# Activities for Fall 2020

#13: September 4th
Speaker: Elliott Gesteau (Caltech)
Title: Operator algebras for the emergence of spacetime.
Abstract: The latest developments in our understanding of quantum gravity have led to a surprising resurgence of operator algebraic techniques. After a very short review of the main ideas of string theory and the AdS/CFT correspondence, I will introduce the notion of entanglement wedge reconstruction, which amounts to describing spacetime as a quantum error-correcting code. It will turn out that the natural mathematical framework to formulate this idea involves some old, foundational notions from the theory of von Neumann algebras like Tomita-Takesaki theory, the Connes cocycle, and their interplay with conditional expectations. I will further explain how these objects carry a spectacular amount of information about the emergent geometry of semiclassical spacetime and its local Lorentz symmetries. I will also introduce a few toy models that illustrate these concepts, and, if time allows, a few open problems in this new, rapidly expanding field, which are physically important, but also surprisingly subtle mathematically. No prior knowledge of physics will be required, apart from having heard of black holes, and a vague familiarity with the Einstein equation.
Background reading: https://arxiv.org/pdf/1811.05482.pdf, https://arxiv.org/pdf/2005.07189.pdf, https://arxiv.org/pdf/2008.04810.pdf, https://arxiv.org/pdf/2007.00230.pdf

#14: September 11th
Speaker: Cain Edie-Michell (Vanderbilt University)
Title: Unitary fusion categories generated by an object of small dimension .
Abstract: Fusion categories have generally been accepted as the natural successor to finite groups, and have hence been given the moniker “quantum symmetries”. As was achieved for finite groups, a classification of all fusion categories is one of the big open questions in the field. Of course, this goal is hopelessly out of reach with our current tools. Instead, research has focused on providing partial classifications of fusion categories, where some sort of ‘smallness’ is imposed on the category. In this talk I will describe the program to classify unitary fusion categories generated by an object of small dimension. I will state such a classification, and discuss several open lines of research.
Background reading: This talk should be accessible to anyone with knowledge of unitary fusion categories. My paper https://arxiv.org/abs/1904.08909 may be helpful, but not necessary.

#15: September 18th (Warning: starting 2pm Eastern Time, instead of the regular 1pm)
Speaker: Pablo Sánchez Ocal (Texas A&M)
Title: Interplays of Hochschild cohomology in quantum symmetries
Abstract:  The Hochschild cohomology is a tool for studying associative algebras that has a lot of structure: it is a Gerstenhaber algebra. Partially due to the generality at which it applies, it is usually seen as an umbrella that extends throughout the representation theory of associative algebras, and seldom as a key tool in studying quantum objects. In this talk we will give an unpretentious introduction to this cohomology, we will justify its importance by computing some of the lower degrees, and we will then give explicit applications that advance the understanding of quantum symmetries.
Background reading: A gentle introduction to Hochschild (co)homology is Hochschild cohomology for Algebras by Witherspoon, where its relation to deformation theory is emphasized. An explicit use of Hochschild cohomology in the context of von Neumann algebras by Sinclair and Smith can be found here, and in the context of algebraic quantum field theory by Hawkins can be found here. Some of my work that has applications to quantum symmetries is arXiv:1909.02181.
Notes for the talk:  PSO_QSSS_20200918

#16: September 25th (Back to 1pm Eastern Time)
Speaker: Sean Sanford (Indiana University Bloomington)
Title: Non-Degeneracy Conditions For Braided Finite Tensor Categories
Abstract: In this paper, Shimizu proves the equivalence of several non-degeneracy conditions for tensor categories.  He makes heavy use of a certain canonical Hopf algebra object in the category.  In the semisimple case, this algebra is the familiar sum of objects $X\otimes X^\ast$ as $X$ runs over all simples (this is a categorization of the casimir element of classical representation theory).  In general the algebra is given by a coend, and this formulation has the advantage of being defined by a universal property.  We will discuss how this universal property allows for a satisfying pictorial calculus and, after learning the relevant diagrams, sketch the proof of one of Shimizu’s theorems.
Notes for the talk: CentralHopfTalk_Notes

#17: October 2nd (10 am Eastern Time = 4pm in Germany)
Speaker: César Galindo (Universidad de Los Andes)
Title: Braided Zesting and its applications.
Abstract: In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used to provide categorifications of new fusion rule algebras, modular data, and minimal, modular extensions of super-modular categories. I will present a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.
Background reading: The talk is based on the manuscript https://arxiv.org/abs/2005.05544 a joint work with Colleen Delaney, Julia Plavnik, Eric C. Rowell, and Qing Zhang.
Notes for the talk: talk_zesting_QSSS

#18: October 9th (1 pm Eastern Time)
Speaker: Bowen Shi (UC San Diego)
Title: Entanglement bootstrap: from gapped many-body ground states to emergent laws
Abstract: Quantum many-body systems can be complicated and hard to study. Interestingly, simple and elegant physical laws can emerge, which effectively describe the universal properties of the system at low energies. A famous example is topologically ordered systems in two spatial dimensions. It can host anyons. The emergent law of anyons (in bosonic systems) is expected to be the unitary modular tensor category. In this talk, we describe a recently discovered approach to derive the emergent law of anyons. The underlying axioms are implied by a generic property of ground-state entanglement entropy. A focus is given to gapped domain walls that separate a pair of topologically ordered phases. In this context, a new superselection sector is identified, which currently lacks a categorical description.
Background reading: The original thought of entanglement bootstrap, i.e., deriving emergent physical laws from an assumption on the entanglement, can date back to a talk by my collaborator Isaac Kim in 2015. In a recent paper (arxiv:1906.09376), we derived the fusion rules of anyons with this line of thought. In a follow-up paper (arxiv:1911.01470), the Verlinde formula is derived under the same assumption, which implies the nontrivial mutual braiding statistics of anyons. The generalization of this approach on systems with gapped domain walls (arxiv:2008.11793) appears to uncover a new physics.
Notes for the talk: Entanglement bootstrap talk Math OSU

#19: October 16th
Speaker: Matthew Harper (The Ohio State University)
Title: A Generalization of the Alexander Polynomial from Higher Rank Quantum Groups
Abstract: Murakami and Ohtsuki have shown that the Alexander polynomial is an R-matrix invariant associated with representations V(t) of unrolled restricted quantum sl2 at a fourth root of unity. In this context, the highest weight t\in\C^\times of the representation determines the polynomial variable. In this talk, we discuss an extension of their construction to a link invariant Δ_g, which takes values in n-variable Laurent polynomials, where n is the rank of g.
We begin with an overview of computing quantum invariants and of the sl2 case. Our focus will then shift to g=sl3. After going through the construction, we briefly sketch the proof of the following theorem: For any knot K, evaluating Δ_sl3 at t_1=±1, t2=±1, or t2=±it^{-1} recovers the Alexander polynomial of K. We also compare Δ_sl3 with other invariants by giving specific examples. In particular, this invariant can detect mutation and is non-trivial on Whitehead doubles.

#20: October 23rd
Speaker: Frank Taipe (Université Paris-Saclay)
Title: Quantum Groupoids as Quantum Symmetries
Abstract: In this talk, I will give the motivation behind the theory of quantum groupoids from the point of view of operator algebras. At first, I will do a review of the progress in this theory, and then I will explain how quantum groupoids arise naturally from inclusions of von Neumann algebras. For convenience, I will focus on the case of II_{1} subfactors. If I have enough time, I will explain the Galois correspondence given by Nikshych-Vainerman.
Notes for the talk: Quantum Groupoids as Quantum Symmetries Notes

#21: October 30th
Speaker: Andrés Fontalvo Orozco (University of Zurich)
Title: Traces on modules over  pivotal categories.
Abstract: The modified trace was introduced by Geer and Patureau-Mirand as a generalization of the categorical trace in 2013 enabling us to  define interesting link invariants in the style of Reshetikhin-Turaev in  the non-semisimple setting. In 2018 Beliakova, Blanchet and Gainutdinov  gave an explicit characterization of this  modified traces in terms of  the algebraic data of a unimodular Hopf algebra. In this talk we take a  look at this data an use it to motivate a further generalization of the categorical trace, namely the module trace. We discuss some of its properties and what they can tell us about the modified trace in the non-unimodular case.

#22: November 6th
Speaker: Pieter Spaas (UCLA)
Title: Symmetries of II$_1$ factors tensored with the hyperfinite II$_1$ factor.
Abstract: We will discuss some aspects of Popa’s deformation/rigidity theory, and see how this can be applied to the classification of tensor products of II$_1$ factors with the hyperfinite II$_1$ factor. We will discuss some rigidity phenomena, i.e. absence of certain kinds of symmetries, for these II$_1$ factors. In particular, we will discuss Popa’s result on the uniqueness of the aforementioned tensor product decomposition in the absence of property Gamma, and touch upon some generalizations in the property Gamma regime.
– S. Popa: On Ozawa’s Property for Free Group Factors. Section 5. https://arxiv.org/abs/math/0608451
– A. Ioana and P. Spaas: A class of II$_1$ factors with a unique McDuff decomposition. Sections 1-3. https://arxiv.org/abs/1808.02965
– A. Ioana: Rigidity for von Neumann algebras. Section 3. https://arxiv.org/abs/1712.00151v1
Notes for the talk: Talk_Ohio State, Nov 6, 2020

#23: November 13th
Speaker: Guillermo Sanmarco (Universidad Nacional de Córdoba)
Title: An invitation to Nichols algebras
Abstract: Finite dimensional Nichols algebras of diagonal type are generalizations of (positive parts) of small quantum groups that play a crucial role in the classification of finite dimensional pointed Hopf algebras with abelian group of group-likes. Remarkably, many combinatorial tools of quantum groups were extended to this setting which led to the classification of such Nichols algebras. In this talk we will introduce these tools through examples and show several recent developments in the theory of Nichols algebras.
Notes for the talk: QSSS Guillermo Sanmarco

#24: November 20th
Speaker: Kursat Sozer (University of Lille)
Title: Survey on low dimensional HQFTs
Abstract: Topological quantum field theories (TQFTs), inspired by theoretical physics, produce manifold invariants behaving well under gluing. For every discrete group G, Turaev introduced homotopy quantum field theories (HQFTs) as G-equivariant versions of TQFTs. In this talk, we review low dimensional (1d, 2d, and 3d) HQFTs, the resulting (G-graded) algebraic structures, and the corresponding invariants of manifolds equipped with principal G-bundles.
References: V. Turaev, Homotopy Quantum Field Theories. European Mathematical Society, Tracts in Mathematics 10 (2010).
V. Turaev and A. Virelizier, On 3-dimensional Homotopy Quantum Field Theory, Part I. Int. J. Math. 23 (2012), 01-28.
V. Turaev and A. Virelizier, On 3-dimensional Homotopy Quantum Field Theory, Part II: The surgery approach. Int. J. Math. 25 (2014), 1-66.
V. Turaev and A. Virelizier, On 3-dimensional Homotopy Quantum Field Theory, Part III: Comparison of two approaches. preprint arXiv:1911.10257 (2019)
Notes for the talk: QSSS Seminar Notes, Kursat Sozer 20.11.2020

#25: November 27th (10 am Eastern Time)
Speaker: Jorge Castillejos (Institute of Mathematics, Polish Academy of Science)
Title: A C*-version of property Gamma
Abstract: Property Gamma was introduced by Murray and von Neumann as a way to show the existence of non-hyperfinite II_1 factors. This property also played a key role in Connes’ classification of injective factors and, outside the injective setting, it remains as a key topic for von Neumann algebras. I will discuss a new C*-version of this property that has played a relevant role in the classification and structure of nuclear C*-algebras.