Séminaire de topologie, géométrie et algèbre (archives)

I will describe a geometric chain level construction of a secondary coproduct operation on a suitable chain model for the free loop space of a manifold using the theory of De Rham chains. Such coproduct was described at the level of homology by Goresky and Higston using different methods. It is secondary in the sense that arises from a "1-parameter family of chain level intersections". The chain level theory around this secondary operation is useful for describing certain phenomena in symplectic topology and symplectic field theory. There are analogues of these geometric operations in the algebraic theory of Hochschild complexes of Frobenius algebras described in work by Wahl, Abbaspour, Tradler, Zeinalian, and others. I will discuss a new way of nterpreting both the algebraic loop product and the algebraic secondary coproduct in a single package. This is work in progress with Dingyu Yang (IMJ-PRG) and Zhengfeng Wang (IMJ-PRG).

The "equivalence principle" (EP) says that meaningful statements in mathematics should be invariant under the appropriate notion of equivalence - "sameness" - of the objects under consideration. In set theoretic foundations, the EP is not enforced; e.g., the statement "1 ϵ Nat" is not invariant under isomorphism of sets. In univalent foundations, on the other hand, the equivalence principle has been proved for many mathematical structures. In this introductory talk, I first give an overview of earlier attempts at designing foundations that satisfy some invariance property. Afterwards I give a brief introduction to the univalent foundations (UF) and present results, both by other and myself, on the validity of EP in UF.

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.