Séminaires de topologie algébrique


The seminar takes place usually on Thursday, 11h00 in B203.

Current organisers: Fathi Ben Aribi and Abel Lacabanne

Past organisers: Miradain Atontsa Nguemo and Elia Rizzo (2018-2019), Pedro Vaz (2016-17), Federico Cantero Morán (2015-16), Pedro Boavida de Brito and Paul Arnaud Tsopméné (2014 - 2015), Urtzi Buijs Martín (2013 - 2014).

Past events: https://perso.uclouvain.be/pedro.vaz/seminar.html


Fev 13




Eiichi Piguet-Nakazawa

(Université de Genève)

Traces of operators in quantum Teichmüller theory

Nov 14




Miradain Atontsa Nguemo


Goodwillie Calculus: Characterization of polynomial functors

Oct 11



Cycl 04

Fathi Ben Aribi


The Teichmüller TQFT volume conjecture for twist knots


July 04



Cyclo 03

Najib Idrissi

Université Paris Diderot

Homologie de factorisation et espaces de configuration

March 20



Cyclo 03

Cristina Ana-Maria Palmer-Anghel

University of Oxford

Topological models for Uq(sl2)-quantum invariants

March 13



Cyclo 03

Krzysztof Putyra

University of Zurich

An equivalence between gl(2)-foams and cobordisms

November 22



Cyclo B203-5

Federico Cantero Morán  


Low-dimensional spatial refinements of Khovanov functors

October 19



Cyclo B335

Yves Félix


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

October 11



Cyclo B335

Miradain Atontsa Nguemo


Algebraic characterisation of homogeneous functors

October 04



Cyclo 07

David Méndez

(University of Malaga)

Realising groups in arrow categories of graphs and spaces



Abstracts 2019/20

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.

Abstracts 2018/19

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.