Séminaire de Géométrie Enumérative

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,
IMJPRG 
Intersecting PsiClasses 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 Psiclasses on these tropical Hassett spaces and determine their intersection behaviour. In particular, we show that for a large family of rational vectors – namely the socalled heavy/light vectors – the intersection products of Psiclasses of the associated tropical Hassett spaces agree with their algebrageometric analogue. This talk is based on a joint work with Shiyue Li. 
Jussieu, 1516  413 
18/10/2021 14h 
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 Gbundles 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 Dmodule 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, 1525502 
15/11/2021 14h 
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 nonsingular tropical variety, the real part is a PLmanifold. Moreover, for a nonsingular 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, 1525502 
15/11/2021 15h30 
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, 1516413 
25/11/2021 
Nitin Chidambaram, MPIM Bonn 
Shifted Witten classes and topological recursion The Witten rspin class is an example of a cohomological field theory which is not semisimple, but it can be "shifted" to make it semi simple. PandharipandePixtonZvonkine studied the shifted Witten class and computed it explicitly in terms of tautological classes using the GiventalTeleman classification theorem. I will show that the Rmatrix 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 EynardOrantin topological recursion, and discuss some potential applications. This is based on work in progress with S. Charbonnier, A. Giacchetto and E. GarciaFailde. 
Jussieu, 1516  413 
02/12/2021 
Sebastian Nill, Heidelberg University 
Extended FJRW theory of the quintic threefold in genus zero The LandauGinzburg Amodel 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 rspin 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 rspin 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 FanJarvisRuanWitten (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 GrothendieckRiemannRoch formula still provides us with a symplectic transformation of the twisted Givental cone. An extension of the Ifunction will arise in the nonequivariant limit of the twisted invariants. This extended Ifunction contains a new term already known as the semiperiod. It is a solution of an inhomogeneous PicardFuchs equation with a constant inhomogeneity. This is work in progress. 
Jussieu, 1516  413 
09/12/2021 
Dimitri Zvonkine, CNRS et Université de Versailles 
GromovWitten invariants of complete intersections We present an algorithm that allows one to compute all GromovWitten (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 nonstable rationality bounds from hypersurfaces to complete intersections in characteristic zero. 
Jussieu, 1516  413 
28/01/2022 14h 
Alexander Thomas, Institut MaxPlanck, Bonn 
Topological field theories from Hecke algebras We describe a construction which to a surface and a IwahoriHecke 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 noncommutative 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 
GromovWitten 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 GromovWitten 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 cutoff 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 cutoff 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 pairofpants The pairofpants 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. 
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 rth 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 rth root of L. More precisely, for a positive integer r we define the stack of rth roots of L, which is a finite flat cover of S of degree r^{2g}. It carries a universal rth root of L (as a log line bundle), and the locus where this rth root is (logarithmically) trivial defines a lift of DR(L) to the stack of rth 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 
bdeformed 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 oneparameter deformation of Hurwitz numbers, the ``bdeformed Hurwitz numbers''. The GouldenJackson bconjecture from 1996 (and variants) asserts that these numbers are well defined (positive) and have to do with the enumeration of maps on nonoriented surface. I will talk about recent progress towards the conjecture, and other developments related to bdeformed "monotone" Hurwitz numbers and \betaensembles of random matrices. 
Jussieu, 1516  413 