Archives Séminaire de topologie algébrique

Elia Rizzo (UCLouvain), June 8

Title: Categorification at a prime root of unity

The Jones polynomial is a topological invariant of links or tangles in the three-sphere that takes values in the ring Z[q, q^-1] and it can be obtained from the representation theory of the quantum group Uq (sl2) at a generic value of q, other than using its famous combinatorial construction.

Specializing q = ζ to a root of unity allows to transform the Jones polynomial into a 3-manifold invariant that takes values in the ring Z[ζ] and which still possesses a representation-theory interpretation that involves the quantum group Uζ (sl2) at a root of unity.
While for q generic the Jones polynomial and its representation-theory interpretation have been categorified, when q is a root of unity we are able to categorify only some forms of the quantum group and some of its representations. In particular, a categorification of the 3-manifold invariant is still missing.

The first step toward the categorification at a root of unity was given by Khovanov, who has been able to categorify the ground ring Z[ζ], when ζ is a prime root of unity, and introduced the framework of hopfological algebra. This hopfological algebra is a generalization of homological algebra which can be associated to finite dimensional hopf algebras and that gives us a family of categories we can use to categorify modules over Z[ζ], always for ζ a prime root of unity.
The goal of the first part of the talk is to give an introduction on hopfological algebra and to show how it is used to categorify the ring Z[ζ], for ζ a prime root of unity. In the second part we want to give an idea on how to categorify modules over Z[ζ] and give a flavour of the recent categorification of the Baby Verma module, that is, a specific finitely generated module over Uζ (sl2).

Mario Fuentes (University of Malaga), Mar 23

Title: Algebraic models for classifying spaces

Given a space X, we can associate it the monoid Aut(X) of self homotopy equivalences of X. Similarly, we also consider the monoid Aut*(X) of pointed homotopy equivalences. From these two spaces we get the fibration sequence X -> B Aut*(X) -> B Aut(X) which is universal: this means that any fibration with fiber X can be obtained from the universal one. Therefore we have a highly interesting object in topology. The goal of this talk is to study this object from the point of view of the Rational Homotopy Theory. I will present a classic result in this area and how it can be extended to a non-simply connected setting. The main tool for doing this are the recently developed Lie models and the use of complete Lie algebras of derivations.

Arthur Soulié (University of Glasgow), Jan 18

Title: Autour des représentations homologiques des groupes de tresses et des groupes de difféotopie

Je vais décrire des méthodes générales de constructions de représentations homologiques des groupes de tresses et des groupes de difféotopie. Elles généralisent notamment les constructions célèbres de Lawrence et Bigelow pour les groupes de tresses. Je présenterai également des résultats d'irréducibilité, de polynomialité et des propriétés sur les noyaux de ces représentations. Cet exposé traite d'un travail en collaboration avec Martin Palmer.

Cristina Anghel-Palmer (University of Geneva), Dec 7

Title : Uq(sl2)-quantum invariants via the intersection of two lagrangians in a symmetric power of a surface

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of invariants. In this context, the quantum group U q (sl(2)) leads to the sequence of coloured Jones polynomials, and the same quantum group at roots of unity gives the coloured Alexander polynomials.
We construct a unified topological model for these two sequences of quantum invariants. More specifically, we prove that the N th coloured Jones and N th coloured Alexander invariants are different specialisations of a state sum of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.
Secondly, we present a globalised model without state sums. We show that one can read off both coloured Jones and coloured Alexander polynomials of colour N from a graded intersection between two explicit Lagrangians in a symmetric power of the punctured disk.

Jules Martel (Université de Dijon), Nov 25

Titre : Représentations homologiques de groupes modulaires: applications à la topologie quantique

A chaque algèbre de Lie semi-simple, Drinfel'd a associé une algèbre "quantique" appelée groupe quantique. La théorie des modules sur ces algèbres a été largement utilisée (d'abord par Reshetikhin--Turaev) pour produire des invariants topologiques tels que : représentations de tresses, polynôme de Jones des nœuds ou encore les théories topologiques des champs quantiques (TQFTs) (qui produisent des invariants de 3-variétés, et des représentations de groupes modulaires de surfaces). Ces constructions reposent sur un bagage algébrique (les groupes quantiques) si bien que leur contenu topologique est souvent mystérieux et finalement le sujet de beaucoup de conjectures de la topologie quantique. Nous savons retrouver les modules (de Verma) des groupes quantiques à partir des homologies d'espaces de configurations, ce qui constitue un modèle homologique pour les représentations quantiques de tresses qui s'étend aux invariants de nœuds associés. Nous espérons également retrouver les ingrédients des TQFTs. Dans cet exposé, je présenterai en détails le cas sl(2) : les modules de Verma et leur tressage à partir des homologies d'espaces de configurations. Je dirai quelques mots sur la généralisation à toute algèbre de Lie (j.w. S. Bigelow). Si le temps le permet je discuterai les conséquences et les généralisations aux invariants quantiques tels que les invariants de Jones de nœuds ou les représentations quantiques de groupes modulaires de surfaces.

Geoffroy Horel (Université Paris 13), Nov 16

Titre : Tour de Goodwillie-Weiss algébrique et invariants de noeuds type fini

J'introduirai la tour de Goodwillie-Weiss algébrique pour les espaces de nœuds. Cette tour est un invariant de nœuds très puissant mais dont la construction est assez technique. Il a été montré par Volic que pour des noeuds dans R^3, cette tour est isomorphe à l'invariant de type fini universel au sens de Vassiliev. Dans un travail en commun avec Pedro Boavida de Brito et Danica Kosanovic, nous avons une conjecture précise sur la tour de Goodwillie-Weiss pour des noeuds dans une variété de dimension 3 quelconque qui généralise le résultat de Volic. Nous avons également une preuve de ce résultat dans le cas d'une variété de la forme (surface)xR. J'exposerai quelques unes des idées derrière cette conjecture et ce résultat.

Grégoire Naisse (MPIM Bonn), Nov 24

Title: A construction of odd Khovanov homology from classical algebraic topology

Slides

Teams

Khovanov homology is a well-studied link homology theory categorifying the Jones polynomial. There exists many constructions of it (higher rep. theory, algebraic geometry, ...), but it was originally combinatorially defined using a TQFT. It is well-known that cohomology of nice even-dimensional manifolds give Frobenius algebras, and thus TQFTs. In particular Khovanov homology TQFT comes from the complex projective line CP^1.
By constrast, odd Khovanov homology, a mysterious cousin of (even) Khovanov homology defined by Osvath, Rasmussen and Szabo also categorifying the Jones polynomial, is by far less understood. In particular, ORS construction is ad hoc. In a attempt to understand better its origin, we gave a construction of odd Khovanov homology based on the geometry of the *real* projective line RP^1.
My goal for this talk will be to explain to you how one gets a TQFT from the cohomology of a nice even-dimensional manifold, and how this can be generalized to an odd dimensional one, giving a chronological TQFT. For this, we will need to explore some classical results of algebraic topology (in the first part of the talk) and how graded monoidal categories work (in the second part).

No knowledge of link homology theory is needed for this talk.

This talk based on a joint work with Jens N. Eberhardt and Arik Wilbert.

Léo Schelstraete (UCLouvain), Oct 13

Title: Supercategorification and Odd Khovanov Homology

Slides

In 2000, Khovanov discovered a new invariant of knots, now called Khovanov homology. As the name suggests, it associates to any knot a certain graded homology. In some sense, this construction gives a categorification of the Jones polynomial: indeed, the Jones polynomial turns out to be the Euler characteristic of Khovanov homology. On the other hand, the Jones polynomial is a quantum invariant, as it can be derived through the representation theory of the quantum algebra Uq(sl2). Later works by Webster and Lauda, Queffelec and Rose showed that the two approaches can be unified: one can use a categorification of the representation theory of Uq(sl2) to derive Khovanov homology.

In 2013, Oszváth, Rasmussen and Szabó constructed a different version of Khovanov homology using exterior algebras, called odd Khovanov homology. This construction also categorifies the Jones polynomial, and over Z/2Z the resulting invariant coincides with (even) Khovanov homology. However, the two homologies are distinct. In March 2020, Naisse and Putyra extended odd Khovanov homology to tangles. At the time being though, no representation theory construction similar to the even case is known. Exploring this question is the goal of my current research.

The end goal of this talk is to introduce the invariant of knots developed in my master's thesis, defined thanks to a categorification of a quantum algebra and conjectured to coincide with odd Khovanov homology. This construction uses the categorical notion of 2-supercategories, hence the name supercategorification. The first part of the talk (~35 min) aims to introduce Khovanov homology and the idea of categorification and categorified quantum algebras, only giving hints toward my work. Prerequisites are at the master's level (notably a sense of what is the homology of a topological space and what is a category). The second part of the talk (~25 min) delves deeper into the construction of odd Khovanov homology through supercategorification.

Note that no acquaintance with quantum algebras is needed to follow any part of the talk.

Jacques Darné (UCLouvain), Jun 15

Title: Group filtrations and invariants: braids and generalizations

Slides

Milnor invariants of degree at most d distinguish braids up to elements of the (d+1)-th term of the lower central series of the pure braid group. The goal of this talk will be to present a proof of this classical result. This will involve talking about group filtrations and how they encode invariants such as finite-type invariants of braids and their generalizations. We will also discuss results and conjectures about similar statements in various contexts.

Eiichi Piguet-Nakazawa (Université de Genève), Feb 13

Title: Traces of operators in quantum Teichmüller theory

Quantum Teichmüller theory allowed to construct unitary projective representations of mapping class group of punctured surfaces in infinite dimension Hilbert spaces. Although the notion of trace of an unitary operator is, a priori, not well-defined, it is expected that the trace of the associated quantum operators provides invariants of mapping tori.

In the first part of the talk, we recall some basic notions of hyperbolic geometry such as ideal triangulations, angle structures and some results about these topics. Then, we will see a method (introduced by Floyd and Hatcher), with the example of the figure-eight knot complement, to construct ideal triangulations of once-punctured torus bundles using only the information of the monodromy.

In the second part of the talk, we explain the main tools of quantum Teichmüller theory and Teichmüller TQFT, and we will see that it is possible to give an interpretation of the trace, in the case of a pseudo-Anosov monodromy of the once-punctured torus, using the Teichmüller TQFT.

Miradain Atontsa Nguemo (UCLouvain), Nov 14

Title: Goodwillie Calculus: Characterization of polynomial functors

The calculus of functors can be seen as the categorification of the calculus of Newton and Leibniz. Namely, this consists of approximating a functor F:C -----> D with a sequence {PnF: C-------> D}n of "polynomial functors". As for Taylor series, PnF is expressed in terms of some "derivatives" d1F, ..., dnF.

Classically, Goodwillie and others have developed the theory when C and D are each either the category of topological spaces and the category of spectra. In this talk, I will extend this construction to the category of chain complexes and the category of differential graded Lie algebras (or more generally algebras over an arbitrary operad). I will then give a more explicit description of PnF.

Fathi Ben Aribi (UCLouvain), Oct 11

Title: The Teichmüller TQFT volume conjecture for twist knots

(joint work with E. Piguet-Nakazawa)

In 2014, Andersen and Kashaev defined an infinite-dimensional TQFT from quantum Teichmüller theory. This Teichmüller TQFT yields an invariant of triangulated 3-manifolds, in particular knot complements.

The associated volume conjecture states that the Teichmüller TQFT of an hyperbolic knot complement contains the volume of the knot as a certain asymptotical coefficient, and Andersen-Kashaev proved this conjecture for the first two hyperbolic knots.

In this talk I will present the construction of the Teichmüller TQFT and how we approached this volume conjecture for the infinite family of twist knots, by constructing new geometric triangulations of the knot complements.

In the first part I will introduce the main objects and state our results, and I will explain their proofs in more detail in the second part.

No prerequisites in Quantum Topology are needed.

Najib Idrissi, Université Paris Diderot, Jul 04

Title: Homologie de factorisation et espaces de configuration

L'homologie de factorisation est une théorie homologique pour les variétés structurées (orientées, parallélisées...) qui trouve ses origines dans les théories topologique et conformes des champs (Beilinson–Drinfeld, Salvatore, Lurie, Ayala–Francis, Costello–Gwilliam...). Après l'avoir définie et donné une idée de ses propriétés, j'expliquerai comment on peut la calculer sur ℝ grâce au modèle de Lambrechts--Stanley des espaces de configuration et/ou grâce à des complexes de graphes dans le cas des variétés fermées parallélisées, des variétés fermées orientées, et des variétés à bord parallélisées. [En partie en collaboration avec R. Campos, J. Ducoulombier, P. Lambrechts, T. Willwacher]

Cristina Ana-Maria Palmer-Anghel, Univesrity of Oxford, Mar 20

title: Topological models for Uq(sl2)-quantum invariants

The theory of quantum invariants for knots started with the Jones polynomial. After that, Reshetikhin and Turaev introduced an algebraic method which having as input a quantum group, leads to a link invariant. The representation theory of Uq(sl(2)) at generic q leads through this method to a family of link invariants {JN(L, q)}_{N∈\mathbb{N}} called coloured Jones polynomials. The first term of this sequence is the original Jones polynomial. On the other hand, the quantum group Uq(sl(2)) at roots of unity leads to a sequence of invariants, called coloured Alexander polynomials, having the original Alexander invariant as the first term. On the topological side, R. Lawrence introduced a sequence of braid group representations based on the homology of coverings of configurations spaces and using these, Bigelow and Lawrence gave a homological model for the original Jones polynomial. We present a topological model for all coloured Jones polynomials, showing that they can be described as graded intersection pairings between two homology classesin a covering of the configuration space in the punctured disc. If time permits, we will show some directions towards a topological description for Uq(sl(2)) quantum invariants at roots of unity.

Krzysztof Putyra (UZH), march 13

title: An equivalence between gl(2)-foams and cobordisms

The original construction of the Khovanov homology of a link can be seen as a formal complex in the category of flat tangles and surfaces between them. There is a way how to associate a chain map with a link cobordism, but only up to a sign. Blanchet has fixed this by introducing the category of gl(2)-foams, certain singular cobordisms between planar trivalent graphs. Originating in the representation theory of quantum groups, foams are usually thought as algebraic objects. In my talk I will bring topology back by interpreting foams as two surfaces transverse to each other. This description leads to a quick proof that gl(2)-foams can be evaluated, a construction of a natural basis of foams, and an explicit equivalence between the category of gl(2)-foams and cobordisms between flat tangles. This is a joint work with Anna Beliakova, Matthew Hogancamp and Stephan Wehrli.

Federico Cantero Morán (BGSMath), Nov 22

Title: Low-dimensional spatial refinements of Khovanov functors

Khovanov spectra of links were introduced in 2014 by Lipshitz and Sarkar. They are link invariants up to stable homotopy, and their cohomology is isomorphic to Khovanov homology. Their construction was later simplified in a joint work with Lawson. In this talk we will explain how to give a further simplification in the first non-trivial quantum gradings. This is joint work with Marithania Silvero.

Yves Félix (UCL), Oct 19

Title : The rational homotopy of mapping spaces - an introduction to the subject

We will begin with a description of the approaches of Haefliger and Brown-Szczarba, and ends with a presentation of Lie models in the spirit of Lazarev-Markl, Getzler and Berglund)

Miradain Atontsa Nguemo (UCL) , Oct 11

Title: Algebraic characterization of homogeneous functors

Goodwillie developped, in a serie of papers in 1990, 1992 and 2003, an approach to give polynomial approximations of functors F: Top ----> Top. In the Taylor tower that he has constructed:

F ... ----->P_nF -----> P_n-1 F ------> .... ------> P_1F

he was able to characterize explicitely the homotopy fibres D_nF of the maps in the tower. In this talk, I will give an overall idea of the Goodwillie approximation for functors F: C ----> D, when C and D are any of the model categories: "Chain complexes", "DGL=differential graded Lie algebras", "Alg_O=Algebras over the operad O". I will end by giving the characterization of homogeneous functors from C to D.

David Méndez (University of Malaga), Oct 04

Title: Realising groups in arrow categories of graphs and spaces

Whenever an algebraic structure arises in any mathematical context, it is natural to ask how to characterise the algebraic objects that may appear in that context. For instance, given any category, we may ask ourselves which groups can appear as automorphism groups of objects in that category. The Kahn realisation problem goes in that direction: it asks which groups may appear as the group of self-homotopy equivalences (that is, the automorphism group in HoTop) of a space X. A partial solution was attained by Costoya-Viruel in 2014, showing that every finite group arises as self-homotopy equivalences of a space (which is in fact rational). In this talk we consider the more general problem of realising groups in arrow categories of a given category C, Arr(C). In Arr(C), objects are arrows in C and morphisms between two arrows are commutative squares, thus given by two arrows in C. In this context, we ask: Given G1, G2 groups and H a subgroup of G1xG2, is there an arrow f:X1--->X2 in C such that Aut(Xi)=Gi and Aut(f)=H? We build a solution in the category of Graphs and show how to translate it to spaces, by means of algebraic models of rational homotopy types of spaces.