# Geometry, Algebra, Mathematical Physics and Topology Research Group

Our interests sweep a broad range of topics, from algebra, geometry, topology, including operator algebras, and non-commutative geometry in pure mathematics, to algebraic and conformal quantum field theory and integrable statistical mechanics in mathematical physics.

The main areas of research within the current group are:

### Pure mathematics

- Algebraic Geometry
- DG categories and derived categories associated to algebraic varieties
- Operator algebras and non-commutative geometry
- Subfactors and planar algebras
- Orbifolds and the McKay correspondence in Algebraic Geometry and Subfactor Theory
- Categorification problems, Mirror symmetry, Moduli spaces
- Quiver representations in Algebraic Geometry and Subfactor Theory
- K-theory - including twisted and equivariant versions
- Quantum symmetries: subfactors, tensor categories, Hopf algebras, quantum groups;
- Enumerative Combinatorics.

### Mathematical physics

- Algebraic Quantum Field Theory
- Conformal Field Theory
- Statistical Mechanics: classical and quantum, integrable systems.

## Head of Group

## Professor David E Evans

Research Professor of Mathematics

- Welsh speaking
*Email:*- evansde@cardiff.ac.uk
*Telephone:*- +44 (0)29 2087 4522

## Academic staff

## Dr Mathew Pugh

Senior Lecturer

- Welsh speaking
*Email:*- pughmj@cardiff.ac.uk
*Telephone:*- +44 (0)29 2087 6862

## Current events

All seminars are held in Room M2.06 on Thursdays at 15:10 unless otherwise stated. All are welcome.

Programme organiser and contact: Dr Ulrich Pennig.

Date | Speaker | Seminar |
---|---|---|

31 January 2019 | Vincenzo Morinelli (Tor Vergata, Rome) | To be announced. |

13 December 2018 | Ashley Montanaro (Bristol) | To be announced. |

6 December 2018 | Matthew Buican (Queen Mary, London) | To be announced. |

15 November 2018 | Vladimir Dotsenko (Trinity College Dublin) | To be announced. |

18 October 2018 | Paul Mitchener (Sheffield) |
The Gelfand-Naimark theorem on C*-algebras, which asserts that a C*-algebra, defined axiomatically, is the same thing as a closed sub-algebra of the algebra of bounded linear operators on a Hilbert space, is well-known. Of course, in some cases, for example, mathematical physics, the concern is with unbounded operators such as position and momentum in quantum mechanics. In this talk, we explore a set of axioms for a mathematical object analogous to a C*-algebra, but for unbounded operators. In particular, our axioms are such that an analogue of the Gelfand-Naimark theorem holds. |

11 October 2018 | Fabian Hebestreit (Bonn |
Twisted K-theory was originally invented to serve as the K-theoretic analogue of singular (co)homology with local coefficients and by design gives explicit Thom- and PoincarĂ© duality isomorphisms. In this formulation it admits a direct description in terms of KK-theory of certain section algebras and thus has tight connections for instance to the geometry of scalar curvature. Modern homotopy theory on the other hand provides a universally twisted companion for every coherently multiplicative cohomology theory by means of parametrised spectra. This construction has very appealing formal properties and, indeed, applied to K-theory allows for much more general twists than those afforded by the operator algebraic one. Necessarily then, such twisted companions are defined in a much more formal manner and thus in general not easily tied to geometry. The goal of my talk is to briefly explain the category of the title, that naturally houses both constructions and then sketch that, indeed, a suitable restriction of the universal one reproduces the operator theoretic version of twisted K-theory. Time permitting, I shall also sketch how our work strengthens recent results of Dardalat and Pennig, describing the more exotic twists of K-theory via self-absorbing C*-algebras. |

4 October 2018 | Stuart White (Glasgow) |
Recent years have seen repeated striking progress in the structure and classification of simple nuclear C*-algebras. I’ll try and survey what the state of the art is, focusing on recent developments. I’ll try and keep the talk self contained, starting out with what these `simple nuclear C*-algebras’ are and why anyone wants to classify them anyway. |

28 June 2018 | Lorenzo De Biase (Cardiff) |
Ordinary braid group C. Generalised braids are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In this talk I will present some progress that have been made towards extending the result of Khovanov and Thomas to the categorification of the generalised braid category.^{n} |