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Seminars (OAS)

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Event When Speaker Title Presentation Material
OASW04 11th January 2017
13:30 to 15:00
David Penneys Introduction to subfactor theory
In this series of lectures, I'll give an introduction to the modern theory of subfactors initiated by Vaughan Jones. We'll begin with invariants for subfactors, like the index, the principal graph, and the standard invariant. We'll then discuss Jones' planar algebras as an elegant and powerful tool for the construction and classification of subfactors. The standard invariant can also be seen as a unitary 2-category, and the categorical framework has been very important for recent results. Finally, we'll discuss the classification of 'small' examples from several viewpoints.
OASW04 11th January 2017
15:00 to 16:30
Yasu Kawahigashi Subfactors, tensor categories and conformal field theory
I will give introductory discussions on type III factors, the Tomita-Takesaki theory, type III subfactors, tensor categories, braiding, quantum doubles, alpha-induction and local conformal nets.
OASW04 12th January 2017
13:30 to 15:00
David Penneys Introduction to subfactor theory
In this series of lectures, I'll give an introduction to the modern theory of subfactors initiated by Vaughan Jones. We'll begin with invariants for subfactors, like the index, the principal graph, and the standard invariant. We'll then discuss Jones' planar algebras as an elegant and powerful tool for the construction and classification of subfactors. The standard invariant can also be seen as a unitary 2-category, and the categorical framework has been very important for recent results. Finally, we'll discuss the classification of 'small' examples from several viewpoints.
OASW04 12th January 2017
15:00 to 16:30
Yasu Kawahigashi Subfactors, tensor categories and conformal field theory
I will give introductory discussions on type III factors, the Tomita-Takesaki theory, type III subfactors, tensor categories, braiding, quantum doubles, alpha-induction and local conformal nets.
OASW04 13th January 2017
13:30 to 15:00
David Penneys Introduction to subfactor theory
In this series of lectures, I'll give an introduction to the modern theory of subfactors initiated by Vaughan Jones. We'll begin with invariants for subfactors, like the index, the principal graph, and the standard invariant. We'll then discuss Jones' planar algebras as an elegant and powerful tool for the construction and classification of subfactors. The standard invariant can also be seen as a unitary 2-category, and the categorical framework has been very important for recent results. Finally, we'll discuss the classification of 'small' examples from several viewpoints.
OASW04 13th January 2017
15:00 to 16:30
Yasu Kawahigashi Subfactors, tensor categories and conformal field theory
I will give introductory discussions on type III factors, the Tomita-Takesaki theory, type III subfactors, tensor categories, braiding, quantum doubles, alpha-induction and local conformal nets.
OASW04 16th January 2017
13:30 to 15:00
David Penneys Introduction to subfactor theory
In this series of lectures, I'll give an introduction to the modern theory of subfactors initiated by Vaughan Jones. We'll begin with invariants for subfactors, like the index, the principal graph, and the standard invariant. We'll then discuss Jones' planar algebras as an elegant and powerful tool for the construction and classification of subfactors. The standard invariant can also be seen as a unitary 2-category, and the categorical framework has been very important for recent results. Finally, we'll discuss the classification of 'small' examples from several viewpoints.
OASW04 16th January 2017
15:00 to 16:30
Yasu Kawahigashi Subfactors, tensor categories and conformal field theory
I will give introductory discussions on type III factors, the Tomita-Takesaki theory, type III subfactors, tensor categories, braiding, quantum doubles, alpha-induction and local conformal nets.
OASW04 17th January 2017
13:30 to 15:00
Roberto Longo Operator Algebras and Conformal Field Theory
OASW04 17th January 2017
15:00 to 16:30
Stefaan Vaes Representation theory, cohomology and L^2-Betti numbers for subfactors
The standard invariant of a subfactor can be viewed in different ways as a ``discrete group like'' mathematical structure - a lambda-lattice in the sense of Popa, a Jones planar algebra, or a C*-tensor category of bimodules. This discrete group point of view will be the guiding theme of the mini course. After an introduction to different approaches to the standard invariant, I will present joint work with Popa and Shlyakhtenko on the unitary representation theory of these structures, on approximation and rigidity properties like amenability, the Haagerup property or property (T), on (co)homology and $L^2$-Betti numbers. I will present several examples and also discuss a number of open problems on the realization of standard invariants through hyperfinite subfactors.
OASW04 18th January 2017
09:00 to 10:30
Roberto Longo Operator Algebras and Conformal Field Theory
OASW04 18th January 2017
10:30 to 12:00
Stefaan Vaes Representation theory, cohomology and L^2-Betti numbers for subfactors
The standard invariant of a subfactor can be viewed in different ways as a ``discrete group like'' mathematical structure - a lambda-lattice in the sense of Popa, a Jones planar algebra, or a C*-tensor category of bimodules. This discrete group point of view will be the guiding theme of the mini course. After an introduction to different approaches to the standard invariant, I will present joint work with Popa and Shlyakhtenko on the unitary representation theory of these structures, on approximation and rigidity properties like amenability, the Haagerup property or property (T), on (co)homology and $L^2$-Betti numbers. I will present several examples and also discuss a number of open problems on the realization of standard invariants through hyperfinite subfactors.
OASW04 19th January 2017
13:30 to 15:00
Roberto Longo Operator Algebras and Conformal Field Theory
OASW04 19th January 2017
15:00 to 16:30
Stefaan Vaes Representation theory, cohomology and L^2-Betti numbers for subfactors
The standard invariant of a subfactor can be viewed in different ways as a ``discrete group like'' mathematical structure - a lambda-lattice in the sense of Popa, a Jones planar algebra, or a C*-tensor category of bimodules. This discrete group point of view will be the guiding theme of the mini course. After an introduction to different approaches to the standard invariant, I will present joint work with Popa and Shlyakhtenko on the unitary representation theory of these structures, on approximation and rigidity properties like amenability, the Haagerup property or property (T), on (co)homology and $L^2$-Betti numbers. I will present several examples and also discuss a number of open problems on the realization of standard invariants through hyperfinite subfactors.
OASW04 20th January 2017
09:00 to 10:30
Roberto Longo Operator Algebras and Conformal Field Theory
OASW04 20th January 2017
10:30 to 12:00
Stefaan Vaes Representation theory, cohomology and L^2-Betti numbers for subfactors
The standard invariant of a subfactor can be viewed in different ways as a ``discrete group like'' mathematical structure - a lambda-lattice in the sense of Popa, a Jones planar algebra, or a C*-tensor category of bimodules. This discrete group point of view will be the guiding theme of the mini course. After an introduction to different approaches to the standard invariant, I will present joint work with Popa and Shlyakhtenko on the unitary representation theory of these structures, on approximation and rigidity properties like amenability, the Haagerup property or property (T), on (co)homology and $L^2$-Betti numbers. I will present several examples and also discuss a number of open problems on the realization of standard invariants through hyperfinite subfactors.
OASW01 23rd January 2017
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.  
  
 
OASW01 23rd January 2017
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.
OASW01 23rd January 2017
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.
OASW01 23rd January 2017
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.
OASW01 23rd January 2017
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.
OASW01 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).
OASW01 24th January 2017
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.

OASW01 24th January 2017
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.
OASW01 24th January 2017
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.
OASW01 24th January 2017
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.
 
OASW01 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.

OASW01 25th January 2017
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.
OASW01 25th January 2017
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.
OASW01 25th January 2017
14:30 to 15:30
Alexei Semikhatov Screening operators in conformal field models and beyond
OASW01 25th January 2017
16:00 to 17:00
Alice Guionnet tba
OASW01 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.
OASW01 26th January 2017
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.
OASW01 26th January 2017
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.
OASW01 26th January 2017
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.
OASW01 26th January 2017
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.
OASW01 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. 
OASW01 27th January 2017
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.
OASW01 27th January 2017
13:30 to 14:30
Noah Snyder Trivalent Categories
If N
OASW01 27th January 2017
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
OASW01 27th January 2017
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).


OAS 31st January 2017
15:30 to 16:30
Scott Morrison The Temperley-Lieb category in operator algebras and in link homology
The Temperley-Lieb category appears in a fundamental way in both the study of subfactors and in link homology theories. Indeed, the discovery of the importance of the Temperley-Lieb category for subfactors led to the creation of the Jones polynomial, and thence, after a long gestation, Khovanov homology.
OAS 7th February 2017
14:00 to 15:00
Yoh Tanimoto Free products in AQFT
We apply the free product construction to various local algebras in algebraic quantum field theory.

If we take the free product of infinitely many identical irreducible half-sided modular inclusions, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take finitely many Moebius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with a large relative commutant.

(joint work with R. Longo and Y. Ueda)



OAS 9th February 2017
14:00 to 15:00
Vincenzo Morinelli Conformal covariance and the split property
Several important structural properties of a quantum field theory are known to be automatic in the conformal case. The split property is the statistical independence of local algebras associated to regions with a positive (spacelike) separation. We show that in chiral theories when the full conformal (i.e. diffeomorphism) covariance is assumed, then the split property holds. Time permitting, we also provide an example of a two-dimensional conformal net that does not have the split property.

The talk relies on the joint work "Conformal covariance and the split property" with Y. Tanimoto (Uni. of Rome "Tor Vergata"), M. Weiner (Budapest Uni. of Technology and Economics), arXiv:1609.02196.



OAS 14th February 2017
14:00 to 15:00
Masaki Izumi Indecomposable characters of infinite dimensional groups associated with operator algebras
A character of a topological group is a normalized continuous positive definite class function on the group. I'll give an account of recent classification results on characters of infinite dimensional groups associated with operator algebras, including the unitary groups of unital simple AF algebras and II_1 factors.



OAS 16th February 2017
14:00 to 15:00
Stefano Rossi Endomorphisms and automorphisms of the 2-adic ring C*-algebra Q_2
The 2-adic ring C*-algebra is the universal C*-algebra Q_2 generated by an isometry S_2 and a unitary U such that S_2U=U^2S_2 and S_2S_2^*+US_2S_2^*U^*=1. By its very definition it contains a copy of the Cuntz algebra O_2. I'll start by discussing some nice properties of this inclusion, as they came to be pointed out in a recent joint work with V. Aiello and R. Conti. Among other things, the inclusion enjoys a kind of rigidity property, i.e., any endomorphism of the larger that restricts trivially to the smaller must be trivial itself. I'll also say a word or two about the extension problem, which is concerned with extending an endomorphism of O_2 to an endomorphism of Q_2. As a matter of fact, this is not always the case: a wealth of examples of non-extensible endomorphisms (automophisms indeed!) show up as soon as the so-called Bogoljubov automorphisms of O_2 are looked at. Then I'll move on to particular classes of endomorphisms and automorphisms of Q_2, including those fixing the diagonal D_2. Notably, the semigroup of the endomorphisms fixing U turns out to be a maximal abelian group isomorphic with the group of continuous functions from the one-dimensional torus to itself. Such an analysis, though, calls for some knowledge of the inner structure of Q_2.  More precisely, it's vital to prove that C*(U) is a maximal commutative subalgebra. Time permitting, I'll also try to present forthcoming generalizations to broader classes of C*-algebras, on which we're currently working with N. Stammeier as well.



OAS 21st February 2017
14:00 to 15:00
Denjoe O'Connor Membrane Matrix Models and non-perturbative tests of gauge/gravity
I will review how relativistic membranes lead to membrane matrix models
and compare non-perturbative studies of some of these matrix models with
results from gravitational predictions. The principal models of interest
will be the BFSS, BMN and Berkooz-Douglas models.



OAS 23rd February 2017
14:00 to 15:00
Ivan Todorov Herz-Schur multipliers of dynamical systems
Herz-Schur multipliers of a locally compact group, introduced by Haagerup and de Canniere in 1985, have been instrumental in operator algebra theory in a variety of contexts, in particular in the study of approximation properties of group operator algebras. They can be viewed as the invariant part of the Schur multipliers - a class of maps on B(H) with another long list of applications, e.g. in perturbation theory of linear operators. In this talk, which is based on a joint work with A. McKee and L. Turowska, I will introduce operator-valued Schur and Herz-Schur multipliers of arbitrary locally compact groups. The latter give rise to natural maps on C*- and von Neumann algebra crossed products. I will present a characterisation of operator-valued Herz-Schur multipliers as the invariant part of the operator-valued Schur multipliers, and will discuss various special cases which highlight the generality of this class of maps and their potential usefulness in subsequent research.



OAS 28th February 2017
14:00 to 15:00
Evgenios Kakariadis Semigroup actions on operator algebras
We will present a dilation technique from commuting endomorphisms to commuting automophisms on a larger C*-algebra such that the corresponding (minimal) Cuntz-Nica-Pimsner algebras are strong Morita equivalent. Hence we can reduce problems on semigroup actions to problems on group actions. Further consequences of our analysis include the association of the ideal structure/nuclearity/exactness of the Nica-Pimsner algebras with minimality-freeness/nuclearity/exactness of the C*-dynamics.



OAS 2nd March 2017
14:00 to 15:00
Stephen Moore A generalization of the Temperley-Lieb algebra from restricted quantum sl2
The Temperley-Lieb algebra was introduced in relation to lattice models in statistical mechanics, before being rediscovered in the standard invariant of subfactors. Alternatively, the Temperley-Lieb algebra can be constructed as the centralizer of the quantum group Uq(sl2). Recent work in logarithmic conformal field theory has brought interest to a restricted version of this quantum group. We generalize the Temperley-Lieb construction to the restricted case, describing generators and a number of relations, then describe morphisms between modules, including a conjecture for the formula for projections onto indecomposable modules.





OAS 7th March 2017
14:00 to 15:00
Keith Hannabuss T-duality and the condensed matter bulk-boundary correspondence
This talk will start with a brief historical review of the classification of solids by their symmetries, and the more recent K-theoretic periodic table of Kitaev. It will then consider some mathematical questions this raises, in particular about the behaviour of electrons on the boundary of materials and in the bulk. Two rather different models will be described, which turn out to be related by T-duality. Relevant ideas from noncommutative geometry will be explained where needed.



OAS 9th March 2017
14:00 to 15:00
Andreas Aaserud Approximate equivalence of measure-preserving actions
I will talk about measure-preserving actions of countable discrete groups on probability spaces. Classically, one mainly considers two notions of equivalence of such actions, namely conjugacy (or isomorphism) and orbit equivalence, both of which have nice descriptions in the language of von Neumann algebras. I will briefly discuss this classical framework before going into some new notions of equivalence of actions. These are approximate versions of conjugacy and orbit equivalence that were recently introduced and investigated by Sorin Popa and myself, and which can most easily be defined in terms of ultrapowers of von Neumann algebras. I will discuss superrigidity within this new framework, and will also compare approximate conjugacy to (classical) conjugacy for actions of various classes of groups. This talk is based on joint work with Sorin Popa.




OAS 14th March 2017
14:00 to 15:00
Paul Fendley Tutte's golden identity from a fusion category
The chromatic polynomial \chi(Q) can be defined on any graph, such that for Q integer it counts the number of colourings. In statistical mechanics, it is known as the partition function of the antiferromagnetic Potts model on that graph. It has many remarkable properties, and Tutte's golden identity is one of the more unusual ones. For any planar triangulation, it relates \chi(\phi+2) to the square of \chi(\phi+1), where \phi is the golden mean. Tutte's original proof is purely combinatorial. I will give here an elementary proof using fusion categories, which are familiar for example from topological quantum field theory, anyonic quantum mechanics, and integrable statistical mechanics. In this setup, the golden identity follows by simple manipulations of a topological invariant related to the Jones polynomial. I will also mention recent work by Agol and Krushkal on understanding what happens to the identity for graphs on more general surfaces.



OAS 16th March 2017
14:00 to 15:00
Makoto Yamashita Weak Morita equivalence of compact quantum groups
Motivated by the 2-categorical interpretation of constructs in subfactor theory, Müger introduced the notion of weak Morita equivalence for tensor categories. This relation roughly says that the tensor categories have the same quantum double, or the same "representation theory". We give a characterization of this equivalence relation for representation categories of compact quantum groups in terms of certain commuting actions. This extends a similar characterization of monoidal equivalence due to Schauenburg and Bichon-De Rijdt-Vaes. Based on joint work with Sergey Neshveyev.



OAS 21st March 2017
14:00 to 15:00
Yasu Kawahigashi A relative tensor product of rational full conformal field theories
We introduce a relative tensor product of two heterotic rational 2-dimensional conformal field theories with trivial representation theories. Such a conformal field theory has a decomposition characterized by modular invariance, and this gives a generalization of fusion rules of modular invariants.



OASW02 27th March 2017
10:00 to 11:00
Vaughan Jones The semicontinuous limit of quantum spin chains
We construct certain states of a periodic quantum spin chain whose length is a power of two and show that they are highly uncorrelated with themselves under a rotation by one lattice spacing.
OASW02 27th March 2017
11:30 to 12:30
Gus Isaac Lehrer Semisimple quotients of Temperley-Lieb
Co-authors: Kenji Iohara (Universite de Lyon), Ruibin Zhang (University of Sydney)
 
The maximal semisimple quotients of the Temperley-Lieb algebras at roots of unity are fully described by means of a presentation, and the dimensions of their simple modules are explicitly determined. Possible applications to Virasoro limits will be discussed.
OASW02 27th March 2017
13:30 to 14:30
N. Christopher Phillips Operator algebras on L^p spaces
 It has recently been discovered that there are algebras on L^pspaces which deserve to be thought of as analogs of selfadjoint operator algebras on Hilbert spaces (even though there is no adjoint on the algebra of bounded operators on an L^pspace).
 
We have analogs of some of the most common examples of Hilbert space operator algebras, such as the 
AF Algebras, the irrational rotation algebras, group C*-algebras and von Neumann algebras, more general crossed products, the Cuntz algebras, and a few others. We have been able to prove analogs of some of the standard theorems about these algebras. We also have some ideas towards when an operator algebra on an L^p space deserves to be considered the analog of a C*-algebra or a von Neumann algebra. However, there is little general theory and there are many open questions, particularly for the analogs of von Neumann algebras.
 
In this talk, we will try to give an overview of some of what is known and some of the interesting open questions.  
OASW02 27th March 2017
14:30 to 15:30
James Tener A geometric approach to constructing conformal nets
Conformal nets and vertex operator algebras are distinct mathematical axiomatizations of roughly the same physical idea: a two-dimensional chiral conformal field theory. In this talk I will present recent work, based on ideas of André Henriques, in which local operators in conformal nets are realized as "boundary values" of vertex operators. This construction exhibits many features of conformal nets (e.g. subfactors, their Jones indices, and their fusion rules) in terms of vertex operator algebras, and I will discuss how this allows one to use Antony Wassermann's approach to calculating fusion rules in a broad class of examples.
OASW02 27th March 2017
16:00 to 17:00
Thomas Schick Geometric models for twisted K-homology
Co-author: Paul Baum (Penn State University)

K-homology, the homology theory dual to K-theory, can be described in a number of quite distinct models. One of them is analytic, uses Kasparov's KK-theory, and is the home of index problems. Another one uses geometric cycles, going back to Baum and Douglas. A large part of index theory is concerned with the isomorphism between the geoemtric and the analytic model, and with Chern character transformations to (co)homology.

In applications to string theory, and for certain index problems, twisted versions of K-theory and K-homology play an essential role.
 
We will descirbe the general context, and then focus on two new models for twisted K-homology and their applications and relations. These aere again based on geometric cycles in the spirit of Baum and Douglas. We will include in particular precise discussions of the different ways to define and work with twists (for us, classified by elements of the third integral cohomology group of the base space in question).
OASW02 28th March 2017
10:00 to 11:00
Osamu Iyama Preprojective algebras and Calabi-Yau algebras
Preprojective algebras are one of the central objects in representation theory. The preprojective algebra of a quiver Q is a graded algebra whose degree zero part is the path algebra kQ of Q, and each degree i part gives a distinguished class of representations of Q, called the preprojective modules. It categorifies the Coxeter groups as reflection functors, and their structure depends on the trichotomy of quivers: Dynkin, extended Dynkin, and wild. From homological algebra point of view, the algebra kQ is hereditary (i.e. global dimension at most one), and its preprojective algebra is 2-Calabi-Yau.
In this talk, I will discuss the higher preprojective algebras P of algebras A of finite global dimension d. When d=2, then P is the Jacobi algebra of a certain quiver with potential. When A is a d-hereditary algebra, a certain distinguished class of algebras of global dimension d, then its higher preprojective algebra is (d+1)-Calabi-Yau. I will explain results and examples of higher preprojective algebras based on joint works with Herschend and Oppermann. If time permits, I will explain a joint work with Amiot and Reiten on algebraic McKay correspondence for higher preprojective algebras.
OASW02 28th March 2017
11:30 to 12:30
Karin Erdmann Periodicity for finite-dimensional selfinjective algebras
 We give a survey on finite-dimensional selfinjective algebras which are periodic as bimodules, with respect to syzygies, and hence are stably Calabi-Yau. These include preprojective algebras of Dynkin types ADE and deformations, as well a class of algebras which we call mesh algebras of generalized Dynkin type. There is also a classification of the selfinjective algebras of polynomial growth which are periodic. Furthermore, we introduce weighted surface algebras, associated to triangulations of compact surfaces, they are tame and symmetric, and have period 4 (they are 3-Calabi-Yau). They generalize Jacobian algebras, and also blocks of finite groups with quaternion defect groups.
 
In general, for such an algebra, all one-sided simple modules are periodic. One would like to know whether the converse holds: Given a finite-dimensional selfinjective algebra A for which all one-sided simple modules are periodic. It is known that then some syzygy of A is isomorphic as a bimodule to some twist of A by an automorphism. It is open whether then A must be periodic.
OASW02 28th March 2017
13:30 to 14:30
Alastair King Quivers and CFT: preprojective algebras and beyond
I will describe the intimate link between ADE preprojective algebras and Conformal Field Theories arising as SU(2) WZW models and explain some of what happens in the SU(3) case.
OASW02 28th March 2017
14:30 to 15:30
Mathew Pugh Frobenius algebras from CFT
Co-author: David Evans (Cardiff University)

I will describe the construction of certain preprojective algebras and their generalisations arising from WZW models in Conformal Field Theory. For the case of SU(2), these algebras are the preprojective algebras of ADE type. More generally, these algebras are Frobenius algebras, and their Nakayama automorphism measures how far away the algebra is from being symmetric. I will describe how the Nakayama automorphism arises from this construction, and will describe the construction in the SO(3) case.
OASW02 28th March 2017
16:00 to 17:00
Joseph Grant Higher preprojective algebras and higher zigzag algebras
Co-author: Osamu Iyama (Nagoya University)
 
I will give a brief introduction to preprojective algebras and higher preprojective algebras, using examples. Then I will explain some results about certain higher preprojective algebras that were obtained in joint work with Osamu Iyama, including results on periodicity and descriptions of these algebras as (higher) Jacobi algebras of quivers with potential. Finally I will explain how, in simple cases, one can define higher zigzag algebras, generalizing certain algebras studied by Huefrano and Khovanov which appear widely in geometry and representation theory. As in the classical case, one finds interesting relations between spherical twists on the derived categories of these algebras.
OASW02 29th March 2017
09:00 to 10:00
Akhil Mathew Polynomial functors and algebraic K-theory
The Grothendieck group K_0 of a commutative ring is well-known to be a λ-ring: although the exterior powers are non-additive, they induce maps on K_0 satisfying various universal identities. The λ-operations yield homomorphisms on higher K-groups. In joint work in progress with Glasman and Nikolaus, we give a general framework for such operations. Namely, we show that the K-theory space is naturally functorial for polynomial functors, and describe a universal property of the extended K-theory functor. This extends an earlier algebraic result of Dold for K_0. In this picture, the λ-operations come from the strict polynomial functors of Friedlander-Suslin.
OASW02 29th March 2017
10:00 to 11:00
Paul Smith A classification of some 3-Calabi-Yau algebras
This is a report on joint work with Izuru Mori and work of Mori and Ueyama.
A graded algebra A is Calabi-Yau of dimension n if the homological shift A[n] is a dualizing object in the appropriate derived category. For example, polynomial rings are Calabi-Yau algebras. Although many examples are known, there are few if any classification results. Bocklandt proved that connected graded Calabi-Yau algebras are of the form TV/(dw) where TV denotes the tensor algebra on a vector space V and (dw) is the ideal generated by the cyclic partial derivatives of an element w in TV. However, it is not known exactly which w give rise to a Calabi-Yau algebra. We present a classification of those w for which TV/(dw) is Calabi-Yau in two cases: when dim(V)=3 and w is in V^{\otimes 3} and when dim(V)=2 and w is in V^{\otimes 4}.  We also describe the structure of TV/(dw)  in these two cases and show that (most) of them are deformation quantizations of the polynomial ring on three variables. 
OASW02 29th March 2017
11:30 to 12:30
Raf Bocklandt Local quivers and Morita theory for matrix factorizations
We will discuss how to construct a Morita theory for matrix factorizations using techniques in Mirror symmetry developed by Cho, Hong and Lau and tie this to the notion of local quivers in representation theory.
OASW02 30th March 2017
10:00 to 11:00
Andre Henriques Higher twisted K-theory a la Dadarlat and Pennig
I will present aspects of the work of Marius Dadarlat and Ulrich Pennig on the relation between Cuntz algebras and higher twists of K-theory. 
OASW02 30th March 2017
11:30 to 12:30
Ulrich Pennig Equivariant higher twisted K-theory
Twisted K-Theory can be expressed in terms of section algebras of locally trivial bundles of compact operators. However, from the point of view of homotopy theory, this setup just captures a small portion of the possible twists. In joint work with Marius Dadarlat we generalised the classical theory to a C*-algebraic model, which captures the higher twists of K-theory as well and is based on strongly self-absorbing C*-algebras. In this talk I will discuss possible generalisations to the equivariant case, which is joint work with David Evans. In particular, I will first review the construction of the equivariant twist of U(n) representing the generator of its equivariant third cohomology group with respect to the conjugation action of U(n) on itself. Then I will talk about work in progress on a generalisation to (localised) higher twisted K-theory. 
  • A Dixmier-Douady theory for strongly self-absorbing C*-algebras arXiv:1302.4468
  • Unit spectra of K-theory from strongly self-absorbing C*-algebras arXiv:1306.2583

OASW02 30th March 2017
13:30 to 14:30
Ulrich Bunke Homotopy theory with C*-categories
 In this talk I propose a presentable infinity category of C*-categories. It is modeled by a simplicial combinatorial model category structure on the category of C*-categories. This allows to set up a theory of presheaves with values in C*-categories on the orbit category of a group together with various induction and coinduction functors. As an application we provide a simple construction of equivariant K-theory spectra (first constructed by Davis-Lück). We discuss further applications to equivariant coarse homology theories.

Related Links
OASW02 30th March 2017
14:30 to 15:30
Michael Murray Real bundle gerbes
Co-authors: Dr Pedram Hekmati (University of Auckland), Professor Richard J. Szabo (Heriot-Watt University), Dr Raymond F. Vozzo (University of Adelaide)
 
Bundle gerbe modules, via the notion of bundle gerbe K-theory provide a realisation of twisted K-theory. I will discuss the generalisation to Real bundle gerbes and Real bundle gerbe modules which realise twisted Real K-theory in the sense of Atiyah. This is joint work with Richard Szabo, Pedram Hekmati and Raymond Vozzo and forms part of arXiv:1608.06466.

Related Links

OASW02 30th March 2017
16:00 to 17:00
Amihay Hanany tba
OASW02 31st March 2017
10:00 to 11:00
Katrin Wendland Vertex Operator Algebras from Calabi-Yau Geometries
Vertex operator algebras occur naturally as mathematically well tractable ingredients to conformal field theories (CFTs), capturing in general only a small part of the structure of the latter. The talk will highlight a few examples for which this procedure yields a natural route from Calabi-Yau geometries to vertex operator algebras. Moreover, we will discuss a new technique, developed in joint work with Anne Taormina, which transforms certain CFTs into vertex operator algebras and their admissible modules, thus capturing a major part of the structure of the CFT in terms of a vertex operator algebra.
OASW02 31st March 2017
11:30 to 12:30
Simon Gritschacher Coefficients for commutative K-theory
Recently, the study of representation spaces has led to the definition of a new cohomology theory, called commutative K-theory. This theory is a refinement of classical topological K-theory. It is defined using vector bundles whose transition functions commute with each other whenever they are simultaneously defined. I will begin the talk by discussing some general properties of the „classifying space for commutativity in a Lie group“ introduced by Adem-Gomez. Specialising to the unitary groups, I will then show that the spectrum for commutative complex K-theory is precisely the ku-group ring of infinite complex projective space. Finally, I will present some results about the real variant of commutative K-theory.
OASW02 31st March 2017
13:30 to 14:30
Danny Stevenson Pre-sheaves of spaces and the Grothendieck construction in higher geometry
The notion of pre-stack in algebraic geometry can be formulated either in terms of categories fibered in groupoids, or else as a functor to the category of groupoids with composites only preserved up to a coherent system of natural isomorphisms.  The device which lets one shift from one perspective to the other is known as the `Grothendieck construction' in category theory.  
A pre-sheaf in higher geometry is a functor to the ∞-category of ∞-groupoids; in this context keeping track of all the coherent natural isomorphisms between composites becomes particularly acute.  Fortunately there is an analog of the Grothendieck construction in this context, due to Lurie, which lets one `straighten out' a pre-sheaf into a certain kind of fibration.  In this talk we will give a new perspective on this straightening procedure which allows for a more conceptual proof of Lurie's straightening theorem. 

OASW02 31st March 2017
14:30 to 15:30
Yang-Hui He Calabi-Yau volumes and Reflexive Polytopes
We study various geometrical quantities for Calabi-Yau varieties realized as cones over Gorenstein Fano varieties in various dimensions, obtained as toric varieties from reflexive polytopes.
One chief inspiration comes from the equivalence of a-maximization and volume-minimization in for Calabi-Yau threefolds, coming from AdS5/CFT4 correspondence in physics.
We arrive at explicit combinatorial formulae for many topological quantities and conjecture new bounds to the Sasaki-Einstein volume function with respect to these quantities. 
Based on joint work with Rak-Kyeong Seong and Shiing-Tung Yau.

OAS 6th April 2017
13:00 to 14:00
Valeriano Aiello The oriented Thompson group, oriented links, and polynomial link invariants
Recently, Vaughan Jones discovered an unexpected connection between the Thompson groups and knots.  Among other things, he showed that any oriented link can be obtained as the "closure" of elements of the oriented Thompson group $\vec{F}$. By using this procedure we show that certain specializations of some link invariants, notably the Homfly polynomial, are functions of positive type on $\vec{F}$ (up to a renormalization).  As for other specializations, we also show that the corresponding functions are not even bounded (in particular, they are not of positive type). This talk is based on a joint work with Roberto Conti (Sapienza Università di Roma) and Vaughan Jones (Vanderbilt University).



OAS 11th April 2017
14:00 to 15:00
Kenny De Commer I-factorial quantum torsors and Heisenberg algebras of quantized enveloping type
A I-factorial quantum torsor consists of an integrable, free and ergodic action of a locally compact quantum group on a type I-factor. We show how such actions admit a nice duality theory. As an example, we consider a deformed Heisenberg algebra associated to a quantum Borel algebra of a semisimple complex Lie algebra g. We show that, endowed with a *-structure swapping the two quantum Borel algebras inside, it allows a completion into a I-factorial quantum torsor for (an amplification of) the von Neumann algebraic completion of the compact form of the quantized enveloping algebra of g.



OAS 18th April 2017
14:00 to 15:00
Yuki Arano Representation theory of Drinfeld doubles (Part 1)
We study the representation theory of Drinfeld doubles of q-deformations via the analogy with that of complex semisimple Lie groups. As applications, we prove central property (T) for general higher rank q-deformations and a Howe-Moore type theorem for q-deformations. We also review how these properties can be interpreted in terms of tensor categories and subfactors.



OAS 20th April 2017
14:00 to 15:00
Andrew McKee Herz--Schur multipliers and approximation properties
Herz--Schur multipliers of a discrete group have proved useful in the study of $C^*$-algebras, as they can be used to link properties of the group to approximation properties of the reduced group $C^*$-algebra. The development of these ideas has shed light on some $C^*$-algebra properties, and motivated the introduction and study of others.

I will introduce Herz--Schur multipliers, and discuss some of their applications, before describing a generalisation of these functions to multipliers of a $C^*$-dynamical system. In the final part of the talk I will show how the generalised Herz--Schur multipliers can be used to study approximation properties of the reduced crossed product formed by a group acting on a $C^*$-algebra, paralleling the applications of Herz--Schur multipliers; these new tools allow us to study approximation properties of the reduced crossed product without requiring the group to be amenable.




OAS 25th April 2017
14:00 to 15:00
Yuki Arano Representation theory of Drinfeld doubles (Part 2)
We study the representation theory of Drinfeld doubles of q-deformations via the analogy with that of complex semisimple Lie groups. As applications, we prove central property (T) for general higher rank q-deformations and a Howe-Moore type theorem for q-deformations. We also review how these properties can be interpreted in terms of tensor categories and subfactors.



OAS 27th April 2017
14:00 to 15:00
Ingo Runkel Traces in non-semisimple categories
Quite generally, a trace on a k-linear category is a family of functions from the endomorphisms of objects to the underlying field k, subject to cyclicity and possibly other constraints. In some interesting cases these functions may only exist for a subset of all objects. One situation where this may happen are non-semisimple braided finite tensor categories, which have applications in link invariants and in two-dimensional conformal field theory. In this talk I will present some results and conjectures related to such categories.





OAS 2nd May 2017
14:00 to 15:00
Yuki Arano Representation theory of Drinfeld doubles (Part 3)
We study the representation theory of Drinfeld doubles of q-deformations via the analogy with that of complex semisimple Lie groups. As applications, we prove central property (T) for general higher rank q-deformations and a Howe-Moore type theorem for q-deformations. We also review how these properties can be interpreted in terms of tensor categories and subfactors.



OAS 4th May 2017
14:00 to 15:00
Roland Vergnioux Free entropy dimension and the orthogonal free quantum groups
Orthogonal free quantum groups have been extensively studied in the past two decades from the operator algebraic point of view, and were shown to share many analytical properties with the ordinary free groups. In a recent preprint with Michael Brannan, we prove that the associated von Neumann algebras are strongly 1-bounded in the sense of Jung. In particular, they are not isomorphic to free group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the associated quantum Cayley tree, and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.



OAS 9th May 2017
14:00 to 15:00
Yuki Arano Representation theory of Drinfeld doubles (Part 4)
We study the representation theory of Drinfeld doubles of q-deformations via the analogy with that of complex semisimple Lie groups. As applications, we prove central property (T) for general higher rank q-deformations and a Howe-Moore type theorem for q-deformations. We also review how these properties can be interpreted in terms of tensor categories and subfactors.



OAS 16th May 2017
12:45 to 13:45
Dorothea Bahns Rieffel deformation, tempered distributions and the Gabor wavefront set
I will give gentle introduction to microlocal analysis and how to use it to define R^n Rieffel deformed products of tempered distributions. Applications of this include quantum field theory on the noncommutative Moyal space or certain warped products. It also answers the question how to extend the Weyl calculus to tempered distributions. This is joint work with René Schulz.
OAS 16th May 2017
14:00 to 15:00
Hans Wenzl Subfactors related to certain symmetric spaces
One of the first examples of subfactors came from groups and subgroups.  In this talk we construct subfactors which can be considered analogs of the embeddings of O(N) into U(N) in the setting of fusion categories.  These are finite depth subfactors whose indices will go to infinity in the classical limit.  Its principal graph will give the induction-restriction graph of these groups in the limit.



OAS 18th May 2017
14:00 to 15:00
David Kyed L^2-Betti numbers of universal quantum groups
I will report on joint works with Julien Bichon, Sven Raum, Matthias Valvekens and Stefaan Vaes, revolving around the computation of L^2-Betti numbers for universal quantum groups.  Among our main results is the fact that the first L^2-Betti number of the duals of the free unitary quantum groups equals 1, and that all other L^2-Betti numbers vanish.  All objects mentioned in the abstract will be defined, more or less rigorously, during the talk.



OAS 23rd May 2017
14:00 to 15:00
Ying-Fen Lin Nilpotent Lie groups: Fourier inversion and prime ideals
In this talk, I will give a version of the Fourier inversion theorem for connected, simply connected nilpotent Lie groups G = exp(g) by showing that there is a continuous retract from the space of adapted smooth kernel functions defined on a sub-manifold of g^* into the Schwartz functions defined on G.  As an application, I will give a characterisation of a class of invariant prime ideals of L^1(G).




OAS 25th May 2017
14:00 to 15:00
Jonathan Rosenberg H^3 and twisted K-theory for compact Lie groups
The WZW model in physics naturally leads to a study of twisted K-theory for compact Lie groups, which has been studied by Moore-Maldacena-Seiberg, Hopkins, Braun, and Douglas.  We re-examine a few aspects of this subject.  For example, what is the map on H^3 induced by a covering of compact simple Lie groups?  The result is complicated and quite surprising.  Also, what can we learn about twisted K-theory from the connection between Langlands duality and T-duality, studied by Daenzer-Van Erp and Bunke-Nikolaus?  Again, the result is rather surprising.  This is joint work with Mathai Varghese.



OAS 2nd June 2017
16:00 to 17:00
Marius Dadarlat Introduction to continuous fields of C*-algebras and their topological invariants (Part 1)
Continuous fields play the role of bundles of C*-algebras (in the sense of topology). The bundle structure that underlines a continuous fields is typically not locally trivial.

We will start by introducing various classes of examples. Thereafter, we will discuss obstructions to local triviality and give an introduction to topological invariants such as parametrized K-theory. Finally, we will present a generalized Dixmier-Douady theory obtained in joint work with Ulrich Pennig.




OAS 6th June 2017
12:45 to 13:45
Joachim Zacharias Bivariant and Dynamical Versions of the Cuntz Semigroup
 The Cuntz Semigroup is an invariant for C*-algebras combining K-theoretical and tracial information. It can be regarded as a C*-analogue of the Murray-von-Neumann semigroup of projections of a von Neumann algebra. The Cuntz semigroup plays an increasingly important role in the classification of simple C*-algebras. We propose a bivariant version of the Cuntz Semigroup based on equivalence classes of order zero maps between a given pair of C*-algebras. The resulting bivariant theory behaves similarly to Kasparov's KK-theory: it contains the ordinary Cuntz Semigroup as a special case just as KK-theory contains K-theory and admits a composition product. It can be described in different pictures similarly to the classical Cuntz Semigroup and behaves well with respect to various stabilisations. Many properties of the ordinary Cuntz Semigroup have bivariant counterparts. Whilst in general hard to determine, the bivariant Cuntz Semigroup can be computed in some special cases. Moreover, it can be used to classify stably finite algebras in analogy to the Kirchberg-Phillips classification of simple purely infinite algebras via KK-theory. We also indicate how an equivariant version of the bivariant Cuntz Semigroup can be defined, at least for compact groups. If time permits, we discuss recent work in progress on a version of the Cuntz Semigroup for dynamical systems, more precisely, groups acting on compact spaces, with potential applications to classifiability of crossed products. (Joint work with Joan Bosa, Gabriele Tornetta.)



OAS 6th June 2017
14:00 to 15:00
Marius Dadarlat Introduction to continuous fields of C*-algebras and their topological invariants (Part 2)
Continuous fields  play the role of bundles of C*-algebras (in the sense of topology). The bundle structure that underlines a continuous fields is typically not locally trivial.

We will start by introducing various classes of examples. Thereafter, we will discuss obstructions to local triviality and give an introduction
to topological invariants such as parametrized K-theory. Finally, we will present a generalized Dixmier-Douady theory obtained in joint work with Ulrich Pennig.

OAS 8th June 2017
12:45 to 13:45
Emmanuel Germain Pimsner legacy
We will discuss two major contributions of Mihai Pimsner about computations of KK-theory of C*-algebras from the 80's and 90's and show how their scope can be greatly expanded using techniques that are somehow more combinatorial than the classical 'analytical' interpretation.



OAS 8th June 2017
14:00 to 15:00
Marius Dadarlat Introduction to continuous fields of C*-algebras and their topological invariants (Part 3)
Continuous fields play the role of bundles of C*-algebras (in the sense of topology). The bundle structure that underlines a continuous fields is typically not locally trivial.

We will start by introducing various classes of examples. Thereafter, we will discuss obstructions to local triviality and give an introduction to topological invariants such as parametrized K-theory. Finally, we will present a generalized Dixmier-Douady theory obtained in joint work with Ulrich Pennig.

OASW03 12th June 2017
10:00 to 11:00
Vaughan Jones Phase transitions in the semicontinuous limit of a quantum spin chain
A quest for the construction of a conformal field theory directly from a subfactor has taken an unexpected turn involving a "semicontinuous limit” Hilbert space, with Thompson group symmetry, that might be as relevant to critical quantum spin chains as CFT itself. Models give spin chains with various phases governed by the value of a spectral parameter, and a holomorphic dynamical system. The phases are the Fatou connected components of the dynamical system and the phase transistions occur when the spectral parameter crosses the Julia set from one Fatou component to another. 
OASW03 12th June 2017
11:30 to 12:30
Arthur Jaffe On Picture Language
We introduce some recent work on pictures and the development of the quon language (joint work with Zhengwei Liu and Alex Wozniakowski). We describe a pictorial journey from planar algebras and parafermions, through a problem in quantum information.
OASW03 12th June 2017
13:30 to 14:30
Yasu Kawahigashi The relative Drinfeld commutant and alpha-induction
We study relative Drinfeld commutants of a fusion category in another fusion category in terms of half-braidings. We identify half-braidings with minimal central projections of the relative tube algebra and certain sectors related to the Longo-Rehren subfactors. We apply this general machinery to various fusion categories arising from alpha-induction applied to a modular tensor category and compute the relative Drinfeld commutants explicitly.
OASW03 12th June 2017
14:30 to 15:30
Roberto Longo Discussion about the Landauer principle (and bound)
OASW03 12th June 2017
16:00 to 17:00
Stefaan Vaes Rothschild Lecture: Classification of von Neumann algebras
The theme of this talk is the dichotomy between amenability and non-amenability. Because the group of motions of the three-dimensional Euclidean space is non-amenable (as a group with the discrete topology), we have the Banach-Tarski paradox. In dimension two, the group of motions is amenable and there is therefore no paradoxical decomposition of the disk. This dichotomy is most apparent in the theory of von Neumann algebras: the amenable ones are completely classified by the work of Connes and Haagerup, while the non-amenable ones give rise to amazing rigidity theorems, especially within Sorin Popa's deformation/rigidity theory. I will illustrate the gap between amenability and non-amenability for von Neumann algebras associated with countable groups, with locally compact groups, and with group actions on probability spaces.

OASW03 13th June 2017
09:00 to 10:00
Dietmar Bisch Subfactors with infinite representation theory
OASW03 13th June 2017
10:00 to 11:00
Zhengwei Liu Synergy on quon language
We will talk about the discovery of the quon language, which was inspired by ideas in different areas: subfactors, quantum information, CFT, TQFT. New proofs and results were carried out by synergy on the quon language. The Fourier analysis on quons turns out to be powerful to study modular tensor categories. 
OASW03 13th June 2017
11:30 to 12:30
Hubert Saleur Associative algebras and conformal field theories
I will review in this talk the relationships physicists observe/conjecture  between the associative algebras
(such as Temperley-Lieb) that appear in lattice models, and the  conformal field theories (CFT)  that  describe
their  continuum limits. I will then discuss in more detail a possible way to perform "fusion" for affine Temperley-Lieb
modules, and what this may have to do with Operator Product Expansion (OPE) in CFT. 
OASW03 13th June 2017
13:30 to 14:30
Paul Fendley Baxterising using conserved currents
Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess an unusual sort of conserved current. Such currents have been constructed by utilising quantum-group algebras, fermionic and parafermionic operators, and ideas from ``discrete holomorphicity''. I define them generally and naturally using a braided tensor category, a structure familiar from the study of knot invariants and from conformal field theory.  Requiring the existence of the currents provides a simple way of ``Baxterising'', i.e. building a solution of the Yang-Baxter equation out of topological data.  This approach allows many new examples of conserved currents to be found, for example in height models.  Although integrable models found by this construction are critical, I find one non-critical generalisation: requiring a ``shift'' operator in the chiral clock chain yields precisely the Hamiltonian of the integrable chiral Potts chain.


OASW03 13th June 2017
14:30 to 15:30
Gandalf Lechner Yang-Baxter representations of the infinite symmetric group
The Yang-Baxter equation (YBE) lies at the heart of many subjects, including quantum statistical mechanics, QFT, knot theory, braid groups, and subfactors. In this talk, I will consider involutive solutions of the YBE ("R-matrices"). Any such R-matrix defines a representation and an extremal character of the infinite symmetric group as well as a corresponding tower of subfactors. 

Using these structures, I will describe how to find all R-matrices up to a natural notion of equivalence (given by the character and the dimension), how to completely parameterize the set of solutions, and how to decide efficiently whether two given R-matrices are equivalent. Joint work with U. Pennig and S. Wood.
OASW03 13th June 2017
16:00 to 17:00
Kasia Rejner The Quantum Sine-Gordon model in perturbative AQFT
Co-author: Dorothea Bahns (University of Goettingen)

In this talk I will present recent results on the convergence of the formal S-matrix and interacting currents in the Sine-Gordon model in 2 dimensions, obtained directly in Minkowski signature, using a class of Hadamard states. In our approach one starts with a perturbation series obtained from the formalism of perturbative AQFT and then one can prove the convergence of the series by some simple estimates. Our result opens the posibility to use pAQFT methods to study integrable models in 2 dimensions and to construct local observables in such models.

Related Links
OASW03 14th June 2017
09:00 to 10:00
Constantin Teleman Kramer-Wannier and electro-magnetic duality in field theory
A classical duality (Kramer-Wannier) relates the low and high temperature of the 2-dimensional Ising model. It has been generalized to other dimensions and groups other than Z/2 and distilled into Poincare duality combined with the Abelian Fourier transform. In this talk, I describe a vast generalization in the language of topological field theories, which includes non-Abelian examples. Via the notion of boundary field theory, thus is related to a duality of TQFTs, specifically electro-magnetic duality in 3 dimensions. There arises a natural speculation about invertibility of gapped phases in a large class of lattice models. This is joint work (in progress) with Dan Freed. 
OASW03 14th June 2017
10:00 to 11:00
Pedram Hekmati An application of T-duality to K-theory
Co-author: David Baraglia (The University of Adelaide)

T-duality is a discrete symmetry that was discovered by physicists in the context of string theory, but has now matured into a precise mathematical statement. In this talk I will give a brief overview of topological T-duality and explain how it can be used to give a new, surprisingly simple proof of Hodgkin’s famous theorem on the K-theory of compact simply connected Lie groups.
OASW03 14th June 2017
11:30 to 12:30
Giovanni Landi Line bundles over noncommutative spaces
We give a Pimsner algebra construction of noncommutative lens spaces as `direct sums of line bundles' and exhibit them as `total spaces' of certain principal bundles over noncommutative weighted projective spaces. For each quantum lens space one gets an analogue of the classical Gysin sequence relating the KK theory of the total space algebra to that of the base space one. This can be used to give explicit geometric representatives of the K-theory classes of the lens spaces. 
OASW03 15th June 2017
09:00 to 10:00
Antony Wassermann Conformal Field Theory, Operator algebras and symmetric Fuchsian equations
OASW03 15th June 2017
10:00 to 11:00
Robin Hillier Loop groups and noncommutative geometry
I describe a way of expressing loop groups and their representation theories (the Verlinde fusion ring) in terms of spectral triples and KK-theory. The ideas come from operator algebraic conformal field theory and extend to many other conformal field theoretical models.

Co-author: Sebastiano Carpi.
OASW03 15th June 2017
11:30 to 12:30
Ralf Meyer Induced C*-hulls for *-algebras
Let A be a *-algebra that is graded by a group G with fibres A_g for g in G.  Assume that we have found a C*-algebra B_e whose “representations” are “equivalent” to the “integrable” “representations” of the unit fibre A_e.  Call a “representation” of A “integrable”, if its restriction to A_e is “integrable”.  Under some assumptions, the “integrable” “representations” of A are “equivalent ” to the “representations” of a certain C*-algebra B constructed from B_e and the graded *-algebra A.  The C*-algebra B is the section C*-algebra of a Fell bundle over G.  The words in quotation marks have to be interpreted carefully to make this true.  In particular, representations must be understood to take place on Hilbert modules, not just Hilbert spaces, and the equivalence is required natural with respect to induction of representations and isometric intertwiners.  Under some commutativity assumptions, the main result of my lecture has been proved by Savchuk and Schmüdgen, who also give several examples.  A sample of the result concerns Weyl algebras and twisted Weyl algebras in countably many generators.  These come with a canonical grading by the free Abelian group on countably many generators.
OASW03 15th June 2017
13:30 to 14:30
Christian Voigt The string group and vertex algebras
I will describe a categorification of complex Clifford algebras arising from certain categories of twisted modules over fermionic vertex superalgebras. Along the way I'll discuss some background from the theory of unitary vertex algebras, and how the String 2-group fits into the picture.
OASW03 15th June 2017
14:30 to 15:30
Andre Henriques Bicommutant categories
Bicommutant categories are higher categorical analogs of von Neumann algebras. There exist currently two sources of examples of bicommutant categories: unitary fusion categories, and completely rational unitary conformal field theories.

I will give a general introduction on what and why bicommutant categories are. Then I will talk about a recent result with Dave Penneys: Morita equivalent unitary fusion categories have isomorphic associated bicommutant categories.
OASW03 15th June 2017
16:00 to 17:00
Lilit Martirosyan Affine centralizer algebras
I will describe the representations of (affine) centralizer algebras for quantum groups in terms of paths. I will talk about generators for these algebras. As an example, we will consider the Lie algebra $G_2$ and its centralizer algebras. 
OASW03 16th June 2017
09:00 to 10:00
Feng Xu On questions around reconstruction program
OASW03 16th June 2017
10:00 to 11:00
Ching Hung Lam On the Classification of holomorphic vertex operator algebras of central charge 24
In 1993, Schellekens obtained a list of possible Lie algebra structures for the weight one subspaces of holomorphic vertex operator algebras (VOA) of central charge 24. It was also conjectured that the VOA structure of a holomorphic VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one space. Recently, all 71 cases in Schellekens' list have been constructed.  

In this talk, we will discuss the recent progress on the classification of holomorphic vertex operator algebras of central charge 24. In particular, we will discuss the construction of holomorphic VOAs using various types of orbifold constructions. A technique, which we call ``Reverse orbifold construction", will also be discussed. This technique may be used to prove the uniqueness of holomorphic VOAs of central charge 24 if the weight one subspace is not zero. 
OASW03 16th June 2017
11:30 to 12:30
Simon Wood What to expect from logarithmic conformal field theory
Logarithmic conformal field theory is a generalisation of ordinary conformal field theory that allows for logarithmic singularities in correlation functions. This implies the existence of reducible yet indecomposable modules on which the action of the Virasoro L_0 operator is not diagonalisable. In this talk I will recall some of what is known about rational conformal field theory and contrast it with what has been achieved so far in logarithmic conformal field theory.
OASW03 16th June 2017
13:30 to 14:30
Sebastiano Carpi Conformal nets, VOAs and their representations
We discuss some recent results on the connection between conformal nets, VOAs and their representation theories. 
OASW03 16th June 2017
14:30 to 15:30
Terry Gannon The truth about finite group orbifolds
Chiral CFTs (VOAs or conformal nets) are interesting for their representation theory. Orbifolds are a standard method for constructing new chiral CFTs from old ones. Start with a chiral theory with trivial representation theory, and orbifold it by a finite group; the result (called a holomorphic orbifold) has the representation theory given by the twisted Drinfeld double of that finite group, where the twist is a 3-cocycle. In practise it is hard to identify that twist. 

I'll begin my talk by giving some examples of orbifolds. I'll identify a well-known class of holomorphic orbifolds where we now know the twist. I'll relate holomorphic orbifolds to KK-theory as well as the PhD thesis of a certain Vaughan Jones. Then I'll explain how any choice of finite group and 3-cocycle is realized by a chiral CFT. This is joint work with David Evans.
OAS 20th June 2017
13:30 to 14:30
Sven Raum Groups acting on trees: representation theory and operator algebras
This talk summarises different aspects of my work on totally disconnected groups, their representation theory and their operator algebras.  Our guiding examples are non-discrete groups acting on trees.  I will address simplicity, amenability and classification of operator algebras associated with groups acting on trees as well as representation theoretic aspects such as the admissibility conjecture.


OAS 20th June 2017
16:00 to 17:00
Masaki Izumi Group actions on C*-algebras and obstruction theory
This talk gives an account of the roles in group actions on C*-algebras played by the classical obstruction theory for fiber bundles. More specifically, I discuss the following deciding problems:

(1) whether a given 2-cocycle action is equivalent to an ordinary action or not,
(2) whether a given 1-cocycle of an action is an (asymptotic) coboundary or not,
(3) whether two given actions are cocycle conjugate or not.

The idea works well for poly-Z groups on Kirchberg algebras.

This is joint work with Hiroki Matui.



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