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:
- 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.
- Algebraic Quantum Field Theory
- Conformal Field Theory
- Statistical Mechanics: classical and quantum, integrable systems.
Head of Group
Research Professor of Mathematics
- Welsh speaking
- +44 (0)29 2087 4522
- +44 (0)29 2087 5546
- Welsh speaking
- +44 (0)29 2087 6862
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.
31 January 2019
Vincenzo Morinelli (Tor Vergata, Rome)
To be announced.
13 December 2018
Matthew Buican (Queen Mary, London)
To be announced.
6 December 2018
Ashley Montanaro (Bristol)
Quantum algorithms for search problems
Quantum computers are designed to use quantum mechanics to outperform any standard, "classical" computer based only on the laws of classical physics. Following many years of experimental and theoretical developments, it is anticipated that quantum computers will soon be built that cannot be simulated by today's most powerful supercomputers. In this talk, I will begin by introducing the quantum computational model, and describing the famous quantum algorithm due to Grover that solves unstructured search problems in approximately the square root of the time required classically. I will then go on to describe more recent work on a quantum algorithm to speed up classical search algorithms based on the technique known as backtracking ("trial and error"), and very recent work on calculating the level of quantum speedup anticipated when applying this algorithm to practically relevant problems. The talk will aim to give a flavour of the mathematics involved in quantum algorithm design, rather than going into the full details.
The talk will be based on the papers
Quantum walk speedup of backtracking algorithms, Theory of Computing (to appear); arXiv:1509.02374
Tomasz Brzezinski (Swansea)
Recently two approaches to twisting of the real structure of spectral triples were introduced. In one approach, the definition of a twisted real structure of an ordinary spectral triple was presented in [T Brzeziński, N Ciccoli, L Dąbrowski, A Sitarz, Twisted reality condition for Dirac operators, Math. Phys. Anal. Geom. 19 (2016), no. 3, Art. 16]. In the second approach [G Landi, P Martinetti, On twisting real spectral triples by algebra automorphisms, Lett. Math. Phys. 106 (2016), no. 11, 1499–1530] the notion of real structure for a twisted spectral triple was proposed. In this talk we present and compare these two approaches.
22 November 2018
Gandalf Lechner (Cardiff)
The Yang-Baxter equation and extremal characters of the infinite braid group
The Yang-Baxter equation (YBE) is a cubic matrix equation which plays a prominent in several fields such as quantum groups, braid groups, knot theory, quantum field theory, and statistical mechanics. Its invertible normal solutions ("R-matrices") define representations and extremal characters of the infinite braid group. These characters define a natural equivalence relation on the family of all R-matrices, and I will describe a research programme aiming at classifying all solutions of the YBE up to this equivalence.
I will then describe the current state of this programme. In the special case of normal involutive R-matrices, the classification is complete (joint work with Simon and Ulrich). The more general case of R-matrices with two arbitrary eigenvalues is currently work in progress, and I will present some partial results, including a classification of all R-matrices defining representations of the Temperley-Lieb algebra and a deformation theorem for involutive R-matrices
Vladimir Dotsenko (Trinity College Dublin)
Noncommutative analogues of cohomological field theories
Algebraic structures that are usually referred to as cohomological field theories arise from geometry of Deligne-Mumford compactifications of moduli spaces of curves with marked points. I shall talk about some new rather remarkable algebraic varieties that have a lot in common with [genus 0] Deligne-Mumford spaces, and several new algebraic structures that naturally arise from studying those varieties.
Andreas Aaserud (Cardiff)
K-theory of some AF-algebras from braided categories
18 October 2018
Paul Mitchener (Sheffield)
Categories of Unbounded Operators
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 / INI Cambridge)
Twisted K-theory via retractive symmetric spectra joint with Steffen Sagave
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 Poincaré 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)
Classification of simple nuclear C*-algebras
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)
Generalised braid categorification
Ordinary braid group Brn is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continuous transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the space Fln of complete flags in Cn. 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.