Operator Algebras and Noncommutative Geometry
Please note that this project is self-funded.
The application of Operator Algebras and K-theory to understand structural problems in statistical mechanics and conformal quantum field theory.
Project aims and methods
A subfactor can be regarded as a quantum symmetry. An outer group action on a hyperfinite factor can be recovered from the inclusion of the fixed-point algebra in the original factor. A subfactor or an inclusion of one factor in another is then a generalisation of a group. A fundamental question in subfactor theory, modular tensor categories and conformal field theory is whether there is anything beyond that obtained from symmetries of groups, their Drinfeld doubles, quantum deformations or loop groups.
Speculation from the early years of rational CFT, asks if the standard constructions (orbifolds, cosets, simple-current extensions, level-rank duality) applied to the basic theories (lattice compactifications, loop groups) exhaust all rational theories. Applied to the MTC rather than the rational CFT, this would constitute a Tannaka-Krein duality where the dual to the category is a conformal net constructed from the basic examples.
A related question is whether a MTC always arises as the representations of a conformal net of factors. The role of K-theory in understanding the MTC and representation theory of the conformal nets is critical to the project.
Operator Algebras and Noncommutative Geometry has become a central field in mathematics with applications and connections across mathematics and theoretical physics. The project integrates algebra, analysis, geometry and topology and applications in and drawing inspiration from physical problems – statistical mechanics and conformal quantum field theory.
The field is important worldwide with centres in Vanderbilt, Berkeley, UCLA, Harvard, Toronto, Kyoto, Tokyo, Canberra, and Sydney and numerous centres in Europe. The proposer has strong links with all of these – including ongoing collaborations with Gannon in Canada, Grossman at UNSW and Kawahigashi at Tokyo University. Evans has co-ordinated two Marie Curie Research Training Networks in Europe, involving Copenhagen, Oslo, Dublin, Paris, Munster, Gottingen, Rome with collaborators Fields Medallist Connes (College de France), Cuntz (Leibnitz Prize winner, Munster).
These networks provide ample opportunity for further collaboration. An EPSRC supported PDRA Andreas Aaserud has been recruited from UCLA – who obtained his Phd under the supervision of ICM plenary speaker Sorin Popa. Stefaan Vaes (Leuven) was the Rothschild Professor at the 6 month programme on Operator Algebras at the Newton Institute in 2017 organised by Evans, Fields Medalist Jones (Vanderbilt) and Izumi (Kyoto). Vaes gave public lectures at Cambridge and Cardiff University. These networks all give ample opportunity for collaboration.
The School runs a weekly postgraduate seminar, weekly research seminars and the annual Gregynog Mathematics Colloquium which provide the first opportunities for developing communication skills. Outside Wales, there will be opportunities to speak at the annual UK young functional analysis gathering, the European YMC*A meeting, the annual NCGOA Non commutative Geometry summer schools in Vanderbilt. Recent graduate students of the proposer have attended such meetings an also invited to a programme at the Mathematics Research Institute in Oberwolfach, Germany and attended training events in Belfast, Glasgow, Copenhagen and Munster.
The student will be working within the Geometry, Algebra, Mathematical Physics and Topology group which has 7 permanent members. The supervisor has currently one EPSRC award supporting a PDRA, has recently hosted an EPSRC postdoctoral fellow, an ERC Starting Grant, led two EU Marie Curie Networks, and regularly organises workshops at Cardiff and elsewhere. A 6-month programme at the Newton Institute, Cambridge, in 2017 was be led by the proposer. The prestigious Rothschild professor at that programme is based in Leuven, who gave public lectures at Cambridge and Cardiff University.
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