Mathematical Physics ClusterResearch interests include: Aberystwyth UniversityInstitute of Mathematics and Physics

Operator algebras and K-theory Quantum Control TheoryQuantum ProbabilityQuantum Computing

Cardiff UniversitySchool of Mathematics

Operator algebras and non-commutative geometryK-theory, non-commutative dynamical systems.Quantum symmetries: subfactors, tensor categories, Hopf algebras, quantum groups, planar algebrasEnumerative CombinatoricsModular invariants

Cardiff UniversitySchool of Mathematics

Foundations and constructive approaches to Quantum Field Theory: algebraic approach, conformal field theories including boundaries, curved backgrounds, perturbation theory, topological field theories.Classical general relativity: asymptotic charges, Hamiltonian formalism, black holes, AdS/CFT.Statistical Mechanics: classical and quantum, equilibrium and non-equilibrium, integrable systems

Swansea UniversityDepartment of Mathematics,School of Physical Sciences

Quantum Algebra, Hopf algebras, quantum groupsNon-commutative differential geometry, Computability of physical and mathematical systems, Categories and non-associative structuresCo-rings and comodulesHomological algebra, triangulated categories

Swansea UniversityDepartment of Physics, School of Physical Sciences

Quantum Field Theory, CosmologyString Theory Gauge-string correspondenceAdS/CFT strong-coupling dynamics ofsupersymmetric gauge theory Phenomenology beyond the Standard ModelLattice Gauge Field Theory

Symmetry and Independence in Quantum Probability

Strongly coupled dynamics of quantum field theories

Motivation and Aim• Development of a quantized theory of distributional symmetries and invariance principles • Investigation of quantized counterparts of conditional independence in quantum probability• Identification of asymptotic models for scaling limits of interacting quantum dynamical systems

Future Cluster Activities

Dr Claus Köstler,Institute of Math and Physics, AberystwythDr Carlos Nunez,Department of Physics, Swansea

Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC WIMCS

Wales Institute of Mathematical and Computational Sciences

Selected Cluster Activities

Motivation and AimTo develop and exploit techniques to investigate the strongly coupled dynamics of quantum field theories (QFTs)

Large N at SwanseaSwansea University, July 7-10 2009A conference drawing together the world's experts to discuss recent advances in strong coupling dynamics in large N gauge theories.Speakers inlcuded: Thomas Appelquist (Yale), Lawrence Yaffe (Seattle), Mikhail Shifman (Minnesota), …

Cardiff Semester on Noncommutative Geometry and Physics, February – July 2010Organized by David E Evans (Cardiff)

Long term visitors as well as a series of short term events, with 120+ participants in all, including:

WIMCS MP Colloquium,26 February 2010 held at Swansea Computer ScienceSpeakers: Marcos Mariño (Geneva), Richard Thomas (Imperial) Planar Period Cardiff University, 15 February – 5 March 2010Speakers included: S Bigelow (UCSB), J Fuchs (Karlstad), …

Method• To use the fact that certain QFTs are known to have a dual description involving an emergent higher dimensional gravitational theory. In particular, the strong coupling regime of the QFT maps to the classical regime of the dual gravitational theory and many observables can be calculated exactly.• To exploit the fact that the dual gravitational theory is described beyond the classical regime by a QFT in order to understand quantum gravitational effects.• To discretize the QFT by defining it on a spacetime Lattice in order that the functional integral can be calculated numerically on state-of-the-art computers by Monte Carlo techniques.

Outcome• The group has produced new strong coupling results in QFTs with minimal and extended supersymmetry.• The results apply not only in the zero temperature theory, but also at finite temperature and density (important for heavy ion collisions and also in the early universe).• The observables which have been investigated include the spectrum of states and their interactions, the expectation values of important operators.• The group has used these results to formulate new dualities involving some QFTs which may have direct relevance in the interpretation of experimental observation at the Large Hadron Collider.

Quantum field theory on curved space-times and curved target-spacesErwin-Schrodinger Institute, Vienna, March+April 2010 , organized by Stefan Hollands (Cardiff)

Frontiers Lecture of the Learned Society of Wales Cardiff University, 17 January 2011Speaker: Sir Michael Atiyah (Edinburgh)

David E Evans (Cardiff)EU Research Training Network in Noncommutative GeometryAppointments in 2011:

Method• Systematic quantization of classical probability using the language of operator algebras• Investigate to which extent classical de Finetti type results transfer to the quantum world • Investigate the relation of symmetry and independence in the general quantized setting

Selected Results - indicate deep connections between important areas in modern analysis and its applications.

Noncommutative extended de Finetti theorem: 'exchangeability' and 'spreadability imply 'noncommutative conditional independence' in terms of 'commuting squares' Failure of common folklore facts from classical probability: ‘exchangeability' and 'spreadability’ of an infinite sequence of random variables are no longer equivalent in the quantized setting. Braided extended de Finetti theorem: we introduce 'braidability' as a new symmetry intermediate to 'exchangeability‘ and 'spreadability'. It applies to subfactor inclusions with small Jones index

Selected Publications R. Gohm; C. Köstler. Noncommutative independence from the braid group B∞. Comm. Math. Phys. 289 (2009), 435–482. R. Gohm; C. Köstler. Noncommutative independence from characters of the infinite symmetric group S∞. arXiv:1005.5726 [math.OA] (2010).C. Köstler. A noncommutative extended de Finetti theorem. J. Funct. Anal. 258 (2010), 1073–1120. C. Köstler; R. Speicher. A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation. Comm. Math. Phys. 291 (2009), 473–490.

Claus Köstler, Institute of Math and Physics, Aberystwyth Carlos Nunez, Department of Physics, Swansea

Stefan Hollands (Cardiff)ERC Starter Research Grant2011-16

INI-WIMCS follow up meeting to Isaac Newton Institute 2006 programme on Noncommutative Geometryto be held at Cardiff or Gregynog in 2012 - application pending

Organizers David E Evans (Cardiff ), Nigel Higson (Penn State), Shahn Majid (Queen Mary, London)Themes of meeting include: Noncommutative Algebraic GeometryRepresentation Theory aspects of Baum-ConnesFree aspects of Noncommutative Geometry

Noncommutative Geometry and Conformal Field TheoryNoncommutative Geometry and CategorificationNoncommutative Spacetime and CosmologyThe Standard Model and Beyond

London Mathematical Society Spitalfields Day Cardiff University, 17 May 2010. Speakers: Nigel Higson (Penn State), Terry Gannon (Alberta) London Mathematical Society Regional Meeting and Workshop Operator Algebras and Physics, Cardiff University, 21-25 June 2010. Series of lectures by Constantin Teleman and other speakers included Werner Nahm (Dublin)

Annual Meeting of the EU Research Training Networkin Noncommutative Geometry,Cardiff University, 28 June – 2 July 2010Speakers included: J Mickelsson (Helsinki), Y Kawahigashi (Tokyo), A Verbeure (Leuven), J Yngvason (ESI Vienna), …. First Frontiers Lecture of the Learned Society of WalesCardiff University, 28 June 2010Speaker: Dan Voiculescu (Berkeley)

Collaborators include:Juan Maldacena (IAS), Daniel Freedman (MIT)

Alfonso Ramallo (Santiago de Compostela), Nick Dorey (Cambridge) Kostas Skenderis (Amsterdam), Dario Martelli (King's College)

Free de Finetti theorem: ‘quantum exchangeability' yields a beautiful characterization of freeness with amalgamation. This quantum probabilistic symmetry comes from a compact quantum group.Thoma theorem is a quantum de Finetti theorem: we give a new operator algebraic proof of the characterization of extremal characters of the infinite symmetric group.

• Alin Galatan (Bucharest)• Antti Harju (Helsinki)• Grace Kennedy (UCSB)

• Jennifer Maier (Hamburg)• Makoto Yamashita (Tokyo)

• Provide new computational methods for exactly solvable models in quantum statistical physics

• Application of the developed tools to collective quantum effects, quantum control and information