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 of 2022 we will return to in person seminars. Past Events: |

**Upcoming Seminar Talks**

There will be no GAPT seminar on 2nd March 2023, but the following might be interesting to seminar participants:

**Friday, 3rd March 2023**

Operator Algebras in the South of the UK

This is the first meeting of a new regional network to promote research in operator algebras in the South of the United Kingdom. Speakers include Francesca Arici (Leiden), Christian Bönicke (Newcastle), Ian Charlesworth (Cardiff), Kevin Boucher (Southampton) and Samantha Pilgrim (Glasgow).

This is the first meeting of a new regional network to promote research in operator algebras in the South of the United Kingdom. Speakers include Francesca Arici (Leiden), Christian Bönicke (Newcastle), Ian Charlesworth (Cardiff), Kevin Boucher (Southampton) and Samantha Pilgrim (Glasgow).

**Thursday, 9th March 2023 15:10 - 16:00**

Katrin Wendland (Trinity College Dublin)

Quarter BPS states in K3 theories

In conformal field theories with extended supersymmetry, the so-called BPS states play a special role. The net number of such states, counted according to a natural $\mathbb{Z}_2$ grading, is protected under deformations. However, pairs of such states with opposite parity can cease being BPS under deformations.

In this talk we will report on joint work with Anne Taormina, investigating this phenomenon for a particular type of deformations in K3 theories. We propose that the process is channelled by an action of $SU(2)$ which has its origin in the underlying K3 geometry.

Quarter BPS states in K3 theories

In conformal field theories with extended supersymmetry, the so-called BPS states play a special role. The net number of such states, counted according to a natural $\mathbb{Z}_2$ grading, is protected under deformations. However, pairs of such states with opposite parity can cease being BPS under deformations.

In this talk we will report on joint work with Anne Taormina, investigating this phenomenon for a particular type of deformations in K3 theories. We propose that the process is channelled by an action of $SU(2)$ which has its origin in the underlying K3 geometry.

**Thursday, 16th March 2023 15:10 - 16:00**

David Ellis (University of Bristol)

Product-free sets, and the diameter problem, in compact Lie groups.

A subset $S$ of a group $G$ is said to be product-free if there are no solutions to the equation $xy=z$ with x,y and z in $S$. Babai and Sós conjectured in 1985 that any finite group $G$ contains a product-free subset of size at least $c|G|$, where c is an absolute positive constant, but this was disproved in seminal work of Gowers in 2007. Gowers showed that if a finite group G is D-quasirandom (meaning that the smallest dimension of a nontrivial ordinary irreducible representation of G is at least D), then any product-free subset of G has measure at most $D^{-1/3}$. This yields an upper bound of 1/poly(n) on the maximal measure of product-free sets in (for example) $PSL(2,n)$, for n a prime power, and the alternating group $A_n$; constructions of Kedlaya give lower bounds which are also 1/poly(n). For compact connected Lie groups of rank n, however, the (conjectural) bounds on the maximal measure of measurable product-free sets are much stronger. Indeed, Gowers conjectured in 2007 that the maximal measure of a measurable product-free subset of $SU_n$ is at most $\exp(-cn)$ for some absolute positive constant c (though his methods only yield an upper bound of $O(n^{-1/3})$ in this case). We make progress on this conjecture of Gowers, showing that the maximal measure of a measurable product-free subset of $SU_n$, $SO_n$, $\text{Spin}_n$ or $Sp_n$ is at most $\exp(-cn^{1/3})$, where c is an absolute positive constant. We also give new bounds for the diameter problem in these groups. (Recall that if G is a group and S is a subset of G, the diameter of S is the minimal integer k, if it exists, such that $S^k = G$; the diameter problem in a compact connected Lie group G asks for the maximum possible diameter of a subset of G of measure m, for each $m > 0$.) Our techniques are based on non-Abelian Fourier analysis and (new) hypercontractive inequalities on compact connected Lie groups; the latter are obtained by two methods - the first method being via a coupling with Gaussian space, the second being via Ricci curvature and the Bakry-Emery criterion.

Based on joint work with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

Product-free sets, and the diameter problem, in compact Lie groups.

A subset $S$ of a group $G$ is said to be product-free if there are no solutions to the equation $xy=z$ with x,y and z in $S$. Babai and Sós conjectured in 1985 that any finite group $G$ contains a product-free subset of size at least $c|G|$, where c is an absolute positive constant, but this was disproved in seminal work of Gowers in 2007. Gowers showed that if a finite group G is D-quasirandom (meaning that the smallest dimension of a nontrivial ordinary irreducible representation of G is at least D), then any product-free subset of G has measure at most $D^{-1/3}$. This yields an upper bound of 1/poly(n) on the maximal measure of product-free sets in (for example) $PSL(2,n)$, for n a prime power, and the alternating group $A_n$; constructions of Kedlaya give lower bounds which are also 1/poly(n). For compact connected Lie groups of rank n, however, the (conjectural) bounds on the maximal measure of measurable product-free sets are much stronger. Indeed, Gowers conjectured in 2007 that the maximal measure of a measurable product-free subset of $SU_n$ is at most $\exp(-cn)$ for some absolute positive constant c (though his methods only yield an upper bound of $O(n^{-1/3})$ in this case). We make progress on this conjecture of Gowers, showing that the maximal measure of a measurable product-free subset of $SU_n$, $SO_n$, $\text{Spin}_n$ or $Sp_n$ is at most $\exp(-cn^{1/3})$, where c is an absolute positive constant. We also give new bounds for the diameter problem in these groups. (Recall that if G is a group and S is a subset of G, the diameter of S is the minimal integer k, if it exists, such that $S^k = G$; the diameter problem in a compact connected Lie group G asks for the maximum possible diameter of a subset of G of measure m, for each $m > 0$.) Our techniques are based on non-Abelian Fourier analysis and (new) hypercontractive inequalities on compact connected Lie groups; the latter are obtained by two methods - the first method being via a coupling with Gaussian space, the second being via Ricci curvature and the Bakry-Emery criterion.

Based on joint work with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

**Thursday, 20th April 2023 15:10 - 16:00**

Argam Ohanyan (Universität Wien)

**Thursday, 27th April 2023 15:10 - 16:00**

Daniel Berlyne (University of Bristol)

**Thursday, 4th May 2023 15:10 - 16:00**

Owen Tanner (University of Glasgow)

**Thursday, 11th May 2023 15:10 - 16:00**

Liana Heuberger (University of Bath)

**Seminar Talks - Autumn 2022**

**Thursday, 6th October 2022 15:10 - 16:00**

Nick Cavenagh (University of Waikato, New Zealand)

Row-column factorial designs of strength at least 2

joint work with Fahim Rahim

The $q^k$ (full) factorial design with replication $\lambda$ is the multi-set consisting of $\lambda$ occurrences of each element of each $q$-ary vector of length $k$; we denote this by $\lambda\times [q]^k$. An $m\times n$

Row-column factorial designs of strength at least 2

joint work with Fahim Rahim

The $q^k$ (full) factorial design with replication $\lambda$ is the multi-set consisting of $\lambda$ occurrences of each element of each $q$-ary vector of length $k$; we denote this by $\lambda\times [q]^k$. An $m\times n$

*row-column factorial design*$q^k$ of*strength*$t$ is an arrangement of the elements of $\lambda \times [q]^k$ into an $m\times n$ array (which we say is of type $I_k(m,n,q,t)$) such that for each row (column), the set of vectors therein are the rows of an orthogonal array of size $k$, degree $n$ (respectively, $m$), $q$ levels and strength $t$. Such arrays have been used in practice in experimental design. In this context, for a row-column factorial design of strength $t$, all subsets of interactions of size at most $t$ can be estimated without confounding by the row and column blocking factors. In this talk we consider row-column factorial designs with strength $t\geq 2$. The constructions presented use Hadamard matrices and linear algebra.**Thursday, 20th October 2022 15:10 - 16:00**

Arman Sarikyan (University of Edinburgh)

On the Rationality of Fano-Enriques Threefolds

A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

On the Rationality of Fano-Enriques Threefolds

A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

**Thursday, 27th October 2022 15:10 - 16:00**

Ana Kontrec (MPI / Bonn)

Representation theory and duality properties of some minimal affine $\mathcal{W}$-algebras

One of the most important families of vertex algebras are affine vertex algebras and their associated $\mathcal{W}$-algebras, which are connected to various aspects of geometry and physics.

Among the simplest examples of $\mathcal{W}$-algebras is the Bershadsky-Polyakov vertex algebra $\mathcal{W}^k(\mathfrak{g}, f_{min})$, associated to $\mathfrak{g} = sl(3)$ and the minimal nilpotent element $f_{min}$.

In this talk we are particularly interested in the Bershadsky-Polyakov algebra $\mathcal W_k$ at positive integer levels, for which we obtain a complete classification of irreducible modules.

In the case $k=1$, we show that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple affine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. This is joint work with D. Adamovic.

Representation theory and duality properties of some minimal affine $\mathcal{W}$-algebras

One of the most important families of vertex algebras are affine vertex algebras and their associated $\mathcal{W}$-algebras, which are connected to various aspects of geometry and physics.

Among the simplest examples of $\mathcal{W}$-algebras is the Bershadsky-Polyakov vertex algebra $\mathcal{W}^k(\mathfrak{g}, f_{min})$, associated to $\mathfrak{g} = sl(3)$ and the minimal nilpotent element $f_{min}$.

In this talk we are particularly interested in the Bershadsky-Polyakov algebra $\mathcal W_k$ at positive integer levels, for which we obtain a complete classification of irreducible modules.

In the case $k=1$, we show that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple affine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. This is joint work with D. Adamovic.

**Thursday, 3rd November 2022 15:10 - 16:00**

Sergio Giron Pacheco (University of Oxford)

Anomalous actions and invariants of operator algebras.

An anomalous symmetry of an operator algebra $A$ is a mapping from a group $G$ to the automorphism group of $A$ which is multiplicative up to inner automorphisms of $A$. This can be rephrased as the action of a pointed tensor category on $A$. Starting from the basics, I will introduce anomalous actions and discuss some history of their study in the literature. I will then discuss their existence and classification on simple C*-algebras. For these questions, it will be important to consider K-theoretic invariants of the algebras.

Anomalous actions and invariants of operator algebras.

An anomalous symmetry of an operator algebra $A$ is a mapping from a group $G$ to the automorphism group of $A$ which is multiplicative up to inner automorphisms of $A$. This can be rephrased as the action of a pointed tensor category on $A$. Starting from the basics, I will introduce anomalous actions and discuss some history of their study in the literature. I will then discuss their existence and classification on simple C*-algebras. For these questions, it will be important to consider K-theoretic invariants of the algebras.

**Thursday, 10th November 2022 15:10 - 16:00**

Thomas Wasserman (University of Oxford)

The Landau-Ginzburg - Conformal Field Theory Correspondence and Module Tensor Categories

In this talk, I will give a brief introduction to the Landau-Ginzburg - Conformal Field Theory (LG-CFT) correspondence, a prediction from physics. This prediction links aspects of Landau-Ginzburg models, described by matrix factorisations for a polynomial known as the potential, with Conformal Field Theories, described by for example vertex operator algebras. While both sides of the correspondence have good mathematical descriptions, it is an open problem to give a mathematical formulation of the correspondence.

After this introduction, I will discuss the only known realisation of this correspondence, for the potential $x^d$. For even $d$ this is a recent result, the proof of which uses the tools of module tensor categories.

I will not assume prior knowledge of matrix factorisations, CFTs, or module tensor categories. This talk is based on joint work with Ana Ros Camacho.

The Landau-Ginzburg - Conformal Field Theory Correspondence and Module Tensor Categories

In this talk, I will give a brief introduction to the Landau-Ginzburg - Conformal Field Theory (LG-CFT) correspondence, a prediction from physics. This prediction links aspects of Landau-Ginzburg models, described by matrix factorisations for a polynomial known as the potential, with Conformal Field Theories, described by for example vertex operator algebras. While both sides of the correspondence have good mathematical descriptions, it is an open problem to give a mathematical formulation of the correspondence.

After this introduction, I will discuss the only known realisation of this correspondence, for the potential $x^d$. For even $d$ this is a recent result, the proof of which uses the tools of module tensor categories.

I will not assume prior knowledge of matrix factorisations, CFTs, or module tensor categories. This talk is based on joint work with Ana Ros Camacho.

**Thursday, 17th November 2022 15:10 - 16:00**

Jacek Krajczok (University of Glasgow)

On the approximation property of locally compact quantum groups

One of the most widely studied properties of groups is the notion of amenability - in one of its many formulations, it gives us a way of approximation the constant function by functions in the Fourier algebra. The notion of amenability was relaxed in various directions: a very weak form of amenability, called the approximation property (AP), was introduced by Haagerup and Kraus in 1994. It still gives us a way of approximating the constant function by functions in the Fourier algebra, but in much weaker sense. During the talk I'll introduce AP for locally compact quantum groups, discuss some of its permanence properties and relation to w*OAP of quantum group von Neumann algebra. The talk is based on a joint work with Matthew Daws and Christian Voigt.

On the approximation property of locally compact quantum groups

One of the most widely studied properties of groups is the notion of amenability - in one of its many formulations, it gives us a way of approximation the constant function by functions in the Fourier algebra. The notion of amenability was relaxed in various directions: a very weak form of amenability, called the approximation property (AP), was introduced by Haagerup and Kraus in 1994. It still gives us a way of approximating the constant function by functions in the Fourier algebra, but in much weaker sense. During the talk I'll introduce AP for locally compact quantum groups, discuss some of its permanence properties and relation to w*OAP of quantum group von Neumann algebra. The talk is based on a joint work with Matthew Daws and Christian Voigt.

**Thursday, 24th November 2022 15:10 - 16:00**

Konstanze Rietsch (King's College London)

Tropical Edrei theorem

The classical Edrei theorem from the 1950's gives a parametrisation of the infinite upper-triangular totally positive Toeplitz matrices by positive real parameters with finite sum. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group, as was discovered by Thoma who reproved Edrei's theorem in the 1960's. A totally different theorem, related to quantum cohomology of flag varieties and mirror symmetry, gives inverse parametrisations of finite totally positive Toeplitz matrices [R, 06]. The latter theorem has an analogue over the field of Puiseaux series, obtained by Judd and studied further by Ludenbach. In this talk I will explain a new `tropical' version of the Edrei theorem, connecting the finite and infinite theories.

Tropical Edrei theorem

The classical Edrei theorem from the 1950's gives a parametrisation of the infinite upper-triangular totally positive Toeplitz matrices by positive real parameters with finite sum. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group, as was discovered by Thoma who reproved Edrei's theorem in the 1960's. A totally different theorem, related to quantum cohomology of flag varieties and mirror symmetry, gives inverse parametrisations of finite totally positive Toeplitz matrices [R, 06]. The latter theorem has an analogue over the field of Puiseaux series, obtained by Judd and studied further by Ludenbach. In this talk I will explain a new `tropical' version of the Edrei theorem, connecting the finite and infinite theories.

**Thursday, 1st December 2022 15:10 - 16:00**

Kevin Aguyar Brix (University of Glasgow)

Irreversible dynamics and C*-algebras

How do we model the evolution of a system? A symbolic dynamical system is a coding of certain time evolutions that can be represented by finite graphs and that are usually invertible. However, in this talk I want to emphasise irreversible symbolic systems, how and why they are mathematically interesting, and their connections to other fields such as C*-algebras (algebras of bounded operators on Hilbert space). Along the way, I will also discuss the infamous conjugacy problem for shifts of finite type.

Irreversible dynamics and C*-algebras

How do we model the evolution of a system? A symbolic dynamical system is a coding of certain time evolutions that can be represented by finite graphs and that are usually invertible. However, in this talk I want to emphasise irreversible symbolic systems, how and why they are mathematically interesting, and their connections to other fields such as C*-algebras (algebras of bounded operators on Hilbert space). Along the way, I will also discuss the infamous conjugacy problem for shifts of finite type.

**Thursday, 8th December 2022 15:10 - 16:00**

Christiaan van de Ven (Universität Würzburg)

Strict deformation quantization in quantum lattice models

Quantization in general refers to the transition from a classical to a corresponding quantum theory. The inverse issue, called the classical limit of quantum theories, is considered a much more difficult problem. A rigorous and natural framework that addresses this problem exists under the name strict (or C*-algebraic) deformation quantization. In this talk, I will first introduce this concept by means of relevant definitions. Next, I will show its connection with the classical limit of quantum theories, starting with a brief summary of the theory in the context of mean-field quantum theories. Finally, I will discuss the results of a recent work on how strict deformation quantization applies to more realistic models described by local interactions for periodic boundary conditions, e.g., the quantum Heisenberg spin chain.

Strict deformation quantization in quantum lattice models

Quantization in general refers to the transition from a classical to a corresponding quantum theory. The inverse issue, called the classical limit of quantum theories, is considered a much more difficult problem. A rigorous and natural framework that addresses this problem exists under the name strict (or C*-algebraic) deformation quantization. In this talk, I will first introduce this concept by means of relevant definitions. Next, I will show its connection with the classical limit of quantum theories, starting with a brief summary of the theory in the context of mean-field quantum theories. Finally, I will discuss the results of a recent work on how strict deformation quantization applies to more realistic models described by local interactions for periodic boundary conditions, e.g., the quantum Heisenberg spin chain.

**Thursday, 15th December 2022 15:10 - 16:00**

Taro Sogabe (University of Tokyo)

The Reciprocal Kirchberg algebras

In the classical homotopy theory, there is the duality, Spanier Whitehead’s duality, connecting homology and cohomology. In this talk, I would like to explain the Spanier Whiteheads duality for KK-theory which is the homotopy theory for C*-algebras, and I will show that this duality gives a characterization of two unital Kirchberg algebras sharing the same homotopy groups of their automorphism groups.

The Reciprocal Kirchberg algebras

In the classical homotopy theory, there is the duality, Spanier Whitehead’s duality, connecting homology and cohomology. In this talk, I would like to explain the Spanier Whiteheads duality for KK-theory which is the homotopy theory for C*-algebras, and I will show that this duality gives a characterization of two unital Kirchberg algebras sharing the same homotopy groups of their automorphism groups.