# Séminaires : Géométrie énumérative

Analyse Algébrique
Penka Georgieva, Ilia Itenberg.

## Séances à suivre

+ Sybille Rosset A comparison formula in quantum K-theory of flag varieties 30/01/2020 16:00 15-25-502 Jussieu

I will present here a correspondence between well-chosen quantum K-theoretical Gromov-Witten invariants of different flag varieties. I will also discuss how this correspondence implies some finiteness properties of the big quantum K-ring of flag varieties.

+ Séances antérieures

### Séances antérieures

+ Sergey Finashin The first homology of real cubics are generated by real lines 16/01/2020 16:00 15-25-502 Jussieu

In a joint work with V. Kharlamov, we suggest a short proof of O. Benoist and O. Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface X of dimension ≥2 the real lines on X generate the whole group H_1(X(ℝ);ℤ/2).

+ Sybille Rosset A comparison formula in quantum K-theory of flag varieties (reporté) 12/12/2019 16:00 15-25-502 Jussieu

I will present here a correspondence between well-chosen quantum K-theoretical Gromov-Witten invariants of different flag varieties. I will also discuss how this correspondence implies some finiteness properties of the big quantum K-ring of flag varieties.

+ Danilo Lewanski ELSV-type formulae 05/12/2019 16:00 15-25-502 Jussieu

The celebrated ELSV formula expresses Hurwitz numbers in terms of intersection theory of the moduli space of stable curves. Hurwitz numbers enumerate branched covers of the Riemann sphere with prescribed ramification profiles.

Since the original ELSV was found, many more ELSV-type formulae appeared in the literature, especially in connection with Eynard-Orantin topological recursion theory. They connect different conditions on the ramification profiles of the Hurwitz problem with the integration of different cohomological classes which have been studied independently.

We will go through this interplay, focusing on a conjecture proposed by Zvonkine and a conjecture of Goulden, Jackson, and Vakil. In both these conjectures, classes introduced by Chiodo play a key role.

+ Grigory Mikhalkin Separating semigroup of real curves and other questions from a 1-dimensional version of Hilbert's 16th problem 28/11/2019 15:15 15-25-502 Jussieu

Kummer and Shaw have introduced the separating semigroup Sep(S) of a real curve S. The semigroup is made of topological multidegrees of totally real algebraic maps from S to the Riemann sphere and can be considered in the context of a 1-dimensional version of Hilbert's 16th problem. We'll explore this point of view and classify Sep(S) for curves of genera up to four.

+ Yizhen Zhao Landau-Ginzburg/Calabi-Yau correspondence for a complete intersection via matrix factorizations 21/11/2019 16:00 15-25-502 Jussieu

In this talk, I will introduce two enumerative theories coming from a variation of GIT stability condition. One of them is the Gromov-Witten theory of a Calabi-Yau complete intersection; the other one is a theory of a family of isolated singularities fibered over a projective line, which is developed by Fan, Jarvis, and Ruan recently. I will show these two theories are equivalent after analytic continuation. For Calabi-Yau complete intersections of two cubics, I will show that this equivalence is directly related - via Chern character - to the equivalences between the derived category of coherent sheaves and that of matrix factorizations of the singularities. This generalizes Chiodo-Iritani-Ruan's theorem matching Orlov's equivalences and quantum LG/CY correspondence for hypersurfaces.

+ Xavier Blot The quantum Witten-Kontsevich series 15/11/2019 10:30 15-16-413 Jussieu
The Witten-Kontsevich series is a generating series of intersection numbers on the moduli space of curves. In 2016, Buryak, Dubrovin, Guéré and Rossi defined an extension of this series using a quantization of the KdV hierarchy based on the geometry of double ramification cycle. This series, the quantum Witten-Konstevich series, depends on a quantum parameter. When this quantum parameter vanishes, the quantum Witten-Kontsevich series restricts to the Witten-Kontsevich series. In this talk, we will first construct the quantum Witten-Kontsevich series and then present all the known results about its coefficients. Surprisingly, a part of these coefficients are expressed in terms of Hurwitz numbers.
+ Hülya Argüz Real Lagrangians in Calabi--Yau Threefolds 18/10/2019 10:30 15-16-413 Campus Pierre et Marie Curie

We compute the mod 2 cohomology of the real Lagrangians in Calabi--Yau threefolds, using a long exact sequence linking it to the cohomology of the Calabi--Yau. We will describe this sequence explicitly, and as an application will illustrate this computation for the quintic threefold. This is joint work with Thomas Prince and with Bernd Siebert.

+ Yanqiao Ding Genus decreasing phenomenon of higher genus Welschinger invariants 27/09/2019 10:30 Jussieu, salle 15-16-413

Shustin introduced an invariant of del Pezzo surfaces to count real curves of positive genera. By considering the properties of these invariants under Morse transformation, we found a genus decreasing phenomenon for these invariants.  In this talk, we will present a genus decreasing formula for these invariants and discuss possible generalization of it.

+ Eugenii Shustin Singular Welschinger invariants 28/06/2019 10:00 Jussieu, salle 15-16-413
We discuss real enumerative invariants counting real deformations of plane curve singularities. A versal deformation base of a plane curve singularity contains local Severi varieties that parameterize deformations with a given delta-invariant. The local Severi varieties are analytic space germs and their (complex) multiplicities were computed by Beauville, Fantecci-Goettsche-van Straten, and Shende. For the equigeneric locus (local Severi variety corresponding to the maximal delta-invariant), a real multiplicity was introduced by Itenberg-Kharlamov-Sh. as a Welschinger-type signed count of certain equigeneric deformations. We show that similar real multiplicities can be defined for some other local Severi varieties as well as for all equiclassical loci (which count equigeneric deformations with a given number of cusps). We exhibit some examples and state open problems.
+ Albrecht Klemm Topological String on compact Calabi-Yau threefolds 21/06/2019 10:30 Jussieu, salle 15-16-413
We review the world-sheet derivation of the holomorphic anomaly equations fulfilled by the all genus topological string partition function $Z$ on Calabi-Yau 3-folds $M$. Interpreting $Z$ as a wave function on $H_3(M, R)$ these equations can be viewed as describing infinitesimal changes of the symplectic frame. A recursive solution for $Z$ to high genus is provided using modular building blocks obtained by the periods of $M$ as well as constraints on the local expansion of $Z$ near singular loci in the complex moduli space of M in appropriate symplectic frames. Some recent applications of these ideas to elliptic fibred Calabi-Yau spaces are given.
+ Alex Degtyarev Tritangents to sextic curves via Niemeier lattices 14/06/2019 10:30 Jussieu, salle 15-25-502
I suggest a new approach, based on the embedding of the (modified) Néron--Severi lattice to a Niemeier lattice, to the following conjecture: The number of tritangents to a smooth sextic is 72, 66 (each realized by a single curve), or less. The maximal number of real tritangents to a real smooth sextic is 66. (Observed are all counts except 65 and 63.) The computation becomes much easier (linear algebra in well-studied lattices rather than abstract number theory), and it has been completed for all but Leech lattices. At present, I am 99% sure that I can eliminate the Leech lattice, settling the above conjecture.
+ Alexander Alexandrov Constellations, Weighted Hurwitz numbers, and topological recursion (a mathematical physicist's view) 19/04/2019 10:30 Jussieu, salle 15-16-413
In my talk I will discuss some elements of the proof of the topological recursion for the weighted Hurwitz numbers. The main ingredient is the tau-function - the all genera generating function, which is a solution of the integrable KP or Toda hierarchy. My talk is based on a series of joint papers with G. Chapuy, B. Eynard, and J. Harnad.
+ Florent Schaffhauser Topologie des variétés de représentations de groupes fuchsiens 29/03/2019 10:30 Jussieu, salle 15-16-413
Le but de l'exposé est de présenter quelques progrès récents dans l'étude la topologie des variétés de représentations de groupes fuchsiens. On s'intéressera principalement à deux exemples, les fibrés vectoriels sur les courbes algébriques réelles et les composantes de Hitchin pour les groupes fondamentaux orbifold, et on montrera par exemple que la composante de Hitchin d'une surface orientable à bord (introduite par McShane et Labourie en 2009) est homéomorphe à un espace vectoriel dont la dimension est donnée par la même formule que celle obtenue par Hitchin dans le cas des surfaces fermées. Plus généralement, nous verrons que les composantes de Hitchin orbifold fournissent des sous-variétés totalement géodésiques contractiles des composantes de Hitchin classiques (pour toute métrique invariante sous l'action du groupe modulaire), dont on peut calculer la dimension et montrer qu'elles fournissent des exemples d'espaces de Teichmüller supérieurs, au même titre que les composantes de Hitchin associées aux groupes de surfaces.
+ Xujia Chen Kontsevich-type recursions for counts of real curves 15/03/2019 10:30 Jussieu, salle 15-16-413
+ Adrien Sauvaget Masur-Veech volumes and intersection theory on the projectivized Hodge bundle 08/03/2019 10:30 Jussieu, salle 15-16-413
In the 80's Masur and Veech defined the volume of moduli spaces Riemann surfaces endowed with a flat metric with conical singularities. We show that these volumes can be expressed as intersection numbers on the projectivized Hodge bundle over the moduli space of curves (this is a joint work with D. Chen, M. Moeller, and D. Zagier).
+ Paolo Rossi Quadratic double ramification integrals and KdV on the non-commutative torus 22/02/2019 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 Sur les nombres de Betti des espaces de modules de faisceaux semi-stables sur le plan projectif 08/02/2019 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 Scattering diagrammes, indices quantiques et géométrie énumérative réelle 01/02/2019 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 Welschinger weights and Segre indices for real lines on real hypersurfaces 25/01/2019 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 Tropical scheme theory 18/01/2019 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 Constellations, Weighted Hurwitz numbers, and topological recursion (a combinatorialist's view) 14/12/2018 10:30 Jussieu, salle 1516-413
+ Dimitri Zvonkine An introduction to the double ramification hierarchies by Buryak and Rossi 07/12/2018 10:30 Jussieu, salle 1516-413
+ Hülya Argüz Tropical and log corals on the Tate curve 16/11/2018 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 Matrix Factorizations and Vanishing Cycles 19/10/2018 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 Positivité pour la K-théorie quantique de la grassmannienne 12/10/2018 10:30 Jussieu, salle 1516-413