Sorbonne Université CNRS Paris Diderot

Géométrie énumérative

2018 2019

Année 2018- 2019

Organisateurs :
Penka Georgieva, Ilia Itenberg.

Paolo Rossi - Università degli Studi di Padova

Quadratic double ramification integrals and KdV on the non-commutative torus

vendredi 22 février 2019 à 10:30 : Jussieu, salle 15-16-413

It’s a result of Richard Hain that the restriction of the double ramification cycle to the space of compact type curves (i.e. stable curves with no non-separating nodes) is Θg/g !, where Θ is the theta divisor in the universal Jacobian (suitably pulled back to the moduli space itself via the marked points). A natural completion of this class is given by exp(Θ), which gives an infinite rank partial cohomological field theory. To such an object one can attach a double ramification hierarchy (thereby putting into play a second DR cycle, hence the "quadratic" in the title). It is possible to compute this hierarchy and trade its infinite rank for an extra space dimension, hence obtaining an integrable hierarchy in 2+1 dimensions which is the natural extension of the usual KdV hierarchy on a non-commutative torus. Its quantization is also provided, obtaining an integrable (2+1) non-relativistic quantum field theory on the non-commutative torus.

Pierrick Bousseau - ETH Zürich

Sur les nombres de Betti des espaces de modules de faisceaux semi-stables sur le plan projectif

vendredi 8 février 2019 à 10:30 : Jussieu, salle 15-16-413

Je vais présenter un nouvel algorithme, à l’allure tropicale, calculant les nombres de Betti (pour la cohomologie d’intersection) des espaces de modules de faisceaux semi-stables sur le plan projectif. Je finirai par une application à une question a priori sans rapport en théorie de Gromov-Witten.

Thomas Blomme - IMJ-PRG

Scattering diagrammes, indices quantiques et géométrie énumérative réelle

vendredi 1er février 2019 à 10:30 : Jussieu, salle 15-16-413

En géométrie énumérative, l’approche tropicale est parfois fort utile pour calculer effectivement certains invariants de part la nature combinatoire de cette dernière. De plus, sa richesse structurelle permet en fait de calculer bien plus que les invariants qui nous intéressent, et c’est par exemple le cas des polynômes de Block-Göttsche. Dès lors se pose la question de l’interprétation de tels invariants en géométrie classique et de nombreuses restent encore ouvertes. Dans le cas des courbes planes, Mikhalkin propose d’interpréter le polynôme de Block-Göttsche comme un comptage de courbes réelles satisfaisant des conditions de tangence à l’infini en les discriminant suivant la valeur que prend l’aire de leur amibe. Nous allons tenter de poser les bases de ce que pourrait être un analogue en dimension supérieure.

Sergey Finashin - Middle East Technical University, Ankara

Welschinger weights and Segre indices for real lines on real hypersurfaces

vendredi 25 janvier 2019 à 10:30 : Jussieu, salle 15-16-413

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.

Oliver Lorscheid - IMPA

Tropical scheme theory

vendredi 18 janvier 2019 à 10:30 : Jussieu, salle 15-16-413

In 2013, Giansiracusa and Giansiracusa have found a way to use F1-geometry for tropical geometry. More precisely, they define the scheme-theoretic tropicalization of a classical variety and show that the set-theoretic tropicalization can be retrieved as the set of T-rational points.

The scheme-theoretic tropicalization carries more information than the set-theoretic tropicalization. For example, it knows about the Hilbert polynomial of the classical variety and the weights of the (maximal cells of the) set-theoretic tropicalization.

There are hopes that this will be useful for future developments, such as tropical sheaf cohomology, a cohomological approach to intersection theory, flat tropical families, and more.

However, some fundamental problems remain unsolved so far. For example, it is not clear how to approach dimension theory or decompositions into irreducible components. It is not even clear what a good notion of a tropical scheme should be since the class of semiring schemes contains too many and pathological objects.

In this talk we give an introduction to tropical scheme theory and an overview of this circle of ideas.

Guillaume Chapuy - CNRS et IRIF

Constellations, Weighted Hurwitz numbers, and topological recursion (a combinatorialist’s view)

vendredi 14 décembre 2018 à 10:30 : Jussieu, salle 1516-413

Dimitri Zvonkine - Laboratoire de Mathématiques de Versailles

An introduction to the double ramification hierarchies by Buryak and Rossi

vendredi 7 décembre 2018 à 10:30 : Jussieu, salle 1516-413

Hülya Argüz - Imperial College Londres

Tropical and log corals on the Tate curve

vendredi 16 novembre 2018 à 10:30 : Jussieu, salle 1516-413

We will discuss an algebro-geometric approach to the symplectic cohomology ring, in terms of tropical geometry and punctured log Gromov-Witten theory of Abramovich-Chen-Gross-Siebert. During this talk, we will restrict ourselves to the Tate curve, the total space of a degeneration of elliptic curves to a nodal elliptic curve. To understand the symplectic cohomology of the Tate curve (minus its central fiber), we will go through the Fukaya category of the elliptic curve and describe this category using tropical Morse trees introduced by Abouzaid-Gross-Siebert.

Marco Robalo - IMJ-PRG

Matrix Factorizations and Vanishing Cycles

vendredi 19 octobre 2018 à 10:30 : Jussieu, salle 1516-413

In this talk I will describe a joint work with B. Toen, G. Vezzosi and A. Blanc, relating categories of matrix factorisations to sheaves of vanishing cycles. Most of the talk will be a review of the theory of vanishing cycles and matrix factorisations and how they can be related in the theory of motives.

Nicolas Perrin - Université de Versailles

Positivité pour la K-théorie quantique de la grassmannienne

vendredi 12 octobre 2018 à 10:30 : Jussieu, salle 1516-413