Séminaire de topologie, géométrie et algèbre (archives)

We study two different invariants of framed oriented links. Augmentations are rank one representations of a non-commutative algebra, whose definition is motivated by Floer homology. Sheaves in microlocal theory can be thought of as generalizations of link group representations. We will demonstrate two constructions going back and forth between these invariants. We will also tell a motivating story behind the scene, using SFT and microlocalization correspondence in symplectic topology.

Kontsevich's recursion, proved by Ruan-Tian in the early 90s, enumerates rational curves in complex surfaces. Welschinger defined invariant signed counts of real rational curves in real surfaces (complex surfaces with a conjugation) in 2003. Solomon interpreted Welschinger's invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline for adapting Ruan-Tian's homotopy style argument to the real setting. For many symplectic fourfolds, these recursions determine all invariants from basic inputs.

On étudie la version réelle suivante d'un théorème célèbre d'Abhyankar-Moh : quelles applications rationnelles de la droite affine dans le plan affine, dont le lieu réel est un plongement fermé non singulier de R dans R^2, sont équivalentes, à difféomorphisme birationnel du plan près, au plongement trivial ? Dans ce cadre, on montre qu’il existe une infinité de plongements non équivalents. Certains d’entre eux sont détectés pas la non-négativité de la dimension de Kodaira réelle du complémentaire de leur image. Mais nous introduisons aussi un invariant plus fin dérivé des propriétés topologiques de « faux plans réels » particuliers associés à ces plongements. (Travail en commun avec Adrien Dubouloz).

Travail en commun avec J.P. Monnier et R. Quarez. La normalisation faible d'une variété algébrique complexe est une variété intermédiaire entre la variété et sa normalisation, qui est en bijection avec la variété de départ. On développe une notion analogue pour les variétés algébriques réelles en s'appuyant sur l'anneau des fonctions rationnelles continues.

A morsification of a real plane singularity is a real deformation with the maximal possible number of hyperbolic nodes. Morsifications are an important tool for the study of Dynkin diagrams, monodromy, topology of the singularity link and other characteristics of singularities. In this talk I will address the problem of isotopy classification of morfisications of totally real singularities of type (3,k). I will show how to obtain this classification by combinatorial means via dessins d'enfants and how it can be encoded by wiring diagrams. I will also described the classification of these morsifications up to Reidemeister moves.

The local Gromov-Witten theory of curves studied by Bryan and Pandharipande revealed strong structural results for the local GW invariants, which were later used by Ionel and Parker in the proof of the Gopakumar-Vafa conjecture. In this talk I will report on a joint work with Eleny Ionel on the extension of these results to the real setting. Similarly to the classical case, we obtain a complete solution in terms of representation theoretic data using the formalism of an extended Klein TQFT. The local real version of the Gopakumar-Vafa formula is obtained as a corollary.

In a joint work with V. Kharlamov, we explained how one may count real lines on real hypersurfaces (when their number is generically finite) with signs, so that the sum is independent of the choice of a hypersurfaces. These signs were assumed conjecturally to be equal to some multidimensional version of Welschinger weights. After elaborating this version of the weights, we proved this conjecture. We developed also a more geometric way of calculation : using the idea of Segre, who introduced two species of real lines on a cubic surface : hyperbolic and elliptic.

Kronheimer et Mrowka utilisent la théorie de jauge SO(3) pour associer à tout graphe K plongé dans une 3-variété un espace vectoriel J♯(K). Grâce aux travaux de Gabai sur les variétées suturées, ils montrent que si K est dans ℝ²⊂ ℝ³ et sans pont alors J♯(K) est non trivial. Ils conjecturent que dans ce cas la dimension de J♯(K) est égale au nombre de coloriage de Tait du graphe K. Cette conjecture implique le théorème des quatre couleurs. Dans cet exposé, j'expliquerai comment l'évaluation des mousses permet de construire un analogue combinatoire à J♯(K). (En commun avec M. Khovanov)

Motivated by microlocal sheaf theory, Nadler introduced a particularly simple and natural class of singularities for the Lagrangian skeleta of Weinstein manifolds. These singularities are called arboreal, and skeleta with only arboreal singularities are called arboreal skeleta. Unlike generic Lagrangian skeleta, arboreal skeleta have unique Weinstein neighborhoods and offer the hope of reducing pseudo-holomorphic curve invariants of Weinstein manifolds to combinatorics. In work in progress joint with Eliashberg, Nadler and Starkston we study the problem of existence and uniqueness of arboreal skeleta for Weinstein manifolds, which is intimately linked to the problem of simplifying the singularities of Lagrangian and Legendrian fronts.