Geometry, Algebra, Mathematical Physics and Topology Research Group
In line with much of modern mathematics, this group is a blend of pure mathematicians and theoretical physicists.
Our interests sweep a broad range of topics, from algebra, geometry, topology, including operator algebras, and noncommutative geometry in pure mathematics, to algebraic and conformal quantum field theory, quantum information theory, and integrable statistical mechanics in mathematical physics.
The main areas of research within the current group are:
Pure mathematics
 Algebraic and enumerative combinatorics
 Algebraic geometry
 Braid group representations
 Categorification problems, mirror symmetry, moduli spaces
 DG categories and derived categories associated to algebraic varieties
 Ktheory  including twisted and equivariant versions
 Modular tensor and fusion categories
 Operator algebras and noncommutative geometry
 Orbifolds and the McKay correspondence in Algebraic Geometry and Subfactor Theory
 Quantum symmetries: subfactors, tensor categories, Hopf algebras, quantum groups
 Quiver representations in Algebraic Geometry and Subfactor Theory
 Subfactors and planar algebras.
Mathematical physics
 Algebraic Quantum Field Theory
 Conformal Field Theory
 Quantum information
 Statistical mechanics: classical and quantum, integrable systems
 Topological phases of matter.
Head of Group
Academic staff
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 

30 April 2020  Konstanze Rietsch (King's College London) Room M/0.34  To be advised 
23 April 2020  Manjil P. Saikia (Cardiff University) Room M/0.34  To be advised 
26 March 2020  Johannes Hofscheier (University of Nottingham) Room M/0.34  To be advised 
19 March 2020  Tomack Gilmore (UCL) Room M/0.34  To be advised 
5 March 2020  Taro Sogabe (Kyoto University) Room M/0.34  To be advised 
27 February 2020  André Henriques (University of Oxford) Room M/0.34  Constructing conformal field theories 30 years after their initial formulation, checking the Segal axioms of conformal field theory remains an elusive task, even for some of the simplest examples. I will give a gentle introduction on conformal field theory à la Segal, and highlight some of the difficulties. I will then sketch a joint project with James Tener whose goal it is to verify the Segal axioms. 
20 February 2020  Jelena Grbic (University of Southampton) Room M/0.34  Homology theory of superhypergraphs Hypergraphs can be seen as incomplete abstract simplicial complexes in the sense that taking subsets is not a closed operation in hypergraphs. This notion can be extended to Δsets with face operations only partially defined, these objects we name superhypergraphs. In this talk I will set foundations of homology theory of these combinatorial objects. 
6 February 2020  Clemens Koppensteiner (University of Oxford) Room M/0.34  Logarithmic RiemannHilbert Correspondences The classical RiemannHilbert Correspondence provides a deep connection between geometry and topology. In its simplest form it stipulates an equivalence between the categories of vector bundles with a flat connection on a complex manifold and local systems on the topological space underlying the manifold. If one allows the connection to have poles, the situation becomes considerably more subtle. We discuss work of KatoNakayama and Ogus on this "logarithmic" setting. This in turn motivates recent joint work with Mattia Talpo on a further generalisation to logarithmic Dmodules. We discuss what form the conjectural log RiemannHilbert Correspondence should take and the progress that has been achieved so far. We will not assume any familiarity with Dmodules or logarithmic geometry. 
30 January 2020  Yue Ren (Swansea University) Room M/0.34  Tropical algebraic geometry  Algorithms and applications This talk is an introductory overview of the many facets of tropical geometry on the basis of its many applications in and outside mathematics. These include enumerative geometry, linear optimization, phylogenetics in biology, auction theory in economics, and celestial mechanics in physics. Special emphasis will be put on constructive algorithms and the mathematical challenges that they entail.

12 December 2019  Iain Moffatt (Royal Holloway, University of London) Room M/0.34  The Tutte polynomial of a graph and its extensions This talk will focus on graph polynomials, which are polynomial valued graph invariants. Arguably, the most important and best studied graph polynomial is the Tutte polynomial. It is important not only because it encodes a large amount of combinatorial information about a graph, but also because of its applications to areas such as statistical physics (as the Ising and Potts models) and knot theory (as the Jones and homfly polynomials). Because of its importance the Tutte polynomial has been extended to various classes of combinatorial object. For some objects there is more than one definition of a "Tutte polynomial". For example, there are three different definitions for the Tutte polynomial of graphs in surfaces: M. Las Vergnas' 1978 polynomial, B. Bolloba's and O. Riordan's 2002 ribbon graph polynomial, and V. Kruskal's polynomial from 2011. On the other hand, for some objects, such as digraphs, there is no wholly satisfactory definition of a Tutte polynomial. Why is this? Why are there three different Tutte polynomials of graphs in surfaces? Which can claim to be the Tutte polynomial of a graph in a surface? More generally, what does it mean to be the Tutte polynomial of a class of combinatorial objects? In this talk I will describe a way to canonically construct Tutte polynomials of combinatorial objects, and, using this framework, will offer answers to these questions. 
5 December 2019  John Harvey (Swansea)  Estimating the reach of a submanifold The reach is an important geometric invariant of submanifolds of Euclidean space. It is a realvalued global invariant incorporating information about the second fundamental form of the embedding and the location of the first critical point of the distance from the submanifold. In the subject of geometric inference, the reach plays a crucial role. I will give a new method of estimating the reach of a submanifold, developed jointly with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar. 
28 November 2019  Xin Li (Queen Mary, London)  Constructing Cartan subalgebras in all classifiable C∗algebras I will explain how to construct Cartan subalgebras in all classifiable stably finite C∗algebras, and I will discuss the JiangSu algebra as a particular example. 
21 November 2019  Christian Bönicke (Glasgow)  On the Ktheory of ample groupoid algebras It is a difficult problem to compute the Ktheory of a crossed product of a C∗algebra by a groupoid. One approach is given by the BaumConnes conjecture, which asserts that a certain assembly map from the topological Ktheory of the groupoid G with coefficients in a GC∗algebra A into the Ktheory of the associated reduced crossed product is an isomorphism. In this talk I will present a method that allows one to deal with certain questions concerning the left hand side of the assembly map: The GoingDown principle. This principle can be used in two ways, both of which I will illustrate by an example: 1) Obtain results about the BaumConnes conjecture, and 
14 November 2019  Andreas Fring (City University of London)  Nonlocal gauge equivalent integrable systems We demonstrate how new integrable nonlocal systems in space and/or time can be constructed by exploiting certain parity transformations and/or time reversal transformations possibly combined with a complex conjugations. By employing Hirota's direct method as well as DarbouxCrum transformations we construct explicit multisoliton solutions for nonlocal versions of Hirota's equation that exhibit new types of qualitative behaviour. We exploit the gauge equivalence between these equations and an extended version of the continuous limit of the Heisenberg equation to show how nonlocality is implemented in those latter systems and an extended version of the LandauLifschitz equation. 
7 November 2019  Francesca Arici (Leiden University)  Circle and sphere bundles in noncommutative geometry In this talk I will recall how Pimsner algebras of self Morita equivalences can be thought of as total spaces of quantum circle bundles, and the associated six term exact sequence in Ktheory can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles. 
24 October 2019  Ko Sanders (Dublin City University)  Killing fields and KMS states in curved spacetimes In quantum physics, every thermal equilibrium state satisfies the KMS condition. This condition is formulated in terms of the evolution in time. In general relativity, however, there is no preferred time flow on spacetime, but there can be several natural choices of a flow, given by Killing vector fields. Physically relevant examples arise especially in the context of black hole spacetimes, where the Killing fields are often timelike only in some region of spacetime, but they are spacelike or lightlike elsewhere. In this talk, based on work with Pinamonti and Verch, I will address the question for which Killing fields the existence of KMS states can be ruled out, because the KMS condition forces the twopoint distributions of such states to be physically illbehaved. 
17 October 2019  Joan Bosa (Universitat Autònoma de Barcelona)  Classification of separable nuclear unital simple C*algebras. History and final results. Over the last decade, our understanding of simple, nuclear C∗C∗algebras has improved a lot. This is thanks to the the interplay between certain topological and algebraic regularity properties, such as nuclear dimension of C∗C∗algebras, tensorial absorption of suitable strongly selfabsorbing C∗C∗algebras and order completeness of homological invariants. In particular, this is reﬂected in the TomsWinter conjecture. In this talk I will speak about this problem, and explain the general classification of nuclear simple C∗C∗algebras using the ﬁnite nuclear dimension (done in two groundbreaking articles by ElliottGongLinNiu and TikuisisWhiteWinter). If time permits, I will also show some research built up from the classification just explained. 
10 October 2019  Ian Short (The Open University)  Integer tilings and Farey graphs In the 1970s, Coxeter studied certain arrays of integers that form friezes in the plane. He and Conway discovered an elegant way of classifying these friezes using triangulated polygons. Recently, research in friezes has revived, in large part because of connections with cluster algebras and with certain infinite arrays (or tilings) of integers. Here we explain how much of the theory of integer tilings can be interpreted using the geometry and arithmetic of an infinite graph embedded in the hyperbolic plane called the Farey graph. We also describe how other types of integer tilings (such as integer tilings modulo n) can be interpreted using variants of the Farey graph obtained by taking quotients of the hyperbolic plane. 
3 October 2019  Victor Przyalkowski (Steklov/HSE, Moscow)  Hodge minimality of weighted complete intersections We discuss Fano varieties whose Hodge diamonds are close to minimal ones. We discuss several conjectures related to them, and classify those of them who can be represented as smooth Fano weighted complete intersections. It turns out that the minimality has derived categories origin. 
16 May 2019  Alan Thompson (Loughborough University)  To be confirmed 
9 May 2019  n/a  There is no GAPT seminar on this day due to the COW seminar taking place in Cardiff 
11 April 2019  Lorenzo De Biase (Cardiff University)  Generalised braid actions In this talk, after giving some background on autoequivalences of derived categories of smooth projective varieties, I will define the generalised braid category and describe its action on the derived categories of (the cotangent bundles of) full and partial flag varieties. Generalised braids are the braids whose strands are allowed to touch in a certain way. The basic building blocks of their action on flag varieties are spherical and nonsplit Pfunctors together with the twist equivalences they induce. I will describe our present progress and future expectations. This is a joint project with Rina Anno and Timothy Logvinenko. 
4 April 2019  Christian Voigt (Glasgow University)  The Plancherel formula for complex semisimple quantum groups (joint with R. Yuncken) The Plancherel formula for complex semisimple Lie groups, due to GelfandNaimark and HarishChandra, is a basic ingredient in their harmonic analysis. In this talk I’ll present a computation of the Plancherel formula for the quantum deformations of these groups obtained via the Drinfeld double construction. The quantum groups obtained this way have featured prominently in the study of property (T) for tensor categories and subfactors in recent years. While the "quantum" Plancherel formula itself looks very similar to its classical counterpart – and is essentially a deformation thereof  the proof is completely different; it relies on the BGGresolution and an application of the Hopf trace formula. Starting from the classical Plancherel Theorem, I’ll give extensive background/motivation to all of the above, and then outline the key part of our proof. 
21 March 2019  Alvaro Torras Casas (Cardiff University)  InputDistributive Persistent Homology Persistent Homology has been developed as the main tool of Topological Data Analysis, with numerous applications in science and engineering. However, for very large data sets this tool can be very expensive to compute, both in terms of computational time and harddisk memory. We will present a new distributive algorithm which takes part directly on the input data. This has some theoretical difficulties since we need to work within the category of persistence modules. In particular, we will see a solution to the extension problem for the Persistent MayerVietoris spectral sequence. At the end we speculate that this approach might give us more information than ordinary Persistent Homology 
7 March 2019  Farzad Fathizadeh (Swansea University)  Heat kernel expansion of the DiracLaplacian of multifractal RobertsonWalker cosmologies I will talk about a recent work in which we find an explicit formula for each SeeleydeWitt coefficient in the full heat kernel expansion of the DiracLaplacian of a RobertsonWalker metric with a general cosmic expansion factor. We use the FeynmanKac formula and combinatorics of Brownian bridge integrals heavily. The extension of the result to the inhomogeneous case, where the spatial part of the model has a fractal structure, will also be presented. This is joint work with Yeorgia Kafkoulis and Matilde Marcolli. 
28 February 2019  Clelia Pech (University of Kent)  Mirror symmetry for cominuscule homogeneous varieties In this talk reporting on joint work with K. Rietsch and L. Williams, I will explain a new version of the construction by Rietsch of a mirror for some varieties with a homogeneous Lie group action. The varieties we study include quadrics and Lagrangian Grassmannians (i.e., Grassmannians of Lagrangian vector subspaces of a symplectic vector space). The mirror takes the shape of a rational function, the superpotential, defined on a Langlands dual homogeneous variety. I will show that the mirror manifold has a particular combinatorial structure called a cluster structure, and that the superpotential is expressed in coordinates dual to the cohomology classes of the original variety. I will also explain how these properties lead to new relations in the quantum cohomology, and a conjectural formula expressing solutions of the quantum differential equation in terms of the superpotential. If time allows, I will also explain how these results should extend to a larger family of homogeneous spaces called cominuscule homogeneous spaces. 
21 February 2019  Tyler Kelly (University of Birmingham)  Open Mirror Symmetry for LandauGinzburg Models Mirror Symmetry provides a link between different suites of data in geometry. On one hand, one has a lot of enumerative data that is associated to curve counts, telling you about important intersection theory in an interesting moduli problem. On the other, one has a variation of Hodge structure, that is, complex algebrogeometric structure given by computing special integrals. While typically one has focussed on the case where we study the enumerative data for a symplectic manifold, we here will instead study the enumerative geometry of a LandauGinzburg model. A LandauGinzburg model is essentially a triplet of data: an affine variety X [think C^n] with a group G acting on it and a Ginvariant algebraic function W from X to the complex numbers. We will describe what open enumerative geometry looks like for this gadget for the simplest examples (W=x^r) and explain what mirror symmetry means in this context. This is joint work in preparation with Mark Gross and Ran Tessler. 
14 February 2019  Enrico Fatighenti (Loughborough University)  Fano varieties of K3 type and IHS manifolds Subvarieties of Grassmannians (and especially Fano varieties) obtained from sections of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperkähler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi. 
7 February 2019  Ulrich Pennig (Cardiff University)  Exponential functors, Rmatrices and higher twists Rmatrices are solutions to the YangBaxter equation, which was introduced as a consistency equation in statistical mechanics, but has since then appeared in many other research areas, for example integrable quantum field theory, knot theory, the study of Hopf algebras and quantum information theory. Twisted Ktheory on the other hand is a variant of topological Ktheory that allows local coefficient systems called twists. Twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the twisted equivariant Ktheory of the group to the Verlinde ring of the associated loop group. In this talk I will discuss how involutive Rmatrices give rise to a natural generalisation of the twist appearing in this theorem via exponential functors. 
31 January 2019  Vincenzo Morinelli (Tor Vergata, Rome)  Scale and Möbius covariance in twodimensional HaagKastler net The relation between conformal and dilation covariance is a controversial problem in QFT. Although many models which are dilation covariant are indeed conformal covariant a complete understanding of this implication in the algebraic approach to QFT is missing. In this talk we present the following result: Given a twodimensional HaagKastler net which is Poincaredilation covariant with additional properties, we prove that it can be extended to a conformal (Möbius) covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. Time permitting we further discuss counterexamples. Based on a joint work with Yoh Tanimoto arXiv:1807.04707. 
13 December 2018  Matthew Buican (Queen Mary, London)  From 4D Supersymmetry to 2D RCFT via Logarithmic Theories I will discuss recent progress connecting the physics of certain large classes of 4D superconformal field theories with logarithmic conformal field theories. I will then use this connection to discuss a bridge between the physics of these 4D theories and certain more familiar 2D rational conformal field theories. 
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 
29 November  Tomasz Brzezinski (Swansea)  Twisted reality 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 YangBaxter equation and extremal characters of the infinite braid group The YangBaxter 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 ("Rmatrices") define representations and extremal characters of the infinite braid group. These characters define a natural equivalence relation on the family of all Rmatrices, 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 Rmatrices, the classification is complete (joint work with Simon and Ulrich). The more general case of Rmatrices with two arbitrary eigenvalues is currently work in progress, and I will present some partial results, including a classification of all Rmatrices defining representations of the TemperleyLieb algebra and a deformation theorem for involutive Rmatrices 
Wednesday  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 DeligneMumford 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] DeligneMumford spaces, and several new algebraic structures that naturally arise from studying those varieties. 
1  Andreas Aaserud (Cardiff)  Ktheory of some AFalgebras from braided categories 
18 October 2018  Paul Mitchener (Sheffield)  Categories of Unbounded Operators The GelfandNaimark theorem on C*algebras, which asserts that a C*algebra, defined axiomatically, is the same thing as a closed subalgebra of the algebra of bounded linear operators on a Hilbert space, is wellknown. 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 GelfandNaimark theorem holds. 
11 October 2018  Fabian Hebestreit (Bonn / INI Cambridge)  Twisted Ktheory via retractive symmetric spectra joint with Steffen Sagave Twisted Ktheory was originally invented to serve as the Ktheoretic 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 KKtheory 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 Ktheory 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 Ktheory. Time permitting, I shall also sketch how our work strengthens recent results of Dardalat and Pennig, describing the more exotic twists of Ktheory via selfabsorbing 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 Br_{n} is a wellknown algebraic structure which encodes configurations of n nontouching strands (“braids”) up to continuous transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the space Fl_{n} of complete flags in C^{n}. Generalised braids are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be noninvertible, 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. 