# Séminaires : Géométrie énumérative

Analyse Algébrique
Penka Georgieva, Ilia Itenberg.

## Séances à suivre

+ Adrien Sauvaget On spin GW/Hurwitz correspondence 02/12/2022 14:00 15-16-413 Jussieu

Spin GW invariants were introduced by Kiem and Li to determine the GW invariants of surfaces with smooth anti-canonical divisors. This numbers are conjectured to be equal to linear combinations of Spin Hurwitz numbers which can be computed via representation theory: this is the so-called spin GW/Hurwitz correspondence. I will explain that this conjecture is valid if the target is P^1 and in general if we assume a general structure of spin GW invariants.

+ Denis Auroux Fonctions analytiques et homologie de Floer pour les surfaces de Riemann et leurs miroirs 09/12/2022 14:00 15-16-413 Jussieu

Cet exposé concerne la symétrie miroir homologique pour les surfaces de Riemann. Après des exemples élémentaires (cylindre et pantalon), on considérera les décompositions le long de cylindres (thèse de Heather Lee) pour arriver à un résultat de symétrie miroir général. On verra en particulier le lien entre les trajectoires de Floer dans les portions cylindriques d'une surface et les développements en série de Laurent des fonctions analytiques sur le miroir. On esquissera enfin une description des catégories de Fukaya de surfaces singulières que l'on peut considérer comme les miroirs de surfaces de Riemann (travail en collaboration avec Efimov et Katzarkov).

+ Séances antérieures

### Séances antérieures

+ Grigory Mikhalkin Enumeration of curves in ellipsoid cobordisms 25/11/2022 14:00 15-16-413 Jussieu

Ellipsoid cobordisms are special case of toric surfaces. They correspond to quadrilaterals cut from the positive quadrants by two disjoint intervals. Holomorphic curves inside these cobordisms obstruct squeezing of one ellipsoid into another. We fit tropical curves into the so-called SFT-framework, and observe a jumping phenomenon in the resulting enumeration. Based on the joint work with Kyler Siegel.

+ Gabriele Rembado Local wild mapping class group 18/11/2022 14:00 15-25-101 Jussieu

The standard mapping class groups are fundamental groups of moduli spaces/stacks of pointed Riemann surfaces. The monodromy properties of
a large family of nonlinear differential equations, the tame isomonodromy connections, are encoded as the action of the mapping class group on the character varieties of the surface. Recently this story has been extended to wild Riemann surfaces, which generalise pointed Riemann surfaces by adding local moduli at each marked point: the irregular classes. These new parameters control the polar parts of meromorphic connections with wild/irregular singularities, defined on principal bundles, and importantly provide an intrinsic viewpoint on the times' of irregular isomonodromic deformations. The monodromy properties of the wild/irregular isomonodromy connections are then encoded as the action of the resulting wild mapping class group on the wild character varieties of the surface.

In this talk we will explain how to compute the fundamental groups of (universal) spaces of deformations of irregular classes, which bring about cabled versions of (generalised) braid groups. The case of generic meromorphic connections has been understood for some time (and known to underlie the Lusztig symmetries of the quantum group since 2002) so the focus will be the new features such as cabling that occur on the general setting. This is joint work with P. Boalch, J. Douçot and M. Tamiozzo (arXiv:2204.08188, 2208.02575, 2209.12695).

If time allows we will sketch a relation with bundles of irregular conformal blocks in the Wess--Zumino--Witten model, in joint work with G. Felder (arXiv:2012.14793) and G. Baverez (in progress).

+ Alex Degtyarev Lines generate the Picard group of a Fermat surface 07/11/2022 11:00 15-16-413 Jussieu

In 1981, Tetsuji Shioda proved that, for each integer m > 0 prime to 6, the 3m^2 lines contained in the Fermat surface \Phi_m:  z_0^m+z_1^m+z_2^m+z_3^m=0 generate the Picard group of the surface over Q, and he conjectured that the same lines also generate the Picard group over Z. If true, this conjecture would give us a complete understanding of the Néron--Severi lattice of \Phi_m, leading to the computation of a number of more subtle arithmetical invariants. It was not until 2010 that the first numeric evidence substantiating the conjecture was obtained by Schütt, Shioda, and van Luijk and, in similar but slightly different settings, by Shimada and Takahashi. I will discuss a very simple purely topological proof of Shioda's conjecture and try to extend it to the more general so-called Delsarte surfaces, where the statement is not always true, raising a new open question.

If time permits, I will also discuss a few advances towards the generalization of the conjecture to the (2d+1)!! m^{d+1} projective d-spaces contained in the Fermat variety of degree m and dimension 2d; this part is joint with Ichiro Shimada.

+ Tyler Kelly Open Mirror Symmetry for Landau-Ginzburg models 21/10/2022 14:00 15-16-413 Jussieu

A Landau-Ginzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function W:X → C from a quasi-projective variety X with a group G acting on X leaving W invariant. An enumerative theory developed by Fan, Jarvis, and Ruan inspired by ideas of Witten gives FJRW invariants, the analogue of Gromov-Witten invariants for LG models. These invariants are now called FJRW invariants. We define a new open enumerative theory for certain Landau-Ginzburg models. Roughly speaking, this involves computing specific integrals on certain moduli of disks with boundary and interior marked points. One can then construct a mirror Landau-Ginzburg model to a Landau-Ginzburg model using these invariants. If time permits or as interest of the audience guides, I will explain some key features that this enumerative geometry enjoys (e.g., topological recursion relations and wall-crossing phenomena). This is joint work with Mark Gross and Ran Tessler.

+ Maria Yakerson On the cohomology of Quot schemes of infinite affine space 14/10/2022 14:00 15-16-413 Jussieu

Hilbert schemes of smooth surfaces and, more generally, their Quot schemes are well-studied objects, however not much is known for higher dimensional varieties. In this talk, we will speak about the topology of Quot schemes of affine spaces. In particular, we will compute the homotopy type of certain Quot schemes of the infinite affine space, as predicted by Rahul Pandharipande. This is joint work in progress with Joachim Jelisiejew and  Denis Nardin.

+ Ellena Moskovsky Generalising Narayana polynomials using topological recursion 30/09/2022 14:00 15-16-413 Jussieu

Narayana polynomials arise in a number of combinatorial settings and have been proven to satisfy many properties, including symmetry, real-rootedness and interlacing of roots. Topological recursion, on the other hand, is a unifying mathematical framework that has been proven to govern a vast breadth of problems. One relatively unexplored feature of topological recursion is its ability to generalise existing combinatorial problems; one can use this feature of topological recursion to motivate a particular generalisation of Narayana polynomials. In ongoing work-in-progress with Norman Do and Xavier Coulter, we prove that the resultant generalised polynomials satisfy certain recursive and symmetry properties analogous to their original counterparts, while conjecturing that they also satisfy real-rootedness and interlacing.

+ Kendric Schefers Microlocal perspective on homology 26/09/2022 14:00 15-16-413 Jussieu

The difference between the homology and singular cohomology of a space can be seen as a measure of the singularity of that space. This difference as a measure of singularity can be made precise in the case of the special fiber of a map between smooth schemes by introducing the so-called "microlocal homology" of such a map, an object which records the singularities of the special fiber as well as the codirections in which they arise. In this talk, we show that the microlocal homology is in fact intrinsic to the special fiber—independent of its particular presentation by any map—by relating it to an object of -1-shifted symplectic geometry: the canonical sheaf categorifying Donaldson-Thomas invariants introduced by Joyce et al. Time permitting, we will discuss applications of our result to ongoing work relating to the singular support theory of coherent sheaves.

+ Xiaohan Yan Quantum K-theory of flag varieties via non-abelian localization 23/09/2022 14:00 15-16-413 Jussieu

Quantum cohomology may be generalized to K-theoretic settings by studying the "K-theoretic analogue" of Gromov-Witten invariants defined as holomorphic Euler characteristics of sheaves on the moduli space of stable maps. Generating functions of such invariants, which are called the (K-theoretic) ”big J-functions”, play a crucial role in the theory. In this talk, we provide a reconstruction theorem of the permutation-invariant big J-function of partial flag varieties (regarded as GIT quotients of vector spaces) using a family of finite-difference operators, based on the quantum K-theory of their associated abelian quotients which is well-understood. Generating functions of K-theoretic quasimap invariants, e.g. the vertex functions, can be realized in this way as values of various twisted big J-functions. We also discuss properties of the level structures as applications of the method. A portion of this talk is based on a joint work with Alexander Givental (my PhD advisor).

+ Campbell Wheeler Quantum modularity of quantum invariants: bringing q-tilde to q-difference equations 21/06/2022 10:00 15-25-502 Jussieu

Recently, there has been increasing interest in solving q-difference equations associated to quantum invariants of 3-manifolds as q-series. I will discuss an algorithmic approach to constructing such solutions, studied for example by Dreyfus, and relate such solutions to state integrals of Andersen-Kashaev. Along with exact computations of monodromy, this proves quantum modularity of such solutions in examples and, in particular, the quantum modularity of the q-Borel resummation of the coloured Jones polynomial. This is based on joint work with Garoufalidis, Gu and Mariño.

+ Eugenii Shustin Non-nodal real enumerative invariants 16/06/2022 16:00 15-16-413 Jussieu

Rational Gromov-Witten and Welschinger invariants of the plane (or another del Pezo surface) count complex and real rational curves passing through appropriate configuration of points in general position and having nodes as their only singularities. One can count complex curves with non-nodal singularities (cusps etc.) which, under some regularity conditions, always leads to enumerative invariants. In 2006, Welschinger noticed that the count of real plane rational unicuspidal curves does depend on the point constraint, and he suggested a real enumerative invariant counting together the unicuspidal curves, reducible nodal curves, and irreducible curves matching some point and tangency conditions. We address the following question: Does there exist real enumerative invariants counting only rational curves with a given collection of non-nodal singularities and with weights depending only on the topology of the real point set?

Theorem 1. In degree 4, such invariants exist only for curves having either one singular point A_5, or one singular point D_4, or one singular point E_6, or three ordinary cusps, provided that, in the latter case, the constraint consists of four pairs of complex conjugate points. All these invariants are positive.

Theorem 2. For each degree d>4, there exists an enumerative invariant that counts real rational curves with one singularity of order d-1, combined of the transversal local branches of odd orders.

Joint work with O. Bojan.

+ Guillaume Chapuy b-deformed Hurwitz numbers 14/04/2022 16:00 15-16-413 Jussieu

I will talk about the papers arXiv:2109.01499 and arXiv:2004.07824 joint with Maciej Dołęga, and with Valentin Bonzom. By using the deformation of characters of the symmetric group obtained by deforming Schur functions into Jack polynomials, we introduce a one-parameter deformation of Hurwitz numbers, the `b-deformed Hurwitz numbers''. The Goulden-Jackson b-conjecture from 1996 (and variants) asserts that these numbers are well defined (positive) and have to do with the enumeration of maps on non-oriented surfaces. I will talk about recent progress towards the conjecture, and other developments related to b-deformed "monotone" Hurwitz numbers and \beta-ensembles of random matrices.

+ David Holmes The double ramification cycle for the universal r-th root 07/04/2022 16:00 15-16-413 Jussieu

The secret goal of this talk is to explain a little of the magic of log line bundles. The vehicle for this will be a story about double ramification cycles for  roots of a line bundle. Given a family of curves C/S and a line bundle L on C, the double ramification cycle DR(L) is a class on S measuring the set of points in S over which L is trivial (or more precisely, where L is trivial as a log line bundle). The formal goal of this talk is to describe a lift to the universal r-th root of L. More precisely, for a positive integer r we define the stack of r-th roots of L, which is a finite flat cover of S of degree r^{2g}. It carries a universal r-th root of L (as a log line bundle), and the locus where this r-th root is (logarithmically) trivial defines a lift of DR(L) to the stack of r-th roots. Pixton's formula for DR(L) admits a fairly straightforward lift to this setting.

+ Kris Shaw A tropical approach to the enriched count of bitangents to quartic curves 24/02/2022 15:30 15-16-413 Jussieu
Using A1 enumerative geometry Larson and Vogt have provided an enriched count of the 28 bitangents to a quartic curve. In this talk, I will explain how these enriched counts can be computed combinatorially using tropical geometry. I will also introduce an arithmetic analogue of Viro's patchworking for real algebraic curves which, in some cases, retains enough data to recover the enriched counts. This talk is based on joint work with Hannah Markwig and Sam Payne.
+ Ilia Zharkov Lagrangian fibrations of the pair-of-pants 17/02/2022 16:00 15-16-413 Jussieu
The pair-of-pants P is the hypersurface in (C*)^n defined by 1+w_1+...+w_n=0. It is a fundamental building block for many problems in mirror symmetry. I will discuss various Liouville structures on P and a map to the tropical hyperplane which is a Lagrangian torus fibration of P for a particular such structure. I will describe the geometry of the fiber over the origin, which is the Lagrangian skeleton of P.
+ Séverin Charbonnier Statistics of multicurves on combinatorial Teichmüller spaces 10/02/2022 16:00 15-16-413 Jussieu
I will describe several results regarding the statistics of multicurves on bordered surfaces, whose combinatorial lengths are bounded by a cut-off parameter. After a description of the combinatorial Teichmüller spaces, I will first state how such statistics can be computed by geometric recursion, a recursive procedure akin to topological recursion. Second, the asymptotics of the number of multicurves as the cut-off tends to infinity allow to define a function on combinatorial Teichmüller spaces, that is interpreted as the volume of the combinatorial unit ball of measured foliations; it is the combinatorial analogue of Mirzakhani's B function in the hyperbolic setup. It descends to the moduli spaces and the structure of the latter allows to completely determine its range of integrability with respect to the Kontsevich measure. The range shows surprising dependence on the topology of the surface. Along the talk, I will compare the results with those holding in the hyperbolic world. Joint works with J. E. Andersen, G. Borot, V. Delecroix, A. Giacchetto, D. Lewański and C. Wheeler.
+ Dhruv Ranganathan Gromov-Witten theory via roots and logarithms 03/02/2022 16:00 15-16-413 Jussieu
The geometry of logarithmic structures and orbifolds offer two routes to the enumeration of curves with tangencies along a divisor in a projective manifold. The theories are quite different in nature: the logarithmic theory has rich connection to combinatorics and mirror symmetry via tropical geometry, while the orbifold geometry is closer in its formal properties to ordinary Gromov-Witten theory, and is more computable as a consequence. I will discuss the relationship between the theories, and try to give a sense of where and why they differ. I will then outline the ideas behind recent work with Nabijou and work in progress with Battistella and Nabijou, which determines genus 0 logarithmic GW theory via the orbifold geometry.
+ Alexander Thomas Topological field theories from Hecke algebras 28/01/2022 14:00 15-16-413 Jussieu
We describe a construction which to a surface and a Iwahori-Hecke algebra associates an invariant which is a Laurent polynomial. More generally, this construction works for surfaces with boundary and behaves well under gluing, giving a non-commutative topological quantum field theory (TQFT). The invariant polynomial has surprising positivity properties, which are proven using Schur elements. Joint work with Vladimir Fock and Valdo Tatitscheff.
+ Johannes Nicaise Variation of stable birational type and bounds for complete intersections 27/01/2022 16:00 15-16-413 Jussieu
This talk is based on joint work with John Christian Ottem. I will explain a generalization of results by Shinder and Voisin on variation of stable birational types in degenerating families, and how this can be used to extend non-stable rationality bounds from hypersurfaces to complete intersections in characteristic zero.
+ Thomas Blomme Enumération de courbes tropicales dans des surfaces abéliennes 06/01/2022 16:00 15-16-413 Jussieu
La géométrie tropicale est un outil puissant qui permet via l'utilisation d'un théorème de correspondance de ramener des problèmes énumératifs algébriques, par exemple compter le nombre de courbes d'un certain degré passant par un nombre de points convenables, à un problème combinatoire. Ces derniers sont plus simples à appréhender mais parfois compliqués à résoudre. De plus, le passage dans le monde tropical permet de définir de mystérieux invariants dits raffinés, obtenus en comptant les solutions d'un problème énumératif avec des multiplicités polynomiales. Dans cet exposé on s'intéressera à l'énumération de courbes et aux invariants raffinés dans les surfaces abéliennes et dans les fibrés en droites au dessus d'une courbe elliptique.
+ Dimitri Zvonkine Gromov-Witten invariants of complete intersections 09/12/2021 16:00 15-16-413 Jussieu
We present an algorithm that allows one to compute all Gromov-Witten (GW) invariants of all complete intersections. The main tool is Jun Li's degeneration formula that expresses GW invariants of one complete intersection via GW invariants of several simpler complete intersections. The main problem is that the degeneration formula does not apply to primitive cohomology classes. To solve this problem we introduce simple nodal GW invariants, show that they can always be computed by degeneration, and then prove that one can recover all GW invariants with primitive cohomology insertions from simple nodal GW invariants. Joint work with H. Arguz, P. Bousseau, and R. Pandharipande.
+ Sebastian Nill Extended FJRW theory of the quintic threefold in genus zero 02/12/2021 16:00 15-16-413 Jussieu
The Landau-Ginzburg A-model of the quintic threefold has a description in terms of higher spin bundles on stable curves. In genus zero the invariants/correlators of the closed r-spin theory are given by integration of the top Chern class of the Witten bundle over the moduli space of stable curves. By allowing a new twist equal to -1 at one of the marked points, Alexandr Buryak, Emily Clader and Ran Tessler found a rank one extension of the closed r-spin theory in genus zero in 2017. After having a look at this extension, we will see that integration of the fifth power of this top Chern class gives an extension of the Fan-Jarvis-Ruan-Witten (FJRW) theory of the quintic threefold in genus zero. In order to calculate the new invariants, we will mimick the work of Alessandro Chiodo and Yongbin Ruan from 2008 and introduce the Givental formalism. I will sketch how Chiodo's Grothendieck-Riemann-Roch formula still provides us with a symplectic transformation of the twisted Givental cone. An extension of the I-function will arise in the non-equivariant limit of the twisted invariants. This extended I-function contains a new term already known as the semi-period. It is a solution of an inhomogeneous Picard-Fuchs equation with a constant inhomogeneity. This is work in progress.
+ Nitin Chidambaram Shifted Witten classes and topological recursion 25/11/2021 16:00 15-16-413 Jussieu
The Witten r-spin class is an example of a cohomological field theory which is not semi-simple, but it can be "shifted" to make it semi simple. Pandharipande-Pixton-Zvonkine studied the shifted Witten class and computed it explicitly in terms of tautological classes using the Givental-Teleman classification theorem. I will show that the R-matrix of (two specific) shifts can be obtained from two differential equations that are generalizations of the classical Airy differential equation. Using this, I will show that the descendant intersection theory of the shifted Witten classes can be computed using the Eynard-Orantin topological recursion, and discuss some potential applications. This is based on work in progress with S. Charbonnier, A. Giacchetto and E. Garcia-Failde.
+ Kris Shaw Real phase structures on matroid fans 15/11/2021 14:00 15-25-502 Jussieu
In this talk, I will propose a definition of real phase structures on polyhedral complexes, focusing on matroid fans. I’ll explain that in the case of matroid fans, specifying a real phase structure is cryptomorphic to providing an orientation of the underlying matroid. Then I’ll define the real part of a polyhedral complex with a real phase structure. This determines a closed chain in the real part of a toric variety. This connection to toric geometry provides a homological obstruction to the orientability of a matroid. Moreover, in the case when the polyhedral complex is a non-singular tropical variety, the real part is a PL-manifold. Moreover, for a non-singular tropical variety with a real phase structures we can apply the same spectral sequence for tropical hypersurfaces, obtained by Renaudineau and myself, to bound the Betti numbers of the real part by the dimensions of the tropical homology groups. This is partially based on joint work in progress with Johannes Rau and Arthur Renaudineau.
+ Marvin Hahn Intersecting Psi-Classes on tropical Hassett spaces 07/10/2021 16:00 15-16-413 Jussieu
In this talk, we study the tropical intersection theory of Hassett spaces in genus 0. Hassett spaces are alternative compactifications of the moduli space of curves with n marked points induced by a vector of rational numbers. These spaces have a natural combinatorial analogue in tropical geometry, called tropical Hassett spaces, provided by the Bergman fan of a matroid which parametrises certain n marked graphs. We introduce a notion of Psi-classes on these tropical Hassett spaces and determine their intersection behaviour. In particular, we show that for a large family of rational vectors – namely the so-called heavy/light vectors – the intersection products of Psi-classes of the associated tropical Hassett spaces agree with their algebra-geometric analogue. This talk is based on a joint work with Shiyue Li.
+ Andrei Gabrielov Lipschitz geometry of definable surface germs 02/07/2021 14:00 15-16-413 Jussieu
We study outer Lipschitz geometry of surface germs definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). By the finiteness theorems of Mostowski, Parusinski and Valette, any definable family has finitely many outer Lipschitz equivalence classes. Our goal is classification of definable surface germs with respect to the outer Lipschitz equivalence. The inner Lipschitz classification of definable surface germs was described by Birbrair. The outer Lipschitz classification is much more complicated. There is also a third, even more complicated, ambient Lipschitz classification problem. Some initial results in this were obtained by Birbrair, Brandenbursky and Gabrielov. Using the contact equivalence classification of Lipschitz functions ("pizza decomposition") by Birbrair et al. and the theory of abnormal surface germs ("snakes") by Gabrielov and Souza, we obtain a decomposition of a surface germ into normally embedded Holder triangles, unique up to outer Lipschitz equivalence. This triangulation, with some additional data ("pizza toppings") is a complete discrete invariant of an outer Lipschitz equivalence class of surface germs. Joint work with L. Birbrair, A. Fernandes, R. Mendes and E. Souza (UFC Fortaleza, Brazil).
+ Kirsten Wickelgren An arithmetic count of rational plane curves 06/05/2021 16:00 à distance
There are finitely many degree d rational plane curves passing through 3d-1 points, and over the complex numbers, this number is independent of generally chosen points. For example, there are 12 degree 3 rational curves through 8 points, one conic passing through 5, and one line passing through 2. Over the real numbers, one can obtain a fixed number by weighting real rational curves by their Welschinger invariant, and work of Solomon identifies this invariant with a local degree. It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. We develop and compute an A1-degree, following Morel, of the evaluation map on Kontsevich moduli space to obtain an arithmetic count of rational plane curves, which is valid for any field k of characteristic not 2 or 3. This shows independence of the count on the choice of generally chosen points with fixed residue fields, strengthening a count of Marc Levine. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.
+ Oliver Leigh Towards a geometric proof of Zvonkine's r-ELSV formula 25/03/2021 16:00 à distance
A stable map is said to have "divisible ramification" if the order of every ramification locus is divisible by 𝑟 (a fixed positive integer). In this talk I'll review the theory of stable maps with divisible ramification and discuss how this leads to a geometric framework from which to view and prove Zvonkine's 𝑟-ELSV formula. I will also discuss recent results within this framework.
+ Antoine Toussaint Comparaison des orientations complexes des courbes réelles planes pseudo-holomorphes et algébriques (d'après S. Orevkov) 21/01/2021 16:00 à distance
L'existence de courbes pseudo-holomorphes réelles dans P² dont le schéma complexe n'est pas réalisable par une courbe algébrique du même degré était un problème ouvert jusqu'à ce qu'Orevkov propose une construction de telles courbes en tout degré congru à 9 modulo 12. On présentera la preuve que les schémas induits ne sont pas réalisables par des courbes algébriques, notamment grâce à de nouvelles restrictions sur les orientations complexes d'une courbe algébrique réelle séparante.
+ Marco Castronovo Open Gromov-Witten theory and cluster mutations 14/01/2021 16:00 à distance
The wall-crossing heuristic in open Gromov-Witten theory suggests that disk counts with different Lagrangian boundary conditions should be related by simple transformations with a geometric meaning, but examples are scarce above complex dimension two. I will describe examples of Lagrangian tori in complex Grassmannians whose disk counts are related by mutations of a cluster algebra in the sense of Fomin-Zelevinsky.
+ Renata Picciotto Stable maps with fields to a projective variety 07/01/2021 16:00 à distance
It is well-known that genus zero Gromov-Witten invariants of a subvariety Z⊂X can be recovered, in many cases, from invariants of X by studying obstruction bundles. Unfortunately, this result fails in general for higher genus invariants. The moduli space of stable maps with p-fields was first introduced by Huai-Liang Chang and Jun Li, who proved a comparison theorem relating the count of stable maps with p-fields to projective space to higher genus Gromov-Witten invariants of the quintic threefold. The original construction has since seen various generalizations and applications. I will give some background and discuss a very general version of the construction of stable maps with p-fields and of the comparison theorem.
+ Sergej Monavari Donaldson-Thomas type invariants of Calabi-Yau 4-folds 17/12/2020 16:00 à distance
Classically, Donaldson-Thomas invariants are integer valued invariants that virtually count stable coherent sheaves on Calabi-Yau threefolds. On a Calabi-Yau fourfold, higher obstructions prevent the existence of virtual fundamental classes in the sense of Behrend-Fantechi. Nevertheless, Borisov-Joyce (via derived differential geometry) and Oh-Thomas (via deformation theory) constructed virtual fundamental classes in this setting, modulo choices of orientations. We review their constructions and explain how to define naturally numerical, K-theoretic and torus-equivariant invariants. Finally we discuss how, conjecturally, DT/PT/GW/GV invariants are related to each other and show instances where the conjectures have been checked. This is based on joint work with Y. Cao and M. Kool.
+ Alessandro Giacchetto Geometry of combinatorial moduli spaces and multicurve counts 10/12/2020 16:00 à distance
The Teichmüller space of bordered surfaces can be described via metric ribbon graphs, leading to a natural symplectic structure introduced by Kontsevich in his proof of Witten's conjecture. I will show that many tools of hyperbolic geometry can be adapted to this combinatorial setting, and in particular the existence of Fenchel–Nielsen coordinates that are Darboux. As applications of this set-up, I will present a combinatorial analogue of Mirzakhani's identity, resulting in a completely geometric proof of Witten–Kontsevich recursion, as well as Norbury's recursion for the counting of integral points. I will also describe how to count simple closed geodesics in this setting, and how its asymptotics compute Masur–Veech volumes of the moduli space of quadratic differentials. The talk is based on a joint work with J.E. Andersen, G. Borot, S. Charbonnier, D. Lewański and C. Wheeler.
+ Omid Amini Théorie de Hodge pour les variétés tropicales 2 26/11/2020 16:00 à distance
L'objectif de ces deux exposés est de donner un aperçu de nos travaux sur la théorie de Hodge tropicale. Nous montrons que les groupes de cohomologie des variétés tropicales projectives et lisses vérifient le théorème de Lefschetz difficile et les relations de Hodge-Riemann. Nous donnons une description des groupes de Chow des matroïdes en terme de groupes de cohomologie de certaines variétés tropicales projectives et lisses, nos résultats peuvent donc être considérés comme une généralisation du travail d'Adiprasito-Huh-Katz à des variétés tropicales plus générales. Nous prouvons également que les variétés tropicales projectives et lisses vérifient l'analogue dans le cadre tropical de la conjecture de monodromie-poids, confirmant une conjecture de Mikhalkin et Zharkov.
+ Matthieu Piquerez Théorie de Hodge pour les variétés tropicales 1 19/11/2020 16:00 à distance
L'objectif de ces deux exposés est de donner un aperçu de nos travaux sur la théorie de Hodge tropicale. Nous montrons que les groupes de cohomologie des variétés tropicales projectives et lisses vérifient le théorème de Lefschetz difficile et les relations de Hodge-Riemann. Nous donnons une description des groupes de Chow des matroïdes en terme de groupes de cohomologie de certaines variétés tropicales projectives et lisses, nos résultats peuvent donc être considérés comme une généralisation du travail d'Adiprasito-Huh-Katz à des variétés tropicales plus générales. Nous prouvons également que les variétés tropicales projectives et lisses vérifient l'analogue dans le cadre tropical de la conjecture de monodromie-poids, confirmant une conjecture de Mikhalkin et Zharkov.
+ Tony Yue Yu Secondary fan, theta functions and moduli of Calabi-Yau pairs 15/10/2020 16:00 Zoom
We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth, and $\Theta$ is an ample divisor not containing any 0-stratum of $E$, is \emph{unirational}. More precisely: note that $Q$ has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of $(-1)$-curves, we prove the full conjecture. The reference is arXiv:2008.02299 joint with Hacking and Keel.
+ Dimitri Zvonkine Quantum Hall effect and vector bundles over moduli spaces of curves and Jacobians 08/10/2020 16:00 15-16-413 Jussieu
Vector bundles of so-called Laughlin states were introduced by physicists to study the fractional quantum Hall effect. Their Chern classes are related to measurable physical quantities. We will explain how they are related to the vector bundle of theta-functions over the moduli space and to certain vector bundles over the Jacobians. We perform the first steps in the computation of their Chern classes. Work in progress with Semyon Klevtsov.
+ Elba Garcia-Failde Simple maps, topological recursion and a new ELSV formula 01/10/2020 16:00 15-16-413 Jussieu
We call ordinary maps a certain type of graphs embedded on surfaces, in contrast to fully simple maps, which we introduce as maps with non-intersecting disjoint boundaries. It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR). We propose a combinatorial interpretation of the important and still mysterious symplectic transformation which exchanges $x$ and $y$ in the initial data of the TR (the spectral curve). We give elegant formulas for the disk and cylinder topologies which recover relations already known in the context of free probability. For genus zero we provide an enumerative geometric interpretation of the so-called higher order free cumulants, which suggests the possibility of a general theory of approximate higher order free cumulants taking into account the higher genus amplitudes. We also give a universal relation between fully simple and ordinary maps through double monotone Hurwitz numbers, which can be proved either using matrix models or bijective combinatorics. As a consequence, we obtain an ELSV-like formula for double strictly monotone Hurwitz numbers.
+ Johannes Nicaise Stable rationality of complete intersections (exposé reporté à une date ultérieure) 27/03/2020 10:30 15-16-413 Jussieu

After giving an overview of known results about stable rationality of hypersurfaces, I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder. One technique to construct interesting degenerations is the use of tropical geometry; I will demonstrate this technique by proving the stable irrationality of the quartic fivefold.

+ Grigory Mikhalkin Titre à préciser (exposé reporté à une date ultérieure) 26/03/2020 15:30 15-16-413 Jussieu
+ Massimo Pippi Réalisations motivique et l-adique de la catégorie des singularités d'un modèle LG twisté (exposé reporté à une date ultérieure) 13/03/2020 10:30 15-16-413 Jussieu

Un modèle de Landau-Ginzburg twisté est un couple (X,s), où X est un schéma (sur une base S) et s est une sectionne globale d'un fibré en droites L sur X.
Dans cet exposé, nous allons étudier la réalisation motivique (et l-adique) de la catégorie de singularités attachées à un modèle de Landau-Ginzburg twisté. Pour faire ça, on devra introduire un formalisme de cycles évanescents approprié. Tous ça, ainsi qu'un théorème du a D.Orlov et à J.Burke-M.Walker, nous permettra de calculer la réalisation l-adique de la catégorie des singularités de la fibre spécial d'un schéma régulier sur un anneau noetherien, local régulier de dimension n. Cette formule généralise un résultat du à A.Blanc-M.Robalo-B.Toën-G.Vezzosi, qui a fortement inspiré ce travail.

+ Karim Adiprasito From toric varieties to embedding problems and l^2 vanishing conjectures 27/02/2020 16:00 15-25-502 Jussieu

I will survey a rather intruiging approach to some problems in geometric topology that start by reformulating them as problems in intersection theory. I will start by explaining, on a specific problem, biased pairing theory, which studies the way that the Hodge-Riemann bilinear relation degenerates on an ideal, and review how this limits for instance the complexity of simplicial complex embeddable in a fixed manifold. I will then discuss a conjecture of Singer concerning the vanishing of l^2 cohomology on non-positively curved manifolds, and use biased pairing theory to relate it to Hodge theory on a Hilbert space that arises as the limit of Chow rings of certain complex varieties.

+ Conan Leung Geometry of Maurer-Cartan equation 06/02/2020 15:30 15-25-502 Jussieu

Motivated from Mirror Symmetry near large complex structure limit, a dgBV algebra will be constructed associated to a possibly degenerate Calabi-Yau variety equipped with local thickening data. Using this, we prove unobstructedness of smoothing of degenerated Log CY satisfying Hodge-deRham degeneracy property.

+ 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.

+ 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