Tresses, noeuds et applications

Montpellier, du 9 au 11 juin 2008

Programme

Emploi du temps : download

Minicours :

• Michael Farber (Durham) : Configuration spaces and robot motion planning algorithms
  In my talks I will describe two topics linking topology of configuration spaces with computer science and engineering.
  Firstly, I will give an introduction to the theory of topological complexity of robot motion planning algorithms. A special emphasize will be made on the problem of navigation in projective spaces and in various braid spaces; the latter spaces are relevant to the task of simultaneous control of multiple objects avoiding collision.
  Secondly, I will discuss topology of configuration spaces of mechanical linkages. The results will include a complete classification of these manifolds in terms of their length vectors and a solution of Walker's conjecture. I will also describe a mixed probabilistic-topological approach to topology of linkages motivated by potential applications in molecular biology and statistics.
  
  
• Raphael Rouquier (Oxford) : Categorification of braid groups
  Soergel bimodules provide the appropriate setting for the categorification of braid groups. Following Khovanov and Stroppel, we will explain how to deduce invariants of knots and links. Via a duality of Schur-Weyl type, this can be understood using higher representations of Lie algebras. These representations should give rise to four-dimensional topological quantum field theories.
  
  

Exposés :

• Jon Berrick (Singapore) : Torsion and rigidity for braid and mapping class groups
  Abstract. Although braid and mapping class groups are closely related to other classical families of groups such as symmetric groups, automorphism groups of free groups and linear groups, in some ways they exhibit different behaviour with regard to stabilization.
  
  
• Jean-Marie Droz (Zurich): Practical computation of knot Floer homology
  An overview of the computer implementation of a method of Anna Beliakova to compute knot Floer homology.
  
  
• Vincent Florens (Pau) : The Alexander representation of the tangle category
  Joint work with S.Bigelow. We construct a natural extension of the Burau representation to the tangle category. This gives a functorial construction of the Alexander polynomial, in terms of intersections in homology of configuration spaces of points in the disk.
  
  
• Jean Fromentin (Caen) : The cycling normal form in dual braid monoids
  We introduce a new normal form for the dual braid monoid. It is based on a natural operation expressing every dual braid on n strands in terms of a certain finite sequence of dual braids on (n-1) strands. We deduce an inductive characterization of the Dehornoy ordering of the dual braid monoids, and explicitly compute the associated order types.
  
  
• Toshitake Kohno (Tokyo) : Braids, Drinfel'd associator and rational homotopy
  It can be shown by rational homotopy theory that unipotent representationof the braid group can be realized as the monodromy representation of a nilpotent logarithmic connection. I will describe the algorithm for an explicit description of the connection by means of the Drinfel'd associator defined over rational numbers.
  
  
• Alexandre Kosyak (Kiev): Representations of the braid group $B_3$ and the highest weight modules of U(sl_2) and U_q(sl_2)
  
  
• Ivan Marin (Paris 7) : Krammer representations for complex braid groups
  Complex braid groups are generalizations of Artin groups, which are associated to complex reflection groups and to the complement of their hyperplane arrangement. We introduce a new local system on this complement, whose monodromy defines a representation of the complex braid group which generalizes the Krammer representation of Artin groups. This provides a good candidate in view of proving the linearity of the complex braid groups, as well as related group-theoretic properties.
  
  
• Michael Polyak (Technion) : Tree-like and 1-loop invariants of string links
  
  
• Paul Turner (Fribourg/Edinburgh) : Coloured posets and their homology with applications to knot theory
  Motivated by the definition of Khovanov homology I will present the notion of a coloured poset and describe a homology functor for these objects. I will explain how this is related to, and indeed generalises, Khovanov's construction. I will then outline a theory of bundles for coloured posets, producing in certain cases a Leray-Serre type spectral sequence. The latter will then be used to produce a new spectral sequence for computing the Khovanov homology of a knot.
  
  
• Geordie Williamson (Freiburg) : Knot homology and equivariant cohomology
  Khovanov-Rozansky link homology is a categorical knot invariant: starting with any knot (or link) one obtains an invariant consisting of a triply graded vector space. Moreover, taking the Euler characteristic yields the HOMFLYPT polynomial. I will describe one way of constructing Khovanov-Rozansky homology (due to Khovanov) in terms of Soergel bimodules and the Rouquier complex and explain joint work with Ben Webster in which we reinterpret the steps in its construction geometrically. Along the way we calculate the cohomology of certain smooth orbit closures in a reductive complex algebraic group.