Séminaire de topologie, géométrie et algèbre (archives)

The Jones representation of the mapping class group of the punctured sphere is constructed by formulating irreducible linear representations of braid groups that factor through Hecke algebras. In this talk we introduce the Jones representation and we show that the normal closure of the m-th power of a half-twist has infinite index in the mapping class group of a punctured sphere. As a corollary we show that the normal closure of a power of a Dehn twist has infinite index in the hyperelliptic mapping class group of a closed surface of genus at least two.

La dualité miroir cohomologique est la propriété $h^{p,q}(X)=h^{3-p,q}(X')$, où $X$ et $X'$ sont deux solides de Calabi-Yau. Elle se manifeste dans le cas de la construction dite de Borcea-Voisin comme une conséquence de la symétrie miroir des surfaces K3 avec involution anti-symplectique. Il s'agit de l'une des premières manifestations de symétrie miroir entre solides de Calabi-Yau, qu'on aimerait bien comprendre dans un cadre unifié. On espère aussi d'aller au delà du simple constat $h^{p,q}(X)=h^{3-p,q}(X')$, vers un énoncé qui met en jeu les nombres -à ce jour presque complètement inconnus- des courbes tracés sur les solides de Calabi-Yau. Dans ce travail, en collaboration avec Kalachnikov et Veniani, on généralise et on démontre la dualité miroir cohomologique pour les couples de type Borcea-Voisin en dimensions quelconque. Comme dans le cas standard, ces couples dérivent de couples miroir de Calabi-Yau avec involution. La méthode est une variante du modèle de Landau-Ginzburg et de la correspondance Landau-Ginzburg/Calabi-Yau. Les modèles de Landau-Ginzburg encodent les informations cruciales des variétés de Calabi-Yau et, dans le cadre classique, jouent le rôle de véhicule entre variétés miroir. Dans ce travail, ces modèles reflètent également la géométrie du lieu fixe de l'involution. On découvre donc au passage des énoncés nouveaux de symétrie miroir qui concernent les courbes sextiques dans P2, les surfaces octiques dans P3, ou les solides de degré 10 dans P4, etc.

Families of groups such as symmetric groups, braid groups, general linear groups, mapping class groups of 2- or 3-dimensional manifolds, or Higman-Thompson groups share the following stability phenomenon: the homology of the nth group in the sequence is isomorphic to that of the (n+1)st group in a range of degrees increasing with n. This phenomemon is called homological stability.

In this series of talks, I will give an introduction to homological stability, showing what the above examples have in common. I'll explain through the framework of homogeneous categories how the question of stability boils down to the question of high connectivity of certain simplicial complexes and give an idea of how these connectivity results are proved in different examples.

Families of groups such as symmetric groups, braid groups, general linear groups, mapping class groups of 2- or 3-dimensional manifolds, or Higman-Thompson groups share the following stability phenomenon: the homology of the nth group in the sequence is isomorphic to that of the (n+1)st group in a range of degrees increasing with n. This phenomemon is called homological stability.

In this series of talks, I will give an introduction to homological stability, showing what the above examples have in common. I'll explain through the framework of homogeneous categories how the question of stability boils down to the question of high connectivity of certain simplicial complexes and give an idea of how these connectivity results are proved in different examples.

Families of groups such as symmetric groups, braid groups, general linear groups, mapping class groups of 2- or 3-dimensional manifolds, or Higman-Thompson groups share the following stability phenomenon: the homology of the nth group in the sequence is isomorphic to that of the (n+1)st group in a range of degrees increasing with n. This phenomemon is called homological stability.

In this series of talks, I will give an introduction to homological stability, showing what the above examples have in common. I'll explain through the framework of homogeneous categories how the question of stability boils down to the question of high connectivity of certain simplicial complexes and give an idea of how these connectivity results are proved in different examples.

I will give an introduction to Legendrian contact homology, which is an invariant of Legendrian submanifolds that is defined by using pseudo-holomorphic disk techniques. In particular, I will explain how one can define this homology with integer coefficients by orienting the moduli spaces of the pseudo-holomorphic disks. I will also discuss how one can make this invariant more easy to compute by replacing the pseudo-holomorphic disks with gradient flow trees, and how the moduli spaces of these trees can be oriented in a computable way.

Talk 1: Triangulated surfaces in triangulated categories

Abstract: The symmetries inherent in the structure constants of a Frobenius algebra can be used to associate certain numerical invariants of oriented surfaces. These numbers behave nicely when chopping surfaces into pieces - in technical terms they form a 2-dimensional open topological field theory. In particular, the invariant of a given surface can be computed in terms of a chosen triangulation. In this talk, we explain how certain symmetries in the foundations of homological algebra behave like a Frobenius algebra to the extent that they define invariants of oriented surfaces.

Based on joint work with Mikhail Kapranov. "

" Talk 2: Relative Calabi-Yau structures [Salle Eole, à 14:00]

Abstract: The basic operation of oriented cobordism is to glue two oriented manifolds along a common boundary component to produce a new oriented manifold. In this talk, we discuss a generalization of this procedure to noncommutative geometry: we introduce the concept of a Calabi-Yau structure on a functor of differential graded categories which should be interpreted as an analog of an oriented manifold with boundary. As an application of the resulting theory, we show that topological Fukaya categories of surfaces give rise to a 2D TFT with values in Calabi-Yau cospans of differential graded categories.

Based on joint work in progress with Chris Brav. "

Recall that the geometric dimension $gd(G)$ of a group $G$ is the smallest dimension of a space on which $G$ acts in such a way that fixed point sets of finite subgroups are contractible. For many prominent classes of groups (e.g. for amenable groups, lattices in classical Lie groups, mapping class groups, groups of outer automorphisms of free groups...) one has equality between the geometric dimensions and the virtual cohomological dimension. On the other hand, there are some examples showing that these two notions of dimension might well differ. I will present some new examples of this phenomenon. This is joint work with Dieter Degrijse.