**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.

**Seminar Talks - Spring 2022**

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

Nikon Kurnosov (UCL)

Geometry of non-Kähler holomorphically symplectic manifolds

Irreducible homomorphically symplectic (IHS) manifolds are building blocks for Calabi-Yau manifolds by the Beauville-Bogomolov decomposition theorem. From differential geometric point of view they are complex manifolds which possess three Kähler complex structures and holomorphic simplectic form. For algebraic geometers they are higher-dimensional analogs of K3-surface. Todorov conjectured that there no non-Kähler simply-connected analogs of IHS. Later, Bogomolov and Guan have constructed a counterexample to this conjecture. Bogomolov's construction emphasizes the analogy of K3 surface with the Kodaira surface. In this talk we will discuss properties of such manifolds, many of those are the same as for hyperkähler manifolds.

The talk is based on joint works with F.Bogomolov, A.Kuznetsova, M.Verbitsky and E.Yasinsky.

Geometry of non-Kähler holomorphically symplectic manifolds

Irreducible homomorphically symplectic (IHS) manifolds are building blocks for Calabi-Yau manifolds by the Beauville-Bogomolov decomposition theorem. From differential geometric point of view they are complex manifolds which possess three Kähler complex structures and holomorphic simplectic form. For algebraic geometers they are higher-dimensional analogs of K3-surface. Todorov conjectured that there no non-Kähler simply-connected analogs of IHS. Later, Bogomolov and Guan have constructed a counterexample to this conjecture. Bogomolov's construction emphasizes the analogy of K3 surface with the Kodaira surface. In this talk we will discuss properties of such manifolds, many of those are the same as for hyperkähler manifolds.

The talk is based on joint works with F.Bogomolov, A.Kuznetsova, M.Verbitsky and E.Yasinsky.

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

Magdalena Musat (Københavns Universitet)

On the convex structure of unital quantum channels, factorizability and the Connes Embedding Problem

Factorizable quantum channels, introduced by C. Anantharaman-Delaroche within the framework of operator algebras, have recently found important applications in the analysis of quantum information theory, revealing new infinite dimensional phenomena, and leading also to reformulations of the celebrated Connes Embedding Problem. In recent work with M. Rørdam, we show that (infinite dimensional) von Neumann algebras are, indeed, needed to describe such channels. Along the way, I will explain some of the several facets of the Connes Embedding Problem, with particular emphasis on the related interplay between operator algebras and quantum information theory.

On the convex structure of unital quantum channels, factorizability and the Connes Embedding Problem

Factorizable quantum channels, introduced by C. Anantharaman-Delaroche within the framework of operator algebras, have recently found important applications in the analysis of quantum information theory, revealing new infinite dimensional phenomena, and leading also to reformulations of the celebrated Connes Embedding Problem. In recent work with M. Rørdam, we show that (infinite dimensional) von Neumann algebras are, indeed, needed to describe such channels. Along the way, I will explain some of the several facets of the Connes Embedding Problem, with particular emphasis on the related interplay between operator algebras and quantum information theory.

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

Christian Hagendorf (Université catholique de Louvain)

The supersymmetric eight-vertex model

The topic of this talk is the eight-vertex model on the square lattice. We consider this model at its so-called "supersymmetric" or "combinatorial point", where it presents surprising links to supersymmetry, enumerative combinatorics and special functions. To illustrate these links, we discuss the model's transfer matrix. Its largest eigenvalue has a particularly simple form in the case of an odd number of vertical lines and periodic boundary conditions along the horizontal direction. We show that several properties of the corresponding eigenvectors can be expressed in terms of special polynomials introduced by Rosengren, Zinn-Justin, and recently related to the enumeration of three-colourings of the square lattice by Hietala.

This talk is based on joint work with Jean Liénardy and Sandrine Brasseur and ongoing work with Hjalmar Rosengren.

The supersymmetric eight-vertex model

The topic of this talk is the eight-vertex model on the square lattice. We consider this model at its so-called "supersymmetric" or "combinatorial point", where it presents surprising links to supersymmetry, enumerative combinatorics and special functions. To illustrate these links, we discuss the model's transfer matrix. Its largest eigenvalue has a particularly simple form in the case of an odd number of vertical lines and periodic boundary conditions along the horizontal direction. We show that several properties of the corresponding eigenvectors can be expressed in terms of special polynomials introduced by Rosengren, Zinn-Justin, and recently related to the enumeration of three-colourings of the square lattice by Hietala.

This talk is based on joint work with Jean Liénardy and Sandrine Brasseur and ongoing work with Hjalmar Rosengren.

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

Severin Bunk (University of Oxford)

Higher symmetries of gerbes

Gerbes are geometric objects describing the third integer cohomology group of a manifold and the $B$-field in string theory. Like line bundles, they admit connections and gauge symmetries. In contrast to line bundles, however, there are now isomorphisms between gauge symmetries: the gauge group of a gerbe is a smooth 2-group. Starting from a hands-on example, I will explain gerbes and some of their properties. The main topic of this talk will then be the study of symmetries of gerbes on a manifold with $G$-action, and how these symmetries assemble into smooth 2-group extensions of $G$. In the last part, I will survey how this construction can be used to provide a new smooth model for the string group, via a theory of $\infty$-categorical principal bundles and group extensions.

Higher symmetries of gerbes

Gerbes are geometric objects describing the third integer cohomology group of a manifold and the $B$-field in string theory. Like line bundles, they admit connections and gauge symmetries. In contrast to line bundles, however, there are now isomorphisms between gauge symmetries: the gauge group of a gerbe is a smooth 2-group. Starting from a hands-on example, I will explain gerbes and some of their properties. The main topic of this talk will then be the study of symmetries of gerbes on a manifold with $G$-action, and how these symmetries assemble into smooth 2-group extensions of $G$. In the last part, I will survey how this construction can be used to provide a new smooth model for the string group, via a theory of $\infty$-categorical principal bundles and group extensions.

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

Anne-Laure Thiel (University of Caen)

A Soergel category for cyclic groups

The category of Soergel bimodules plays an essential role in (higher) representation theory, to construct categorical braid group actions and homological invariants of knots. After having briefly recalled the definition and some facts about this category, the aim of this talk will be to present some of its generalizations. I will focus on a Soergel-like category attached to a cyclic group. I will give a complete description of this category through a classification of its indecomposable objects and study its split Grothendieck ring. This gives rise to an algebra which is an extension of the Hecke algebra of the cyclic group, that can be presented by generators and relations. If time permits, I will mention some partial results about a diagrammatic description of this category: how Catalan numbers appear in this context and how the Temperley-Lieb algebra can describe certain morphism spaces in this category. This is joint work with Thomas Gobet.

A Soergel category for cyclic groups

The category of Soergel bimodules plays an essential role in (higher) representation theory, to construct categorical braid group actions and homological invariants of knots. After having briefly recalled the definition and some facts about this category, the aim of this talk will be to present some of its generalizations. I will focus on a Soergel-like category attached to a cyclic group. I will give a complete description of this category through a classification of its indecomposable objects and study its split Grothendieck ring. This gives rise to an algebra which is an extension of the Hecke algebra of the cyclic group, that can be presented by generators and relations. If time permits, I will mention some partial results about a diagrammatic description of this category: how Catalan numbers appear in this context and how the Temperley-Lieb algebra can describe certain morphism spaces in this category. This is joint work with Thomas Gobet.

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

Luca Pol (Universität Regensburg)

Torsion model for tensor-triangulated categories

In this talk I will discuss how one can construct a model for (sufficiently well-behaved) tensor triangulated categories built from the data of local and torsion objects. The idea is to mirror constructions in commutative algebra such as torsion and localization at prime ideals. I will then discuss two interesting examples arising from algebra and equivariant stable homotopy theory. This is joint work with S. Balchin, J.P.C. Greenlees and J. Williamson.

Torsion model for tensor-triangulated categories

In this talk I will discuss how one can construct a model for (sufficiently well-behaved) tensor triangulated categories built from the data of local and torsion objects. The idea is to mirror constructions in commutative algebra such as torsion and localization at prime ideals. I will then discuss two interesting examples arising from algebra and equivariant stable homotopy theory. This is joint work with S. Balchin, J.P.C. Greenlees and J. Williamson.

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

Emily Norton (University of Kent)

Calibrated representations of cyclotomic Hecke algebras at roots of unity

The cyclotomic Hecke algebra is a "higher level" version of the Iwahori-Hecke algebra of the symmetric group. It depends on a collection of parameters, and its combinatorics involves multipartitions instead of partitions. We are interested in the case when the parameters are roots of unity. In general, we cannot hope for closed-form character formulas of the irreducible representations. However, a certain type of representation called "calibrated" is more tractable: those representations on which the Jucys-Murphy elements act semisimply. We classify the calibrated representations in terms of their Young diagrams, give a multiplicity-free formula for their characters, and homologically construct them via BGG resolutions. This is joint work with Chris Bowman and José Simental.

Calibrated representations of cyclotomic Hecke algebras at roots of unity

The cyclotomic Hecke algebra is a "higher level" version of the Iwahori-Hecke algebra of the symmetric group. It depends on a collection of parameters, and its combinatorics involves multipartitions instead of partitions. We are interested in the case when the parameters are roots of unity. In general, we cannot hope for closed-form character formulas of the irreducible representations. However, a certain type of representation called "calibrated" is more tractable: those representations on which the Jucys-Murphy elements act semisimply. We classify the calibrated representations in terms of their Young diagrams, give a multiplicity-free formula for their characters, and homologically construct them via BGG resolutions. This is joint work with Chris Bowman and José Simental.

**Thursday, 31st March 2022 15:10 - 16:00**

Ramona Wolf (ETH Zürich)

From subfactors to conformal field theories via lattice models

There is a long-standing conjecture that to every subfactor, there exists a corresponding conformal field theory. Although a considerable amount of evidence has been collected to support this conjecture, the general statement has not yet been proven or disproven. In the study of this correspondence, it might be helpful to lend methods from physics: Since every subfactors gives rise to two (unitary) fusion categories, one can imagine attempting to construct counterpart CFTs via lattice models built directly from these categories. In this talk, I will explain how this construction works and discuss problems and challenges that occur when carrying it out for relevant examples.

From subfactors to conformal field theories via lattice models

There is a long-standing conjecture that to every subfactor, there exists a corresponding conformal field theory. Although a considerable amount of evidence has been collected to support this conjecture, the general statement has not yet been proven or disproven. In the study of this correspondence, it might be helpful to lend methods from physics: Since every subfactors gives rise to two (unitary) fusion categories, one can imagine attempting to construct counterpart CFTs via lattice models built directly from these categories. In this talk, I will explain how this construction works and discuss problems and challenges that occur when carrying it out for relevant examples.

**Thursday, 28th April 2022 15:10 - 16:00**

Matina Trachana (Cardiff University)

On a solution of the multidimensional truncated matrix-valued moment problem

We consider the truncated multidimensional matrix-valued moment problem. We present necessary and sufficient conditions for a given truncated $p \times p$ Hermitian matrix-valued multisequence to have a minimal positive semidefinite matrix-valued representing measure. The support of the representing measure is also described. Moreover, we shall discuss the bivariate quadratic matrix-valued moment problem. The talk is based on joint work with David Kimsey.

On a solution of the multidimensional truncated matrix-valued moment problem

We consider the truncated multidimensional matrix-valued moment problem. We present necessary and sufficient conditions for a given truncated $p \times p$ Hermitian matrix-valued multisequence to have a minimal positive semidefinite matrix-valued representing measure. The support of the representing measure is also described. Moreover, we shall discuss the bivariate quadratic matrix-valued moment problem. The talk is based on joint work with David Kimsey.

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

Anne-Sophie Kaloghiros (Brunel University London)

1-dimensional K-moduli spaces of Fano 3-folds

Recent advances in the study of K-stability have shown that there is a projective good moduli space of K-polystable Q-Gorenstein smoothable Q-Fano varieties of dimension $n$ and volume $V$.

The classification of smooth Fano 3-folds is due to Iskovshikh, Mori and Mukai and dates back to the 80s. While the classification is non-modular, it contains rich information on the geometry of Fano 3-folds. Fano 3-folds offer a rich source of examples in which we can explicitly construct and understand some K-moduli spaces.

In this talk, I will discuss some examples of 1-dimensional K-moduli spaces of Fano 3-folds.

This is joint work with Abban, Cheltsov, Jiao, Martinez-Garcia, Papazachariou and Sarikyan.

1-dimensional K-moduli spaces of Fano 3-folds

Recent advances in the study of K-stability have shown that there is a projective good moduli space of K-polystable Q-Gorenstein smoothable Q-Fano varieties of dimension $n$ and volume $V$.

The classification of smooth Fano 3-folds is due to Iskovshikh, Mori and Mukai and dates back to the 80s. While the classification is non-modular, it contains rich information on the geometry of Fano 3-folds. Fano 3-folds offer a rich source of examples in which we can explicitly construct and understand some K-moduli spaces.

In this talk, I will discuss some examples of 1-dimensional K-moduli spaces of Fano 3-folds.

This is joint work with Abban, Cheltsov, Jiao, Martinez-Garcia, Papazachariou and Sarikyan.

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

Simona Paoli (University of Aberdeen)

From Homotopy Theory to Higher categories.

Topological spaces can be studied by breaking them into building blocks, called $n$-types, using a classical construction in homotopy theory, the Postnikov decomposition. The desire to model algebraically the building blocks of spaces was one of the motivations for the development of higher groupoids, generalizing the fundamental groupoid of a space. In this talk I will first illustrate how this naturally leads to the need to encode weakly associative and weakly unital compositions of higher morphisms in a higher groupoid and I will discuss the challenges that this poses.

More generally, structures arising in mathematical physics, namely topological quantum field theories, call for the need to define a notion of higher category, in which higher morphisms are not necessarily invertible.

The precise formalization of the notions of higher groupoids and higher categories can be achieved through several combinatorial machineries. I will introduce one of the approaches arising from homotopy theory, based on the notion of multisimplicial sets. I will finally briefly discuss why this approach is promising in terms of proving a long-standing open conjecture in higher category theory.

From Homotopy Theory to Higher categories.

Topological spaces can be studied by breaking them into building blocks, called $n$-types, using a classical construction in homotopy theory, the Postnikov decomposition. The desire to model algebraically the building blocks of spaces was one of the motivations for the development of higher groupoids, generalizing the fundamental groupoid of a space. In this talk I will first illustrate how this naturally leads to the need to encode weakly associative and weakly unital compositions of higher morphisms in a higher groupoid and I will discuss the challenges that this poses.

More generally, structures arising in mathematical physics, namely topological quantum field theories, call for the need to define a notion of higher category, in which higher morphisms are not necessarily invertible.

The precise formalization of the notions of higher groupoids and higher categories can be achieved through several combinatorial machineries. I will introduce one of the approaches arising from homotopy theory, based on the notion of multisimplicial sets. I will finally briefly discuss why this approach is promising in terms of proving a long-standing open conjecture in higher category theory.