Séminaires de topologie algébrique

Louvain-La-Neuve

The seminar usually takes place on thursday, 11h00.

Current organisers: Miradain Atontsa Nguemo and Elia Rizzo

Past organisers: 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

 

2018/19

Oct 04        

2018

11h00-12h00   

Cyclo 07

David Méndez 

(University of Malaga)

Realising groups in arrow categories of graphs and spaces

Oct 11  

2018

11h00-12h00  

Cyclo B335

Miradain Atontsa Nguemo

UCL

Algebraic characterisation of homogeneous functors

Oct 19  

2018

11h00-12h00   

Cyclo B335

Yves Félix 

UCL

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

Nov 22  

2018

11h00-12h00   

Cyclo B203-5

Federico Cantero Morán  

BGSMath

  Low-dimensional spatial refinements of Khovanov functors

March 13

2019

11h00-12h00

Cyclo 03

Krzysztof Putyra

University of Zurich

An equivalence between gl(2)-foams and cobordisms

March 20

2019

11h00-12h00

Cristina Ana-Maria Palmer-Anghel

University of Oxford

Topological models for Uq(sl2)-quantum invariants

 

 

Abstracts 2018/19

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.
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.
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)
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.
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.
Cristina Ana-Maria Palmer-Anghel, Univesrity of Oxford, march 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.