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 spring semester 2021 the GAPT seminar will be a virtual seminar on Zoom due to the Coronavirus pandemic. Please contact me to get the Zoom login data for the talk. Past Events: |

**Upcoming Seminar Talks**

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

Nikon Kurnosov (UCL)

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

Christian Hagendorf (Université catholique de Louvain)

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

Severin Bunk (Oxford)

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

Luca Pol (Regensburg)

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

Emily Norton (University of Kent)

**Thursday, 5th May 2022 15:10 - 16:00**

Anne-Sophie Kaloghiros (Brunel University London)

**Thursday, 12th May 2022 15:10 - 16:00**

Simona Paoli (University of Aberdeen)

**Seminar Talks - Autumn 2021**

**Thursday, 21st October 2021 15:10 - 16:00**

Sara Azzali (Universität Greifswald)

The Baum-Connes correspondence for the pure braid group

The Baum-Connes conjecture can be seen as a far reaching generalisation of the Atiyah--Singer index theorem. Given a locally compact group $G$, the conjecture predicts an isomorphisms between a topological and an analytic object constructed from $G$. One of the main motivations of the Baum-Connes conjecture comes from the case of discrete groups, where it implies the Novikov conjecture on the homotopy invariance of higher signatures. We first give an introduction to the topic, then we look at examples of explicit computations of the left and right hand side of the Baum-Connes correspondence. In particular we discuss certain braid groups. For this class of groups the conjecture is known to be true in by results of Oyono-Oyono, Chabert-Echterhoff, Schick.

The talk is based on joint work with Sarah Browne, Maria Paula Gomez, Lauren Ruth and Hang Wang.

The Baum-Connes correspondence for the pure braid group

The Baum-Connes conjecture can be seen as a far reaching generalisation of the Atiyah--Singer index theorem. Given a locally compact group $G$, the conjecture predicts an isomorphisms between a topological and an analytic object constructed from $G$. One of the main motivations of the Baum-Connes conjecture comes from the case of discrete groups, where it implies the Novikov conjecture on the homotopy invariance of higher signatures. We first give an introduction to the topic, then we look at examples of explicit computations of the left and right hand side of the Baum-Connes correspondence. In particular we discuss certain braid groups. For this class of groups the conjecture is known to be true in by results of Oyono-Oyono, Chabert-Echterhoff, Schick.

The talk is based on joint work with Sarah Browne, Maria Paula Gomez, Lauren Ruth and Hang Wang.

**Thursday, 4th November 2021 15:10 - 16:00**

Mihály Weiner (Budapest University of Technology and Economics) -

From quantum fields to local algebras: strong locality

A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. Unfortunately, there have been very few direct methods proposed for showing strong locality of a quantum field. Among them, linear energy bounds are the most widely used.

The problem is especially interesting in the conformal chiral setting, where there are plenty of existing unitary VOA models; i.e. quantum fields given in a rather concrete manner. However, a chiral conformal field of conformal weight $d > 2$ cannot admit linear energy bounds. Nevertheless, we prove that if a chiral conformal field satisfies an energy bound of degree $d−1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry.

As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $W_3$-algebra is strongly local. For central charge $c>2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.

**on Zoom**From quantum fields to local algebras: strong locality

**without**linear energy boundsA family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. Unfortunately, there have been very few direct methods proposed for showing strong locality of a quantum field. Among them, linear energy bounds are the most widely used.

The problem is especially interesting in the conformal chiral setting, where there are plenty of existing unitary VOA models; i.e. quantum fields given in a rather concrete manner. However, a chiral conformal field of conformal weight $d > 2$ cannot admit linear energy bounds. Nevertheless, we prove that if a chiral conformal field satisfies an energy bound of degree $d−1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry.

As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $W_3$-algebra is strongly local. For central charge $c>2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.

**Thursday, 11th November 2021 15:10 - 16:00**

Nils Carqueville (Universität Wien) -

Fully extended spin TQFTs

I shall discuss topological quantum field theories on 2-dimensional bordisms with spin structures. After reviewing the functorial approach by Atiyah and Segal, I will explain what it means for TQFTs to be "extended to the point". The general theory will be illustrated by and applied to state sum models, Landau-Ginzburg models, and truncated 3-dimensional sigma models. (Based on joint work with Lóránt Szegedy, and with Ilka Brunner and Daniel Roggenkamp.)

**on BigBlueButton**Fully extended spin TQFTs

I shall discuss topological quantum field theories on 2-dimensional bordisms with spin structures. After reviewing the functorial approach by Atiyah and Segal, I will explain what it means for TQFTs to be "extended to the point". The general theory will be illustrated by and applied to state sum models, Landau-Ginzburg models, and truncated 3-dimensional sigma models. (Based on joint work with Lóránt Szegedy, and with Ilka Brunner and Daniel Roggenkamp.)

**Thursday, 18th November 2021 15:10 - 16:00**

Federico Barbacovi (University College London)

Categorical dynamics

A topological dynamical system is given by a topological space $X$ and a continuous map $f : X \to X$. To such a couple $(X,f)$ one can associate an invariant called the topological entropy of $f$, which is a non-negative extended real number. A famous theorem of Gromov and Yomdin says that when $X$ is a smooth, projective variety, and $f$ is a holomorphic automorphism, the topological entropy of $f$ can be computed by looking at the action of $f$ on the (algebraic part of the) cohomology. For the derived-category-minded geometer, it is therefore natural to ask whether such result is the shadow of some statement that holds at the level of derived categories.

In this talk I will introduce the notion of categorical dynamical system as defined by Dimitrov-Haiden-Katzarkov-Kontsevich and I will report on joint work with Jongmyeong Kim in which we tackle Kikuta-Takahashi’s conjecture (which will be introduced during the talk) both in the negative direction (producing new counterexamples) and positive direction (giving condition that ensure that it holds - at least in a weak sense).

**- on Zoom**Categorical dynamics

A topological dynamical system is given by a topological space $X$ and a continuous map $f : X \to X$. To such a couple $(X,f)$ one can associate an invariant called the topological entropy of $f$, which is a non-negative extended real number. A famous theorem of Gromov and Yomdin says that when $X$ is a smooth, projective variety, and $f$ is a holomorphic automorphism, the topological entropy of $f$ can be computed by looking at the action of $f$ on the (algebraic part of the) cohomology. For the derived-category-minded geometer, it is therefore natural to ask whether such result is the shadow of some statement that holds at the level of derived categories.

In this talk I will introduce the notion of categorical dynamical system as defined by Dimitrov-Haiden-Katzarkov-Kontsevich and I will report on joint work with Jongmyeong Kim in which we tackle Kikuta-Takahashi’s conjecture (which will be introduced during the talk) both in the negative direction (producing new counterexamples) and positive direction (giving condition that ensure that it holds - at least in a weak sense).

**Thursday, 25th November 2021 15:10 - 16:00**

Fiona Torzewska (University of Leeds) -

Motion groupoids & mapping class groupoids

The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group.

In this talk I will construct for each manifold $M$ its motion groupoid $Mot_M$, whose object class is the power set of $M$. I will also give a construction of a mapping class groupoid $MCG_M$ associated to a manifold $M$ with the same object class. I will give examples which frame questions that inform the modelling of topological phases, such as questions about the skeletons of these categories.

For each manifold $M$ I will construct a functor $F\colon Mot_M \to MCG_M$ and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of $M$ is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in \mathbb{N}$.

**in person**Motion groupoids & mapping class groupoids

The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group.

In this talk I will construct for each manifold $M$ its motion groupoid $Mot_M$, whose object class is the power set of $M$. I will also give a construction of a mapping class groupoid $MCG_M$ associated to a manifold $M$ with the same object class. I will give examples which frame questions that inform the modelling of topological phases, such as questions about the skeletons of these categories.

For each manifold $M$ I will construct a functor $F\colon Mot_M \to MCG_M$ and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of $M$ is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in \mathbb{N}$.

**Thursday, 2nd December 2021 15:10 - 16:00**

David Penneys (The Ohio State University) -

Fusion categories in mathematics and physics

Classically, the notion of symmetry is described by a group. In recent decades, we have seen the emergence of quantum mathematical objects whose symmetries are best described by tensor categories. Fusion categories simultaneously generalize the notion of a finite group and its category of finite dimensional complex representations, and we think of these objects as encoding quantum symmetries. We will give a basic introduction to the theory of fusion categories and describe applications to some areas of mathematics and physics, namely operator algebras and theoretical condensed matter.

**on Zoom**Fusion categories in mathematics and physics

Classically, the notion of symmetry is described by a group. In recent decades, we have seen the emergence of quantum mathematical objects whose symmetries are best described by tensor categories. Fusion categories simultaneously generalize the notion of a finite group and its category of finite dimensional complex representations, and we think of these objects as encoding quantum symmetries. We will give a basic introduction to the theory of fusion categories and describe applications to some areas of mathematics and physics, namely operator algebras and theoretical condensed matter.

**Thursday, 9th December 2021 15:10 - 16:00**

Corey Jones (North Carolina State University)

Braided tensor categories from von Neumann algebras.

In the setting of algebraic quantum field theory, the superselection sectors of a net of von Neumann algebras naturally form a braided tensor category. In this talk we will explain an analogous construction which assigns a braided tensor category to a single von Neumann algebra, despite the absence of spatial degrees of freedom. Our construction builds on the work of Connes, Jones and Popa, and extends Connes' $\chi(M)$ invariant. We will highlight several parallels to the theory of conformal nets, and show that for any finite depth inclusion $N\subseteq M$ of non-Gamma $II_1$ factors, the braided tensor category associated to the enveloping algebra $M_{\infty}$ is the Drinfeld center of the original standard invariant.

Based on joint work with Quan Chen and David Penneys.

Braided tensor categories from von Neumann algebras.

In the setting of algebraic quantum field theory, the superselection sectors of a net of von Neumann algebras naturally form a braided tensor category. In this talk we will explain an analogous construction which assigns a braided tensor category to a single von Neumann algebra, despite the absence of spatial degrees of freedom. Our construction builds on the work of Connes, Jones and Popa, and extends Connes' $\chi(M)$ invariant. We will highlight several parallels to the theory of conformal nets, and show that for any finite depth inclusion $N\subseteq M$ of non-Gamma $II_1$ factors, the braided tensor category associated to the enveloping algebra $M_{\infty}$ is the Drinfeld center of the original standard invariant.

Based on joint work with Quan Chen and David Penneys.