Geometry, Algebra, Mathematical Physics and Topology Research Group

Knizhnik-Zamolodchikov Equations

In this talk, I aim to give an accessible introduction to the much-celebrated Knizhnik-Zamolodchikov (KZ) Equations. These are a set of ODEs, arising (somewhat unexpectedly!) from representations of affine Lie algebras. The solutions to these equations have proven useful in a diverse range of fields within Mathematical Physics. However, the main focus of this talk will be to talk about the various mathematical objects and the key elements of underlying Lie algebra representation theory that are needed to understand the KZ equations. We give an outline of an elegant derivation which makes use of the Sugawara-Segal construction.

26 March 2020

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.

Homology theory of super-hypergraphs

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 Δ-sets  with face operations only partially defined, these objects we name super-hypergraphs. In this talk I will set foundations of homology theory of these combinatorial objects.

Logarithmic Riemann-Hilbert Correspondences

The classical Riemann-Hilbert 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 Kato-Nakayama and Ogus on this "logarithmic" setting. This in turn motivates recent joint work with Mattia Talpo on a further generalisation to logarithmic D-modules. We discuss what form the conjectural log Riemann-Hilbert Correspondence should take and the progress that has been achieved so far. We will not assume any familiarity with D-modules or logarithmic geometry.

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.

We will conclude the talk with possible future applications that are the basis of my UKRI fellowship in Swansea, made possible by recent advances in both theory and software.

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.

John Harvey (Swansea)

Estimating the reach of a submanifold

The reach is an important geometric invariant of submanifolds of Euclidean space. It is a real-valued 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.

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 Jiang-Su algebra as a particular example.

Christian Bönicke (Glasgow)

On the K-theory of ample groupoid algebras

It is a difficult problem to compute the K-theory of a crossed product of a C∗-algebra by a groupoid. One approach is given by the Baum-Connes conjecture, which asserts that a certain assembly map from the topological K-theory of the groupoid G with coefficients in a G-C∗-algebra A into the K-theory 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 Going-Down principle. This principle can be used in two ways, both of which I will illustrate by an example:

1) Obtain results about the Baum-Connes conjecture, and
2) in cases where the conjecture is known to hold, prove something about the K-theory of crossed products.

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 K-theory can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles.
After reviewing some results in this direction, I will report on work in progress concerning the construction of higher dimensional quantum sphere bundles in terms of Cuntz–Pimsner algebras of sub-product systems.

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.

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 non-split P-functors 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 Gelfand-Naimark and Harish-Chandra, 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 BGG-resolution 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)

Input-Distributive 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 hard-disk 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 Mayer-Vietoris 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 Dirac-Laplacian of multifractal Robertson-Walker cosmologies

I will talk about a recent work in which we find an explicit formula for each Seeley-deWitt coefficient in the full heat kernel expansion of the Dirac-Laplacian of a Robertson-Walker metric with a general cosmic expansion factor. We use the Feynman-Kac 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 Landau-Ginzburg 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 algebro-geometric 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 Landau-Ginzburg model. A Landau-Ginzburg model is essentially a triplet of data: an affine variety X [think C^n] with a group G acting on it and a G-invariant 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, R-matrices and higher twists

R-matrices are solutions to the Yang-Baxter 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 K-theory on the other hand is a variant of topological K-theory 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 K-theory of the group to the Verlinde ring of the associated loop group. In this talk I will discuss how involutive R-matrices 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 two-dimensional Haag-Kastler 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 two-dimensional Haag-Kastler net which is Poincare-dilation 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
Applying quantum algorithms to constraint satisfaction problems (with Earl Campbell and Ankur Khurana); arXiv:1810.05582

29 November
2018

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 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

Wednesday
14 November 2018
13:10

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.

1
November
2018

Andreas Aaserud (Cardiff)

K-theory of some AF-algebras from braided categories

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⊗A→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 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.

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 Br_n 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 Fl_n 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.