Séminaire de Géométrie Enumérative Contacts : Penka Georgieva Ilia Itenberg Année 2021 - 2022 Temps : Jeudi à 16h Lieu : Jussieu, 1516 - 413 Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université

 Date Orateur Titre et résumé Lieu 07/10/2021 Marvin Hahn, IMJ-PRG Intersecting Psi-Classes on tropical Hassett spaces 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. Jussieu,1516 - 413 18/10/202114h Vivek Shende, Syddansk Universitet Localization of Fukaya categories and quantizing the Hitchin system For a complex curve C and reductive group G, the space of G-bundles on C has been of much interest to many mathematicians. For the purposes of the geometric Langlands correspondence, one wishes to construct certain Hecke eigensheaves' over this space. It has long been expected (and in some cases known) that these should arise from quantization of fibers of Hitchin's integrable system, this being the map h: T*Bun(C, G) --> A which, for G = GL(n), records the spectral curve of a Higgs bundle. Historically this means that one tries to associate a D-module on Bun(C, G) to each fiber of h. More recently, the fact that Langlands dual groups give rise to dual Hitchin fibrations has led to the expectation that geometric Langlands duality should be some sort of homological mirror symmetry. In this talk we will take a step towards making this precise: recent results on the localization of wrapped Fukaya categories allow us to use Floer theory to associate a constructible sheaf on Bun(C, G) to a fiber of the Hitchin fibration. (More precisely, we may do for smooth fibers, in components of Bun(C, G) where there are no strictly semistable Higgs bundles, and should assume G connected center). We don't yet know how to check that we have eigensheaves, but can check some expected properties: our sheaves have the expected endomorphisms, rank, microstalks on certain components, and sheaves from different fibers are orthogonal. Jussieu,1525-502 15/11/202114h Kris Shaw, University of Oslo Real phase structures on matroid fans 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. Jussieu,1525-502 15/11/202115h30 Lucía López de Medrano, UNAM Topologie des variétés tropicales Il a été récemment montré que le nombre de Betti supérieur des variétés tropicales peut dépasser les bornes supérieures de ceux des variétés complexes de même dimension et de même degré. En effet, contrairement aux variétés complexes, les bornes supérieures des premiers nombres de Betti pour les variétés tropicales dépendent également de la codimension. Dans cet exposé, nous rappellerons les constructions maximales connues à ce jour et montrerons que dans le cas des courbes tropicales cubiques, cette construction est maximalement optimale. Travail en commun avec Benoît Bertrand et Erwan Brugallé. Jussieu,1516-413 25/11/2021 Nitin Chidambaram, MPIM Bonn Shifted Witten classes and topological recursion 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. Jussieu,1516 - 413 02/12/2021 Sebastian Nill, Heidelberg University Extended FJRW theory of the quintic threefold in genus zero 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. Jussieu,1516 - 413 09/12/2021 Dimitri Zvonkine, CNRS et Université de Versailles Gromov-Witten invariants of complete intersections 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. Jussieu,1516 - 413 06/01/2022 Thomas Blomme, Université de Genève Enumération de courbes tropicales dans des surfaces abéliennes 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. Jussieu,1516 - 413 27/01/2022 Johannes Nicaise, Imperial College Londres et KU Leuven Variation of stable birational type and bounds for complete intersections 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. Jussieu,1516 - 413 28/01/202214h Alexander Thomas, Institut Max-Planck, Bonn Topological field theories from Hecke algebras 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. Jussieu,1516 - 413 03/02/2022 Dhruv Ranganathan, University of Cambridge Gromov-Witten theory via roots and logarithms 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. Jussieu,1516 - 413 10/02/2022 Séverin Charbonnier, IRIF Statistics of multicurves on combinatorial Teichmüller spaces 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. Jussieu,1516 - 413 17/02/2022 Ilia Zharkov, Kansas State University Lagrangian fibrations of the pair-of-pants The pair-of-pants P is the hypersurface in (ℂ*)n defined by 1+w1+...+wn=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. Jussieu,1516 - 413 24/02/2022 15h30 Kris Shaw, University of Oslo A tropical approach to the enriched count of bitangents to quartic curves 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. Jussieu,1516 - 413 07/04/2022 David Holmes, University of Leiden The double ramification cycle for the universal r-th root 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 r2g. 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. Jussieu,1516 - 413 14/04/2022 Guillaume Chapuy, CNRS et IRIF b-deformed Hurwitz numbers 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 surface. 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. Jussieu,1516 - 413