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Timetable (OASW01)

Structure of operator algebras: subfactors and fusion categories

Monday 23rd January 2017 to Friday 27th January 2017

Monday 23rd January 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
10:00 to 11:00 Sorin Popa
On rigidity in II1 factor framework
II1 factors appear naturally from a multitude of data (groups, group actions, operations such as free products, etc). 
This leads to two types of rigidity phenomena in this framework:  
1. W*-rigidity, aiming at recovering the building data from the isomorphism class of the algebra. 
2. Restrictions on the symmetries of the II1 factor (like the index of its subfactors). 
We will discuss some old and new results in this direction, and the role of deformation-rigidity techniques 
in obtaining them.  
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Stephen Bigelow
A diagrammatic approach to Ocneanu cells
Kuperberg's SU(3) spider has "web" diagrams with oriented strands and trivalent vertices. A closed web evaluates to a real number, which can be thought of as a weighted sum of certain ways to "colour" the faces of the web. The weighting here is defined using Ocneanu cells, which were explicitly calculated in a 2009 paper by Evans and Pugh. I will describe a diagrammatic way to recover their calculation in the simplest case of the A series. Each strand of a web becomes a parallel pair of coloured strands, and each vertex becomes three coloured strands that connect up the three incoming pairs of coloured strands.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Corey Jones
Operator Algebras in rigid C*-tensor categories
In this talk, we will describe a theory of operator algebra objects in an arbitrary rigid C*-tensor category C.  Letting C be the category of finite dimensional Hilbert spaces, we recover the ordinary theory of operator algebras.  We will explain the philosophy and motivation for this framework, and how it provides a unified perspective on various aspects of the theories of rigid C*-tensor categories, quantum groups, and subfactors.  This is based on joint work with Dave Penneys.
14:30 to 15:30 David Jordan
Dualizability and orientability of tensor categories
A topological field theory is an invariant of oriented manifolds, valued in some category C, with many pleasant properties.  According to the cobordism hypothesis, a fully extended -- a.k.a. fully local -- TFT is uniquely determined by a single object of C, which we may think of as the invariant assigned by the theory to the point.  This object must have strong finiteness properties, called dualizability, and strong symmetry properties, called orientability.

In this talk I'd like to give an expository discussion of several recent works "in dimension 1,2, and 3" -- of Schommer-Pries, Douglas--Schommer-Pries--Snyder, Brandenburg-Chivrasitu-Johnson-Freyd, Calaque-Scheimbauer -- which unwind the abstract notions of dualizability and orientability into notions very familiar to the assembled audience:  things like Frobenius algebras, fusion categories, pivotal fusion categories, modular tensor categories.  Finally in this context, I'll discuss some work in progress with Adrien Brochier and Noah Snyder, which finds a home on these shelves for arbitrary tensor and pivotal tensor categories (no longer finite, or semi-simple), and for braided and ribbon braided tensor categories.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Yusuke Isono
On fundamental groups of tensor product II_1 factors
We study a stronger notion of primeness for II_1 factors, which was introduced in my previous work. Using this, we prove that if G and H are groups which are realized as fundamental groups of II_1 factors, then so are groups GH and G \cap H.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 24th January 2017
10:00 to 11:00 George Elliott
The classification of unital simple separable C*-algebras with finite nuclear dimension
As, perhaps, a climax to forty years of work by many people, the class of algebras in the title (assumed also to satisfy the UCT, which holds in all concrete examples and may be automatic) can now be classified by means of elementary invariants (the K-groups and tracial simplex).
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Stuart White
The structure of simple nuclear C*-algebras: a von Neumann prospective
I'll discuss aspects of structure of simple nuclear C*-algebras ( in particular the Toms-Winter regularity conjecture) drawing parallels with results for injective von Neumann algebras.

12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Wilhelm Winter
Structure and classification of nuclear C*-algebras: The role of the UCT
The question whether all separable nuclear C*-algebras satisfy the Universal Coefficient Theorem remains one of the most important open problems in the structure and classification theory of such algebras. It also plays an integral part in the connection between amenability and quasidiagonality. I will discuss several ways of looking at the UCT problem, and phrase a number of intermediate questions. This involves the existence of Cartan MASAS on the one hand, and certain kinds of embedding problems for strongly self-absorbing C*-algebras on the other.
14:30 to 15:30 Sam Evington
W$^*$-Bundles and Continuous Families of Subfactors
W$^*$-bundles were first introduced by Ozawa, motivated by work on the Toms-Winter Conjecture and, more generally, the classification of C$^*$-algebras.

I will begin with a brief introduction to W$^*$-bundles, explaining how they combine the measure theoretic nature of tracial von Neumann algebras with the topological nature of C$^*$-algebras. I will then discuss the relationship between the triviality problem for W$^*$-bundles and the Toms-Winter Conjecture. Finally, I will present my work with Ulrich Pennig on locally trivial W$^*$-bundles and my ongoing work on expected subbundles of W$^*$-bundles inspired by subfactor theory.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Koichi Shimada
A classification of real-line group actions with faithful Connes--Takesaki modules on hyperfinite factors
  We classify certain real-line-group actions on (type III) hyperfinite factoers, up to cocycle conjugacy. More precisely, we show that an invariant called the Connes--Takesaki module completely distinguishs actions which are not approximately inner at any non-trivial point. Our classification result is related to the uniqueness of the hyperfinite type III_1 factor, shown by Haagerup, which is equivalent to the uniquness of real-line-group actions with a certain condition on the hyperfinite type II_{\infty} factor. We classify actions on hyperfinite type III factors with an analogous condition. The proof is based on Masuda--Tomatsu's recent work on real-line-group actions and the uniqueness of the hyperfinite type III_1 factor.
Wednesday 25th January 2017
10:00 to 11:00 Stefaan Vaes
Classification of free Araki-Woods factors
Co-authors: Cyril Houdayer (Université Paris Sud) and Dimitri Shlyakhtenko (UCLA).
Free Araki-Woods factors are a free probability analog of the type III hyperfinite factors. They were introduced by Shlyakhtenko in 1996, who completely classified the free Araki-Woods factors associated with almost periodic orthogonal representations of the real numbers. I present a joint work with Houdayer and Shlyakhtenko in which we completely classify a large class of non almost periodic free Araki-Woods factors. The key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states on a von Neumann algebra.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Dima Shlyakhtenko
Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions
Co-authors: Sorin Popa (UCLA) and Stefaan Vaes (Leuven)

We introduce L$^2$-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II$_1$ factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups Γ, we recover the ordinary (co)homology of Γ. For Cartan subalgebras, we recover Gaboriau's L$^2$-Betti numbers for the associated equivalence relation. In this common framework, we prove that the L$^2$-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the L$^2$-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Arnaud Brothier
Crossed-products by locally compact groups and intermediate subfactors.
I will present examples of an action of a totally disconnected group G on a factor Q such that intermediate subfactors between Q and the crossed-product correspond to closed subgroups of G. This extends previous work of Choda and Izumi-Longo-Popa. I will discuss about the analytical difference with the case of actions of discrete groups regarding the existence of conditional expectations or operator valued weights. Finally I will talk about intermediate subfactors in the context of actions of Hecke pairs of groups. This is a joint work with Rémi Boutonnet.
14:30 to 15:30 Alexei Semikhatov
Screening operators in conformal field models and beyond
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Alice Guionnet
19:30 to 22:00 Formal Dinner at Emmanuel College
Thursday 26th January 2017
10:00 to 11:00 Benjamin Doyon
Conformal field theory out of equilibrium
Non-equilibrium conformal field theory is the application of methods of conformal field theory to states that are far from equilibrium. I will describe exact results for current-carrying steady-states that occur in the partitioning protocol: two baths (half-lines) are independently thermalized at different temperatures, then joined together and let to evolve for a large time. Results include the exact energy current, the exact scattering map describing steady-state averages and correlations of all fields in the energy sector (the stress-energy tensor and its descendants), and the full scaled cumulant generating function describing the fluctuations of energy transport. I will also explain how, in space-time, the steady state occurs between contact discontinuities beyond which lie the asymptotic baths. If time permits, I will review how these results generalize to higher-dimensional conformal field theory, and to non-conformal integrable models. This is work in collaboration with Denis Bernard.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Alina Vdovina
Buildings and C*-algebras
We will give an elementary introduction to the theory of buildingsfrom a geometric point of view. Namely, we present buildings as universal coversof finite polyhedral complexes. It turns out that the combinatorial structure of these complexesgives rise to a large class of higher rank Cuntz-Krieger algebras, which K-theory can be explicitly computed.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Claus Kostler
An elementary approach to unitary representations of the Thompson group F
I provide an elementary construction of  unitary representations of the Thompson group F.  Further I will motivate this new approach by recent results on distributional symmetries in noncommutative probability.  My talk is based on joined work with Rajarama Bhat, Gwion Evans, Rolf Gohm and Stephen Wills.
14:30 to 15:30 Rolf Gohm
Braids, Cosimplicial Identities, Spreadability, Subfactors
Actions of a braid monoid give rise to cosimplicial identities. Cosimplicial identities for morphisms of (noncommutative) probability spaces lead to spreadable processes for which there is a (noncommutative) de Finetti type theorem. This scheme can be applied to braid group representations from subfactors. We discuss results and open problems of this approach. This is joint work with G. Evans and C. Koestler.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Alexei Davydov
Modular invariants for group-theoretical modular data
Group-theoretical modular categories is a class of modular categories for which modular invariants can be described effectively (in group-theoretical terms). This description is useful for applications in conformal field theory, allowing classification of full CFTs with given chiral halves being holomorphic orbifolds. In condensed matter physics it can be used to classify possible boson condensations. It also provides ways of studying braided equivalences between group-theoretical modular categories. The class of modular categories can be used to provide examples of counter-intuitive behaviour of modular invariants: multiple physical realisations of a given modular invariant, non-physicality of some natural modular invariants. The talk will try to give an overview of known results and open problems.
Friday 27th January 2017
10:00 to 11:00 Julia Plavnik
On gauging symmetry of modular categories
Co-authors: Shawn X. Cui ( Stanford University), César Galindo (Universidad de los Andes), Zhenghan Wang (Microsoft Research, Station QUniversity of CaliforniaSanta Barbara)
A very interesting class of fusion categories is the one formed by modular categories. These categories arise in a variety of mathematical subjects including topological quantum field theory, conformal field theory, representation theory of quantum groups, von Neumann algebras, and vertex operator algebras. In addition to the mathematical interest, a motivation for pursuing a classification of modular categories comes from their application in condensed matter physics and quantum computing.
Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. In this talk, we will present a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a unitary modular category (UMC) with a symmetry group G, gauging is a 2-step process: first extend the UMC to a G-crossed braided fusion category and then take the equivariantization of the resulting category. This is an useful tool to construct new modular categories from given ones. We will show through concrete examples which are the ingredients involved in this process. In addition, if time allows, we will mention some classification results and conjectures associated to the notion of gauging. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Pinhas Grossman
Algebras, automorphisms, and extensions of quadratic fusion categories
To a finite index subfactor there is a associated a tensor category along with a distinguished algebra object. If the subfactor has finite depth, this tensor category is a fusion category. The Brauer-Picard group of a fusion category, introduced by Etingof-Nikshych-Ostrik, is the (finite) group of Morita autoequivalences. It contains as a subgroup the outer automorphism group of the fusion category. In this talk we will decribe the Brauer-Picard groups of some quadratic fusion categories as groups of automorphisms which move around certain algebra objects. Combining this description with an operator algebraic construction, we can classify graded extensions of the Asaeda-Haagerup fusion categories. This is joint work with Masaki Izumi and Noah Snyder.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Noah Snyder
Trivalent Categories
If N
14:30 to 15:30 Henry Tucker
Eigenvalues of rotations and braids in spherical fusion categories
Co-authors: Daniel Barter (University of Michigan), Corey Jones (Australian National University)

Using the generalized categorical Frobenius-Schur indicators for semisimple spherical categories we have established formulas for the multiplicities of eigenvalues of generalized rotation operators. In particular, this implies for a finite depth planar algebra, the entire collection of rotation eigenvalues can be computed from the fusion rules and the traces of rotation at finitely many depths. If the category is also braided these formulas yield the multiplicities of eigenvalues for a large class of braids in the associated braid group representations. This provides the eigenvalue multiplicities for braids in terms of just the S and T matrices in the case where the category is modular.

Related Links
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 David Penneys
Operator algebras in rigid C*-tensor categories, part II
In this talk, we will first define a (concrete) rigid C*-tensor category. We will then highlight the main features that are important to keep in mind when passing to the abstract setting. I will repeat a fair amount of material on  C*/W* algebra objects from Corey Jones' Monday talk. Today's goal will be to prove the Gelfand-Naimark theorem for C*-algebra objects in Vec(C). To do so, we will have to understand the analog of the W*-algebra B(H) as an algebra object in Vec(C). In the remaining time, we will elaborate on the motivation for the project from the lens of enriched quantum symmetries. This talk is based on joint work with Corey Jones (arXiv:1611.04620).

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons