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 in room M/0.34. Speakers give 50 minute talks. Past Events: |

**Upcoming Seminar Talks**

**Thursday, 25th June**

**10:00 - 11:00 (UK time) Zoom**

David Ridout (University of Melbourne) -

A new approach to W-algebras

W-algebras are a class of vertex algebras that find many applications in both mathematics and mathematical physics. While some classes of examples are fairly well-understood, they are still quite mysterious. Here, I want to review some of what's known and then sketch (in a not-too-technical fashion) a promising new approach to W-algebras and their representations based on old work of Semikhatov and recent work of Adamović.

**moved online - please contact me to get the Zoom password**A new approach to W-algebras

W-algebras are a class of vertex algebras that find many applications in both mathematics and mathematical physics. While some classes of examples are fairly well-understood, they are still quite mysterious. Here, I want to review some of what's known and then sketch (in a not-too-technical fashion) a promising new approach to W-algebras and their representations based on old work of Semikhatov and recent work of Adamović.

**Thursday, 4th June 2020 15:10 - 16:00 Zoom**

Owen Tanner (Cardiff University - Online Project Viva) -

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.

**moved online - please contact me to get the Zoom password**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.

**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)

**Seminar Talks - Autumn 2019**

**Thursday, 3rd October 15:10 - 16:00 M/0.34**

Victor Przyalkowski (Steklov/HSE, Moscow)

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.

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.

**Thursday, 10th October 15:10 - 16:00 M/0.34**

Ian Short (The Open University)

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.

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.

**Thursday, 17th October 15:10 - 16:00 M/0.34**

Joan Bosa (Universitat Autònoma de Barcelona)

Classification of separable nuclear unital simple C*-algebras. History and final results.

Over the last decade, our understanding of simple, nuclear $C^*$-algebras has improved a lot. This is thanks to the the interplay between certain topological and algebraic regularity properties, such as nuclear dimension of $C^*$-algebras, tensorial absorption of suitable strongly self-absorbing $C^*$-algebras and order completeness of homological invariants. In particular, this is reﬂected in the Toms-Winter conjecture. In this talk I will speak about this problem, and explain the general classification of nuclear simple $C^*$-algebras using the ﬁnite nuclear dimension (done in two groundbreaking articles by Elliott-Gong-Lin-Niu and Tikuisis-White-Winter). If time permits, I will also show some research built up from the classification just explained.

Classification of separable nuclear unital simple C*-algebras. History and final results.

Over the last decade, our understanding of simple, nuclear $C^*$-algebras has improved a lot. This is thanks to the the interplay between certain topological and algebraic regularity properties, such as nuclear dimension of $C^*$-algebras, tensorial absorption of suitable strongly self-absorbing $C^*$-algebras and order completeness of homological invariants. In particular, this is reﬂected in the Toms-Winter conjecture. In this talk I will speak about this problem, and explain the general classification of nuclear simple $C^*$-algebras using the ﬁnite nuclear dimension (done in two groundbreaking articles by Elliott-Gong-Lin-Niu and Tikuisis-White-Winter). If time permits, I will also show some research built up from the classification just explained.

**Thursday, 24th October 15:10 - 16:00 M/0.34**

Ko Sanders (Dublin City University)

Killing fields and KMS states in curved spacetimes

In quantum physics, every thermal equilibrium state satisfies the KMS condition. This condition is formulated in terms of the evolution in time. In general relativity, however, there is no preferred time flow on spacetime, but there can be several natural choices of a flow, given by Killing vector fields. Physically relevant examples arise especially in the context of black hole spacetimes, where the Killing fields are often timelike only in some region of spacetime, but they are spacelike or lightlike elsewhere.

In this talk, based on work with Pinamonti and Verch, I will address the question for which Killing fields the existence of KMS states can be ruled out, because the KMS condition forces the two-point distributions of such states to be physically ill-behaved.

Killing fields and KMS states in curved spacetimes

In quantum physics, every thermal equilibrium state satisfies the KMS condition. This condition is formulated in terms of the evolution in time. In general relativity, however, there is no preferred time flow on spacetime, but there can be several natural choices of a flow, given by Killing vector fields. Physically relevant examples arise especially in the context of black hole spacetimes, where the Killing fields are often timelike only in some region of spacetime, but they are spacelike or lightlike elsewhere.

In this talk, based on work with Pinamonti and Verch, I will address the question for which Killing fields the existence of KMS states can be ruled out, because the KMS condition forces the two-point distributions of such states to be physically ill-behaved.

**Thursday, 7th November 15:10 - 16:00 M/0.34**

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.

Based on (past and ongoing) joint work with G. Landi and J. Kaad.

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.

Based on (past and ongoing) joint work with G. Landi and J. Kaad.

**Thursday, 14th November 15:10 - 16:00 M/0.34**

Andreas Fring (City University of London)

Nonlocal gauge equivalent integrable systems

We demonstrate how new integrable nonlocal systems in space and/or time can be constructed by exploiting certain parity transformations and/or time reversal transformations possibly combined with a complex conjugations. By employing Hirota's direct method as well as Darboux-Crum transformations we construct explicit multi-soliton solutions for nonlocal versions of Hirota's equation that exhibit new types of qualitative behaviour. We exploit the gauge equivalence between these equations and an extended version of the continuous limit of the Heisenberg equation to show how nonlocality is implemented in those latter systems and an extended version of the Landau-Lifschitz equation.

Nonlocal gauge equivalent integrable systems

We demonstrate how new integrable nonlocal systems in space and/or time can be constructed by exploiting certain parity transformations and/or time reversal transformations possibly combined with a complex conjugations. By employing Hirota's direct method as well as Darboux-Crum transformations we construct explicit multi-soliton solutions for nonlocal versions of Hirota's equation that exhibit new types of qualitative behaviour. We exploit the gauge equivalence between these equations and an extended version of the continuous limit of the Heisenberg equation to show how nonlocality is implemented in those latter systems and an extended version of the Landau-Lifschitz equation.

**Thursday, 21st November 15:10 - 16:00 M/0.34**

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.

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.

**Thursday, 28th November 15:10 - 16:00 M/0.34**

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.

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.

**Thursday, 5th December 15:10 - 16:00 M/0.34**

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.

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.

**Thursday, 12th December 15:10 - 16:00 M/0.34**

Iain Moffatt (Royal Holloway, University of London)

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.

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.

**Seminar Talks - Spring 2020**

**Thursday, 30th January 15:10 - 16:00 M/0.34**

Yue Ren (Swansea University)

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.

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.

**Thursday, 6th February 15:10 - 16:00 M/0.34**

Clemens Koppensteiner (University of Oxford)

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.

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.

**Thursday, 20th February 15:10 - 16:00 M/0.34**

Jelena Grbic (University of Southampton)

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 $\Delta$-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.

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 $\Delta$-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.

**Thursday, 27th February 15:10 - 16:00 M/0.34**

André Henriques (University of Oxford)

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.

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.

**Thursday, 5th March 15:10 - 16:00 M/0.34**

Taro Sogabe (Kyoto University)

A topological invariant for continuous fields of Cuntz algebras

Our interest is in locally trivial bundles of $C^*$-algebras including the Cuntz algebras and their topological invariants. In this talk, I would like to explain the $C^*$-algebras we are interested in, and introduce a topological invariant for bundles of Cuntz algebras which are elements of a generalised cohomology theory also considered by Dadarlat and Pennig.

A topological invariant for continuous fields of Cuntz algebras

Our interest is in locally trivial bundles of $C^*$-algebras including the Cuntz algebras and their topological invariants. In this talk, I would like to explain the $C^*$-algebras we are interested in, and introduce a topological invariant for bundles of Cuntz algebras which are elements of a generalised cohomology theory also considered by Dadarlat and Pennig.

**Thursday, 2nd April 15:10 - 16:00 Zoom**

Tomack Gilmore (UCL) -

Plane partitions, rhombus tilings, lattice paths, and perfect matchings (with some pretty pictures)

Plane partitions were first studied by Gen. Percy Alexander MacMahon at the beginning of the twentieth century and since then these objects have been studied by mathematicians and mathematical physicists alike.

In this talk I will discuss some lovely bijections between plane partitions, rhombus tilings, lattice paths, and perfect matchings, as well as some of the physical principles that appear to govern such tiling models.

**moved online - please contact me to get the Zoom password**Plane partitions, rhombus tilings, lattice paths, and perfect matchings (with some pretty pictures)

Plane partitions were first studied by Gen. Percy Alexander MacMahon at the beginning of the twentieth century and since then these objects have been studied by mathematicians and mathematical physicists alike.

In this talk I will discuss some lovely bijections between plane partitions, rhombus tilings, lattice paths, and perfect matchings, as well as some of the physical principles that appear to govern such tiling models.

**Thursday, 23rd April 15:10 - 16:00 Zoom**

Manjil P. Saikia (Cardiff University)

The Remarkable Sequence 1, 1, 2, 7, 42, 429, ... .

The sequence in the title counts several combinatorial objects, some of which I will describe in this talk. The major focus would be one of these objects, alternating sign matrices (ASMs). ASMs are square matrices with entries in the set {0,1,-1}, where non-zero entries alternate in sign along rows and columns, with all row and column sums being 1. I will discuss some questions that are central to the theme of ASMs, mainly dealing with their enumeration.

**- online - please contact me to get the Zoom password**The Remarkable Sequence 1, 1, 2, 7, 42, 429, ... .

The sequence in the title counts several combinatorial objects, some of which I will describe in this talk. The major focus would be one of these objects, alternating sign matrices (ASMs). ASMs are square matrices with entries in the set {0,1,-1}, where non-zero entries alternate in sign along rows and columns, with all row and column sums being 1. I will discuss some questions that are central to the theme of ASMs, mainly dealing with their enumeration.

**Thursday, 30th April 15:10 - 16:00 Zoom**

Thomas Schick (Georg-August Universität Göttingen)

The codimension 2 index obstruction to positive scalar curvature

We adress the following general question:

Given a (compact without boundary) manifold $M$, does $M$ admit a metric of positive scalar curvature.

Very classically, the Gauss-Bonnet theorem implies that among the (connected orientable compact) surfaces only the 2-sphere has this property. In higher dimensions, the most powerful information uses the Dirac operator and its index, and an old observation of Schroedinger ("Über das Diracsche Elektron im Schwerefeld") coupling scalar curvature to the latter.

We will quickly introduce classical and more modern ("higher") index theory approaches to this problem, and then discuss a special implementation: How and why certain submanifolds of codimension 2 act as a vaccine (or poison, depending on your point of view) and prevent the occurrence of positive scalar curvature metrics. Realistically, there won't be too much time to talk about that.

**- online - please contact me to get the Zoom password**The codimension 2 index obstruction to positive scalar curvature

We adress the following general question:

Given a (compact without boundary) manifold $M$, does $M$ admit a metric of positive scalar curvature.

Very classically, the Gauss-Bonnet theorem implies that among the (connected orientable compact) surfaces only the 2-sphere has this property. In higher dimensions, the most powerful information uses the Dirac operator and its index, and an old observation of Schroedinger ("Über das Diracsche Elektron im Schwerefeld") coupling scalar curvature to the latter.

We will quickly introduce classical and more modern ("higher") index theory approaches to this problem, and then discuss a special implementation: How and why certain submanifolds of codimension 2 act as a vaccine (or poison, depending on your point of view) and prevent the occurrence of positive scalar curvature metrics. Realistically, there won't be too much time to talk about that.