This is the unofficial seminar homepage of the GAPT group at Cardiff University.
The research interests of the GAPT group 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. Usually the seminar takes place every Thursday at 15:10 in room M/0.34. Speakers give 50 minute talks. |

**Wednesday 14th November 13:10 - 14:00 M/2.06**

Vladimir Dotsenko (Trinity College Dublin)

Noncommutative analogues of cohomological field theories

Abstract:

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.

Noncommutative analogues of cohomological field theories

Abstract:

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.

**Thursday 22nd November 15:10 - 16:00 M/0.34**

Gandalf Lechner (Cardiff)

The Yang-Baxter equation and extremal characters of the infinite braid group

Abstract:

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.

The Yang-Baxter equation and extremal characters of the infinite braid group

Abstract:

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.

**Thursday 29th November 15:10 - 16:00 M/0.34**

Tomasz Brzezinski (Swansea)

Twisted reality

Abstract:

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.

Twisted reality

Abstract:

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.

**Thursday 06th December 15:10 - 16:00 M/0.34**

Ashley Montanaro (Bristol)

**Thursday 13th December 15:10 - 16:00 M/0.34**

Matthew Buican (Queen Mary, London)

**Thursday 31st January 15:10 - 16:00 M/0.34**

Vincenzo Morinelli (Tor Vergata, Rome)

**Past Seminar Talks**

**Thursday 4th October 15:10 - 16:00 M/0.34**

Stuart White (Glasgow)

Classification of simple nuclear C*-algebras

Abstract:

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.

Classification of simple nuclear C*-algebras

Abstract:

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.

**Thursday 11th October 15:10 - 16:00 M/0.34**

Fabian Hebestreit (Bonn / INI Cambridge)

Twisted K-theory via retractive symmetric spectra

joint with Steffen Sagave

Abstract:

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.

Twisted K-theory via retractive symmetric spectra

joint with Steffen Sagave

Abstract:

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.

**Thursday 18th October 15:10 - 16:00 M/0.34**

Paul Mitchener (Sheffield)

Categories of Unbounded Operators

Abstract:

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.

Categories of Unbounded Operators

Abstract:

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.

**Thursday 01 November 15:10 - 16:00 M/0.34**

Andreas Aaserud (Cardiff)

K-theory of some AF-algebras from braided categories

Abstract:

In the 1980s, Renault, Wassermann, Handelman and Rossmann explicitly described the K-theory of the fixed point algebras of certain actions of compact groups on AF-algebras as polynomial rings. Similarly, Evans and Gould in 1994 explicitly described the K-theory of certain AF-algebras related to SU(2) as quotients of polynomial rings. In this talk, I will explain how, in all of these cases, the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ (where $A$ denotes the AF-algebra whose K-theory is being computed) arising from a unitary braiding on an underlying C*-tensor category and essentially defined by Erlijman and Wenzl in 2007. I will also give new explicit descriptions of the K-theory of certain AF-algebras related to SU(3), Sp(4) and G$_2$ as quotients of polynomial rings. Finally, I will attempt to explain how this work is motivated by the Freed-Hopkins-Teleman formula for the fusion rings of WZW-models in conformal field theory. This is all based on joint work with David E. Evans.

K-theory of some AF-algebras from braided categories

Abstract:

In the 1980s, Renault, Wassermann, Handelman and Rossmann explicitly described the K-theory of the fixed point algebras of certain actions of compact groups on AF-algebras as polynomial rings. Similarly, Evans and Gould in 1994 explicitly described the K-theory of certain AF-algebras related to SU(2) as quotients of polynomial rings. In this talk, I will explain how, in all of these cases, the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ (where $A$ denotes the AF-algebra whose K-theory is being computed) arising from a unitary braiding on an underlying C*-tensor category and essentially defined by Erlijman and Wenzl in 2007. I will also give new explicit descriptions of the K-theory of certain AF-algebras related to SU(3), Sp(4) and G$_2$ as quotients of polynomial rings. Finally, I will attempt to explain how this work is motivated by the Freed-Hopkins-Teleman formula for the fusion rings of WZW-models in conformal field theory. This is all based on joint work with David E. Evans.