**Seminar Talks - Autumn 2020**

**Thursday, 8th October 2020 14:00 - 15:00 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.

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.

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.

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 and Motion Groups

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.

Representations of Braid Groups and Motion Groups

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.

**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.

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.

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)

Bosonic Ghostbusters

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.

Bosonic Ghostbusters

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.

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

**Zoom**

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

Lie groupoids, Lie algebroids, representations and ideals

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.

Lie groupoids, Lie algebroids, representations and ideals

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.

**Seminar Talks - Spring 2021**

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

Andreas Thom (TU Dresden)

Sofic approximations — what’s the problem?

I am planning to give a general introduction to sofic groups, mention a few applications to fundamental conjectures about groups and group rings, and explain Misha Gromov’s conjecture that all groups are sofic. Finally I want to discuss the notion of stability and present recent progress in the field.

Sofic approximations — what’s the problem?

I am planning to give a general introduction to sofic groups, mention a few applications to fundamental conjectures about groups and group rings, and explain Misha Gromov’s conjecture that all groups are sofic. Finally I want to discuss the notion of stability and present recent progress in the field.

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

Dominic Verdon (University of Bristol)

Frobenius algebras, fibre functors and entanglement-assisted transformations of covariant channels

I will first recall that finite-dimensional $C^*$-algebras and completely positive trace preserving maps precisely correspond to special symmetric Frobenius algebras and counit-preserving CP morphisms in the category of finite-dimensional Hilbert spaces and linear maps. I will extend this result to a Frobenius-algebraic characterisation of $G$-$C^*$-algebras and covariant channels for a compact quantum group $G$, using Tannaka-Krein-Woronowicz duality for fibre functors on the rigid $C^*$-tensor category Rep($G$). I will then introduce unitary pseudonatural transformations (UPTs) between fibre functors, a higher-dimensional generalisation of unitary monoidal natural transformations, and show how these UPTs give rise to reversible entanglement-assisted transformations of covariant classical and quantum channels generalising quantum teleportation and dense coding.

Frobenius algebras, fibre functors and entanglement-assisted transformations of covariant channels

I will first recall that finite-dimensional $C^*$-algebras and completely positive trace preserving maps precisely correspond to special symmetric Frobenius algebras and counit-preserving CP morphisms in the category of finite-dimensional Hilbert spaces and linear maps. I will extend this result to a Frobenius-algebraic characterisation of $G$-$C^*$-algebras and covariant channels for a compact quantum group $G$, using Tannaka-Krein-Woronowicz duality for fibre functors on the rigid $C^*$-tensor category Rep($G$). I will then introduce unitary pseudonatural transformations (UPTs) between fibre functors, a higher-dimensional generalisation of unitary monoidal natural transformations, and show how these UPTs give rise to reversible entanglement-assisted transformations of covariant classical and quantum channels generalising quantum teleportation and dense coding.

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

Karl-Hermann Neeb (FAU Erlangen-Nürnberg)

A representation theoretic perspective on modular theory

Modular theory is an important aspect of the theory of operator algebras and in the theory of local observables in Algebraic Quantum Field Theory (AQFT). It creates a one-parameter group of modular automorphisms from a single state and, sometimes, this group represents the flow of time (the dynamics) in a space-time domain. We study this question from a representation theoretic perspective by asking questions like: Which one-parameter groups of Lie groups can arise in this context as modular groups? This leads us to real standard subspaces of a complex Hilbert space, to antiunitary representations and to nets of standard subspaces and operator algebras on causal symmetric spaces.

This is joint work with Gestur Olafsson (Baton Rouge) and Vincenzo Morinelli (Rome)

A representation theoretic perspective on modular theory

Modular theory is an important aspect of the theory of operator algebras and in the theory of local observables in Algebraic Quantum Field Theory (AQFT). It creates a one-parameter group of modular automorphisms from a single state and, sometimes, this group represents the flow of time (the dynamics) in a space-time domain. We study this question from a representation theoretic perspective by asking questions like: Which one-parameter groups of Lie groups can arise in this context as modular groups? This leads us to real standard subspaces of a complex Hilbert space, to antiunitary representations and to nets of standard subspaces and operator algebras on causal symmetric spaces.

This is joint work with Gestur Olafsson (Baton Rouge) and Vincenzo Morinelli (Rome)

**Thursday, 04th March 2021 15:10 - 16:00 Zoom**

Sara Azzali (Universität Hamburg) -

Discrete group actions and Baum-Connes correspondence for the pure braid group

The Baum-Connes conjecture predicts an isomorphism between two objects associated with a discrete countable group G. The first one is topological in nature and involves a classifying space for proper G-actions; the second one is analytic and involves the $K$-theory of the $C^*$-algebra of G.

We introduce the subject and give some examples of explicit computations, in particular for certain braid groups. For this class of groups the conjecture is known to be true 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.

**talk postponed**Discrete group actions and Baum-Connes correspondence for the pure braid group

The Baum-Connes conjecture predicts an isomorphism between two objects associated with a discrete countable group G. The first one is topological in nature and involves a classifying space for proper G-actions; the second one is analytic and involves the $K$-theory of the $C^*$-algebra of G.

We introduce the subject and give some examples of explicit computations, in particular for certain braid groups. For this class of groups the conjecture is known to be true 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, 11th March 2021 15:10 - 16:00 Zoom**

Jose Perea (Michigan State University)

Quasiperiodicy in data - an applied topology view

The analysis of time-varying systems using topological methods has gained considerable attention over the last few years. This talk will be about quasiperiodic recurrence in time series data; i.e., the superposition of periodic oscillators with non-commensurate frequencies. The sliding window (or time delay) embeddings of such functions can be shown to be dense in high-dimensional tori, and we will discuss techniques to study the persistent homology of such sets. Along the way, we will present a recent Künneth theorem for persistent homology, as well as several applications to data science and engineering.

Quasiperiodicy in data - an applied topology view

The analysis of time-varying systems using topological methods has gained considerable attention over the last few years. This talk will be about quasiperiodic recurrence in time series data; i.e., the superposition of periodic oscillators with non-commensurate frequencies. The sliding window (or time delay) embeddings of such functions can be shown to be dense in high-dimensional tori, and we will discuss techniques to study the persistent homology of such sets. Along the way, we will present a recent Künneth theorem for persistent homology, as well as several applications to data science and engineering.

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

Florencia Orosz Hunziker (University of Colorado)

Tensor categories for non-rational Virasoro vertex algebras

The minimal models are representations of the Virasoro algebra for particular central charges $c_{p,q}$. For these central charges the category of representations has finitely many irreducibles which give rise to a rational conformal field theory. In this talk, we focus on the non rational central charges where the number of irreducible representations is infinite. In particular, we prove that there is a braided tensor category structure on a natural subcategory of representations of the Virasoro algebra for arbitrary central charge. This talk is based on joint work with Thomas Creutzig, Cuibo Jiang, David Ridout and Jinwei Yang.

Tensor categories for non-rational Virasoro vertex algebras

The minimal models are representations of the Virasoro algebra for particular central charges $c_{p,q}$. For these central charges the category of representations has finitely many irreducibles which give rise to a rational conformal field theory. In this talk, we focus on the non rational central charges where the number of irreducible representations is infinite. In particular, we prove that there is a braided tensor category structure on a natural subcategory of representations of the Virasoro algebra for arbitrary central charge. This talk is based on joint work with Thomas Creutzig, Cuibo Jiang, David Ridout and Jinwei Yang.

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

Roberto Conti (Sapienza Università di Roma)

Old and news about Cuntz algebras

The Cuntz algebras form a prominent family of C*-algebras which, for one reason or another, have contributed to shape many aspects of recent research activity, both in the general area of classification of C*-algebras and group actions as well as in Quantum Field Theory, where they played a central role in the development of abstract forms of Tannaka-Krein dualities.

In the talk I’ll focus on some specific topics taken from a long-term research project aimed at understanding as much as possible about their automorphisms (which might be considered as a very dangerous but fascinating trip into a microcosm where algebra, analysis, combinatorics, dynamics, geometry etc are all tied up together in various ways).

Old and news about Cuntz algebras

The Cuntz algebras form a prominent family of C*-algebras which, for one reason or another, have contributed to shape many aspects of recent research activity, both in the general area of classification of C*-algebras and group actions as well as in Quantum Field Theory, where they played a central role in the development of abstract forms of Tannaka-Krein dualities.

In the talk I’ll focus on some specific topics taken from a long-term research project aimed at understanding as much as possible about their automorphisms (which might be considered as a very dangerous but fascinating trip into a microcosm where algebra, analysis, combinatorics, dynamics, geometry etc are all tied up together in various ways).

**Thursday, 22nd April 2021 15:10 - 16:00 Zoom**

Gandalf Lechner, Kirstin Strokorb, Simon Wood (thankfully replacing Ulrich Pennig at short notice, all Cardiff University)

Introduction to Online Tools for Mathematicians

This meeting will be a little different: Since the GAPT group has grown a bit in recent times, I thought it might be a good idea to give our new PhD students and all other interested participants an overview of the online tools that mathematicians use in their daily life, such as the arXiv, LaTeX, mathscinet, Google Scholar, computer algebra systems, etc. There is a lot out there, and I will only be able to cover everything on my list briefly, but the hope is that in the end you will have a good overview about the things that are available.

List of Online Tools for Mathematicians

Introduction to Online Tools for Mathematicians

This meeting will be a little different: Since the GAPT group has grown a bit in recent times, I thought it might be a good idea to give our new PhD students and all other interested participants an overview of the online tools that mathematicians use in their daily life, such as the arXiv, LaTeX, mathscinet, Google Scholar, computer algebra systems, etc. There is a lot out there, and I will only be able to cover everything on my list briefly, but the hope is that in the end you will have a good overview about the things that are available.

List of Online Tools for Mathematicians

**Thursday, 29th April 2021 15:10 - 16:00 Zoom**

Thomas Creutzig (University of Alberta)

Vertex Tensor Categories

Categories of modules of a vertex operator algebra satisfying certain conditions are vertex tensor categories, in particular they are braided tensor categories. If these categories are neither semisimple nor finite, then it is particularly challenging to understand them. I would like to give an overview of this topic, explain its relevance and recent progress.

Vertex Tensor Categories

Categories of modules of a vertex operator algebra satisfying certain conditions are vertex tensor categories, in particular they are braided tensor categories. If these categories are neither semisimple nor finite, then it is particularly challenging to understand them. I would like to give an overview of this topic, explain its relevance and recent progress.

**Thursday, 13th May 2021 15:10 - 16:00 Zoom**

Simon Lentner (Universität Hamburg)

Nichols Algebras, Selberg Integrals and Screening Operators

Nichols Algebras are certain universal algebras associated to braidings. They appear as Borel part of quantum groups, and more generally of pointed Hopf algebras. Local resp. nonlocal screening operators appear in conformal field theory, and their composition is governed by contour integrals over singlevalued resp. multivalued analytic functions, which generalize Selberg integrals

In my talk I give an introduction to Nichols algebras, and then I explain my recent result that the relations in a Nichols algebra predict zeroes in the Selberg integral. As main application, this implies that nonlocal screening operators are described by Nichols algebras, just as local screening operators are usually described by Lie algebras. I will not go deep into this, but I will try to motivate conformal field theory and its connection to quantum groups in an illustrative example.

Nichols Algebras, Selberg Integrals and Screening Operators

Nichols Algebras are certain universal algebras associated to braidings. They appear as Borel part of quantum groups, and more generally of pointed Hopf algebras. Local resp. nonlocal screening operators appear in conformal field theory, and their composition is governed by contour integrals over singlevalued resp. multivalued analytic functions, which generalize Selberg integrals

In my talk I give an introduction to Nichols algebras, and then I explain my recent result that the relations in a Nichols algebra predict zeroes in the Selberg integral. As main application, this implies that nonlocal screening operators are described by Nichols algebras, just as local screening operators are usually described by Lie algebras. I will not go deep into this, but I will try to motivate conformal field theory and its connection to quantum groups in an illustrative example.