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. Speakers give 50 minute talks. In the autumn semester 2020 the GAPT seminar will be a virtual seminar on Zoom due to the Coronavirus pandemic. Past Events: |

**Upcoming Seminar Talks**

**Thursday, 8th October 2020 14:00 - 15:00 (note the different time) Zoom**

César Galindo (Universidad de los Andes / Universität Hamburg)

Braided Zesting and its applications

In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. We will present a complete obstruction theory and parametrisation approach to the construction and illustrate its utility with several examples.

The talk is based on the manuscript https://arxiv.org/abs/2005.05544 a joint work with Colleen Delaney, Julia Plavnik, Eric C. Rowell, and Qing Zhang.

**- please contact me to get the Zoom login data**Braided Zesting and its applications

In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. We will present a complete obstruction theory and parametrisation approach to the construction and illustrate its utility with several examples.

The talk is based on the manuscript https://arxiv.org/abs/2005.05544 a joint work with Colleen Delaney, Julia Plavnik, Eric C. Rowell, and Qing Zhang.

**Thursday, 22nd October 2020 15:10 - 16:00 Zoom**

Jamie Walton (University of Nottingham)

Aperiodic Order: The Mathematics of Systems of Approximate Symmetry

Symmetry is frequently exploited in Mathematics, but there are many situations in which systems exhibit long-range recurrence without precise periodic repetition. A simple example is given by a coding of an irrational circle rotation. With Shechtman's discovery of quasicrystals - physical materials with long-range order but also rotational symmetry precluding the standard periodicity of usual crystals - it seems that "aperiodically ordered" patterns can appear in nature too. In this talk I will introduce the field of Aperiodic Order, which investigates intriguing infinite idealisations of such patterns. A prototypical family of examples is given by Penrose's famous rhomb, or kite and dart tilings. I will then explain what sorts of mathematical structures can be introduced to systemise their study. I will focus on the construction of the tiling space of an aperiodic pattern, through which one may construct fundamental invariants using standard tools from Algebraic Topology.

**- please contact me to get the Zoom login data**Aperiodic Order: The Mathematics of Systems of Approximate Symmetry

Symmetry is frequently exploited in Mathematics, but there are many situations in which systems exhibit long-range recurrence without precise periodic repetition. A simple example is given by a coding of an irrational circle rotation. With Shechtman's discovery of quasicrystals - physical materials with long-range order but also rotational symmetry precluding the standard periodicity of usual crystals - it seems that "aperiodically ordered" patterns can appear in nature too. In this talk I will introduce the field of Aperiodic Order, which investigates intriguing infinite idealisations of such patterns. A prototypical family of examples is given by Penrose's famous rhomb, or kite and dart tilings. I will then explain what sorts of mathematical structures can be introduced to systemise their study. I will focus on the construction of the tiling space of an aperiodic pattern, through which one may construct fundamental invariants using standard tools from Algebraic Topology.

**Thursday, 29th October 2020 15:10 - 16:00 Zoom**

Irakli Patchkoria (University of Aberdeen)

Equivariant Stallings-Swan theorem and cohomology of orbit categories

Given a discrete group $G$ and a family of subgroups $F$, a conjecture in geometric group theory asserts that the geometric dimension of $G$ with respect to $F$ is equal to 1 if and only if the cohomological dimension of the orbit category $O_F(G)$ associated to $F$ is equal to 1. The conjecture is known to be true for certain families: the trivial family (Stallings-Swan), the family of all finite subgroups (Dunwoody), and the family of virtually cyclic subgroups (Degrijse) among others. We prove the conjecture for any family $F$ which does not contain the family of all finite subgroups. As a consequence one obtains a cohomological characterisation of free groups with an action of finite group and invariant basis. This is an equivariant version of the classical Stallings-Swan theorem. We will also discuss applications to the equivariant LS category. The talk will introduce basic concepts at the beginning and should be accessible to a general audience. This is all joint work with Mark Grant and Ehud Meir.

**- please contact me to get the Zoom login data**Equivariant Stallings-Swan theorem and cohomology of orbit categories

Given a discrete group $G$ and a family of subgroups $F$, a conjecture in geometric group theory asserts that the geometric dimension of $G$ with respect to $F$ is equal to 1 if and only if the cohomological dimension of the orbit category $O_F(G)$ associated to $F$ is equal to 1. The conjecture is known to be true for certain families: the trivial family (Stallings-Swan), the family of all finite subgroups (Dunwoody), and the family of virtually cyclic subgroups (Degrijse) among others. We prove the conjecture for any family $F$ which does not contain the family of all finite subgroups. As a consequence one obtains a cohomological characterisation of free groups with an action of finite group and invariant basis. This is an equivariant version of the classical Stallings-Swan theorem. We will also discuss applications to the equivariant LS category. The talk will introduce basic concepts at the beginning and should be accessible to a general audience. This is all joint work with Mark Grant and Ehud Meir.

**Thursday, 5th November 2020 15:10 - 16:00 Zoom**

Eric Rowell (Texas A&M University)

Representations of braid groups appear in many (related) guises, as sources of knot and link invariants, transfer matrices in statistical mechanics, quantum gates in topological quantum computers and commutativity morphisms in braided fusion categories. Regarded as trajectories of points in the plane, the natural generalization of braid groups are groups of motions of links in 3 manifolds. While much of the representation theory of braid groups and motions groups remains mysterious, we are starting to see hints that suggest a few conjectures. I will describe a few of these conjectures and some of the progress towards verification.

**- please contact me to get the Zoom login data**

Representations of Braid Groups and Motion GroupsRepresentations of braid groups appear in many (related) guises, as sources of knot and link invariants, transfer matrices in statistical mechanics, quantum gates in topological quantum computers and commutativity morphisms in braided fusion categories. Regarded as trajectories of points in the plane, the natural generalization of braid groups are groups of motions of links in 3 manifolds. While much of the representation theory of braid groups and motions groups remains mysterious, we are starting to see hints that suggest a few conjectures. I will describe a few of these conjectures and some of the progress towards verification.

**Thursday, 12th November 2020 15:10 - 16:00 Zoom**

Eleonore Faber (University of Leeds)

Infinite constructions: Grassmannian categories of infinite rank and triangulations of an infinity-gon

The homogeneous coordinate ring of the Grassmannian of $k$-dimensional subspaces in $n$-space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmannian cluster category $C(k,n)$, as shown by Jensen, King, and Su in 2016. In particular, the cluster category $C(2,n)$ models triangulations of a regular $n$-gon. A natural question is, if there is some kind of limit construction, i.e., the category "$C(2,\infty)$" and how to model triangulations of a regular "$\infty$-gon".

This talk is about a categorification of the homogeneous coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules/matrix factorizations over a hypersurface singularity. This gives an infinite rank analogue of the categories of Jensen, King, and Su. We show that there is a structure preserving bijection between the generically free rank one modules in a Grassmannian category of infinite rank and the Plücker coordinates in a Grassmannian cluster algebra of infinite rank.

In a special case, when the hypersurface singularity is a curve of countable Cohen-Macaulay type, our category has a combinatorial model by an "infinity-gon" and we can determine triangulations of this infinity-gon.

This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

**- please contact me to get the Zoom login data**Infinite constructions: Grassmannian categories of infinite rank and triangulations of an infinity-gon

The homogeneous coordinate ring of the Grassmannian of $k$-dimensional subspaces in $n$-space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmannian cluster category $C(k,n)$, as shown by Jensen, King, and Su in 2016. In particular, the cluster category $C(2,n)$ models triangulations of a regular $n$-gon. A natural question is, if there is some kind of limit construction, i.e., the category "$C(2,\infty)$" and how to model triangulations of a regular "$\infty$-gon".

This talk is about a categorification of the homogeneous coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules/matrix factorizations over a hypersurface singularity. This gives an infinite rank analogue of the categories of Jensen, King, and Su. We show that there is a structure preserving bijection between the generically free rank one modules in a Grassmannian category of infinite rank and the Plücker coordinates in a Grassmannian cluster algebra of infinite rank.

In a special case, when the hypersurface singularity is a curve of countable Cohen-Macaulay type, our category has a combinatorial model by an "infinity-gon" and we can determine triangulations of this infinity-gon.

This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

**Thursday, 19th November 2020 15:10 - 16:00 Zoom**

Marius Dadarlat (Purdue University)

Obstructions to matrix stability of discrete groups

A discrete countable group is matricially stable if its finite dimensional approximate unitary representations are perturbable to genuine representations in the point-norm topology. We aim to explain in accessible terms why matricial stability for a group $G$ implies the vanishing of the rational even cohomology of $G$ for large classes of groups, including the linear groups.

**- please contact me to get the Zoom login data**Obstructions to matrix stability of discrete groups

A discrete countable group is matricially stable if its finite dimensional approximate unitary representations are perturbable to genuine representations in the point-norm topology. We aim to explain in accessible terms why matricial stability for a group $G$ implies the vanishing of the rational even cohomology of $G$ for large classes of groups, including the linear groups.

**Thursday, 26th November 2020 15:10 - 16:00 Zoom**

Robert Allen (Cardiff University)

Under certain assumptions, I will discuss a conjectural equivalence between the representation theory of vertex algebras and quantum groups. As a specific example, I will talk about the bosonic ghost vertex algebra.

**- please contact me to get the Zoom login data**

Bosonic GhostbustersUnder certain assumptions, I will discuss a conjectural equivalence between the representation theory of vertex algebras and quantum groups. As a specific example, I will talk about the bosonic ghost vertex algebra.

**Thursday, 3rd December 2020 15:10 - 16:00**

**Zoom**

Madeleine Jotz-Lean (Georg-August-Universität Göttingen)

Symmetries are usually understood as the action of a group. However, many mathematical objects have a structure that we recognise as a symmetry, but which is in fact expressed by a groupoid.

This talk motivates the notions of Lie groupoids and Lie algebroids, and explains why the classical notions of representations and ideals are not sufficient in this context.

Representations up to homotopy of Lie algebroids on graded vector bundles are explained, in particular the adjoint representation up to homotopy of a Lie algebroid is described. The graded trace of the powers of the curvature of a connection up to homotopy induces characteristic classes of graded vector bundles. Using these, one can prove obstructions to the existence of a representation up to homotopy on a graded vector bundle. This gives a further interpretation for the classical Pontryagin characters of vector bundles.

As a consequence, an obstruction to the existence of an infinitesimal ideal system in a Lie algebroid is given -- these objects are considered the right notion of ideal in the context of Lie algebroids.

**- please contact me to get the Zoom login data**

Lie groupoids, Lie algebroids, representations and idealsSymmetries are usually understood as the action of a group. However, many mathematical objects have a structure that we recognise as a symmetry, but which is in fact expressed by a groupoid.

This talk motivates the notions of Lie groupoids and Lie algebroids, and explains why the classical notions of representations and ideals are not sufficient in this context.

Representations up to homotopy of Lie algebroids on graded vector bundles are explained, in particular the adjoint representation up to homotopy of a Lie algebroid is described. The graded trace of the powers of the curvature of a connection up to homotopy induces characteristic classes of graded vector bundles. Using these, one can prove obstructions to the existence of a representation up to homotopy on a graded vector bundle. This gives a further interpretation for the classical Pontryagin characters of vector bundles.

As a consequence, an obstruction to the existence of an infinitesimal ideal system in a Lie algebroid is given -- these objects are considered the right notion of ideal in the context of Lie algebroids.

**Thursday, 17th December 2020 15:10 - 16:00 Zoom**

Magdalena E. Musat (University of Copenhagen)

**- please contact me to get the Zoom login data****Postponed Talks**

Johannes Hofscheier (University of Nottingham)

The Cohomology Ring of Toric Bundles

Khovanskii and Pukhlikov observed that the cohomology ring of smooth projective toric varieties is completely described by the Bernstein-Kushnirenko theorem via the volume polynomial on the space of polytopes. In this talk, I will report on joint work with Khovanskii and Monin where we extend this approach to a description of the cohomology ring of equivariant compactifications of torus principal bundles by formulating respective Bernstein-Kushnirenko-type theorems. We conclude the presentation by explaining how this description of the cohomology ring can be used to compute the ring of conditions for horospherical homogeneous spaces.

The Cohomology Ring of Toric Bundles

Khovanskii and Pukhlikov observed that the cohomology ring of smooth projective toric varieties is completely described by the Bernstein-Kushnirenko theorem via the volume polynomial on the space of polytopes. In this talk, I will report on joint work with Khovanskii and Monin where we extend this approach to a description of the cohomology ring of equivariant compactifications of torus principal bundles by formulating respective Bernstein-Kushnirenko-type theorems. We conclude the presentation by explaining how this description of the cohomology ring can be used to compute the ring of conditions for horospherical homogeneous spaces.

Konstanze Rietsch (King's College London)

Mario Berta (Imperial College London)