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Timetable (OASW02)

Subfactors, higher geometry, higher twists and almost Calabi-Yau algebras

Monday 27th March 2017 to Friday 31st March 2017

Monday 27th March 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
10:00 to 11:00 Vaughan Jones (Vanderbilt University); (University of California, Berkeley)
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Gus Isaac Lehrer (The University of Sydney)
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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 N. Christopher Phillips (University of Oregon)
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.  
14:30 to 15:30 James Tener (University of California, Santa Barbara)
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.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Thomas Schick (Georg-August-Universität Göttingen)
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).
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 28th March 2017
10:00 to 11:00 Osamu Iyama (Nagoya University)
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Karin Erdmann (University of Oxford)
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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Alastair King (University of Bath)
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.
14:30 to 15:30 Mathew Pugh (Cardiff University)
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.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Joseph Grant (University of East Anglia)
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.
Wednesday 29th March 2017
09:00 to 10:00 Akhil Mathew (Harvard University)
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.
10:00 to 11:00 Paul Smith (University of Washington)
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. 

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Raf Bocklandt (Universiteit van Amsterdam)
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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Christ's College
Thursday 30th March 2017
10:00 to 11:00 Andre Henriques (University of Oxford); (Universiteit Utrecht)
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. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Ulrich Pennig (Cardiff University)
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

12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Ulrich Bunke (Universität Regensburg)
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
14:30 to 15:30 Michael Murray (University of Adelaide)
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

15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Amihay Hanany (Imperial College London)
Friday 31st March 2017
10:00 to 11:00 Katrin Wendland (Albert-Ludwigs-Universität Freiburg)
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Simon Gritschacher (University of Oxford)
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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Danny Stevenson (University of Adelaide)
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. 

14:30 to 15:30 Yang-Hui He (City University, London); (University of Oxford)
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.

15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons