave function satisfying a few axioms about entanglement and trying to derive the universal properties of the system. A physical reason immersion is helpful is that It uses the quantum state more efficiently by recycling qubits and allows the construction of states on nontrivial topology from a ball. As concrete examples, we discuss the immersed "figure-8" annulus and punctured orientable surfaces, where the problem is already interesting. We conjecture that the information convex set of the immersed "figure-8" annulus always contains an Abelian sector and explain why this conjecture has nontrivial implications. We solve a special case of the conjecture. If time permits, we sketch why immersion may also be relevant in a setup with a chiral (gapless) edge.

Higher Tannaka Reconstruction

Speaker: David Green (OSU)

Date: Apr. 25, 2023

Abstract: We will examine a categorifiable proof of Tannaka reconstruction, and show the corresponding result for fusion 2-categories which admit a fiber functor to 2Vec. In particular we will see that every fusion 2-category which admits a fiber functor is equivalent to the 2-category of modules for a multifusion category. At the end of the talk, we will briefly discuss some future applications towards the classification of various higher algebraic objects.

Generalizing 1-1 graph tangles

Speaker: Puttipong Pongtanapaisan (University of Saskatchewan)

Date: Apr. 20, 2023

Abstract: In this talk, I will discuss graphs with diagrams containing distinguished vertices of degree one. We consider these diagrams up to Kauffman moves applied away from the endpoints. I will demonstrate that these types of open-ended objects are suitable for studying entanglements in proteins. Furthermore, given one of these open-ended objects, one can obtain a closed knotted object in a canonical way, which can be beneficial for fast computations of invariants.

\(L^p\) decomposition of the free group von Neumann algebra

Speaker: Tao Mei (Baylor)

Date: Apr. 18, 2023

Abstract: I plan to discuss the \(L^p\)-unconditional decompositions of the group von Neumann algebras associated with the free group \(\mathbb{F}_n\) and its possible connection to Murray/von Neumann's free group factor problem. I will also introduce a concrete operator-space isomorphism between the noncommutative-\(L^p\) spaces associated with the group von Neumann algebras associated with \(\mathbb{F}_n\) for \(n=2,\infty\) if time permits. This talk is based on joint work with Zhenchuan Liu and Sheng Yin.

Crossed module graded categories and state-sum homotopy invariants of maps

Speaker: Kursat Sozer (McMaster)

Date: Apr. 13, 2023

Abstract: In topology, groups serve as algebraic models for 1-types, which are spaces with vanishing second and higher homotopy groups. Crossed modules, on the other hand, generalize groups and are useful for modeling 2-types. In this talk, we will introduce the concept of a crossed module graded fusion category, which is a generalization of a fusion category graded by a group. We will then use these categories to construct a 3-dimensional state-sum homotopy quantum field theory (HQFT) with a 2-type target. Such an HQFT assigns a scalar to a map defined from a closed oriented 3-manifold to the fixed 2-type and this scalar is invariant under homotopies. Our construction generalizes the state-sum Turaev-Virelizier HQFT with an aspherical target. This is joint work with Alexis Virelizier.

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.

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.



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  • Tensor categories.
  • Subfactors and operator algebras.
  • Higher Categories.
  • Topological phases of matter.


  • Dolor pulvinar etiam.
  • Sagittis adipiscing.
  • Felis enim feugiat.


  1. Dolor pulvinar etiam.
  2. Etiam vel felis viverra.
  3. Felis enim feugiat.
  4. Dolor pulvinar etiam.
  5. Etiam vel felis lorem.
  6. Felis enim et feugiat.





Name Description Price
Item One Ante turpis integer aliquet porttitor. 29.99
Item Two Vis ac commodo adipiscing arcu aliquet. 19.99
Item Three Morbi faucibus arcu accumsan lorem. 29.99
Item Four Vitae integer tempus condimentum. 19.99
Item Five Ante turpis integer aliquet porttitor. 29.99


Name Description Price
Item One Ante turpis integer aliquet porttitor. 29.99
Item Two Vis ac commodo adipiscing arcu aliquet. 19.99
Item Three Morbi faucibus arcu accumsan lorem. 29.99
Item Four Vitae integer tempus condimentum. 19.99
Item Five Ante turpis integer aliquet porttitor. 29.99


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