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Séminaire de Géométrie Enumérative
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Année 2020 - 2021 Temps : Jeudi à 16h Lieu : Jussieu, 1516 - 413 Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université |
| Date | Orateur | Titre et résumé | Lieu |
| 01/10/2020 |
Elba Garcia-Failde,
Institut de Physique Théorique of Paris-Saclay et IHES |
Simple maps, topological recursion and a new ELSV formula 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. |
Jussieu, 1516 - 413 |
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08/10/2020 |
Dimitri Zvonkine, Laboratoire de Mathématiques de Versailles et CNRS |
Quantum Hall effect and vector bundles over moduli spaces of curves and Jacobians 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. |
Jussieu, 1516 - 413 |
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15/10/2020 |
Tony Yue Yu, Laboratoire de Mathématiques d'Orsay et CNRS |
Secondary fan, theta functions and moduli of Calabi-Yau pairs We conjecture that any connected component Q of the moduli space of triples ( X , E = E 1 + ⋯ + E n , Θ ) 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 Θ 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. |
Zoom et Jussieu, 1516 - 413 |
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19/11/2020 |
Matthieu Piquerez, École polytechnique |
Théorie de Hodge pour les variétés tropicales 1 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. |
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26/11/2020 |
Omid Amini, École polytechnique et CNRS |
Théorie de Hodge pour les variétés tropicales 2 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. |
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10/12/2020 |
Alessandro Giacchetto, MPIM Bonn |
Geometry of combinatorial moduli spaces and multicurve counts 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. |
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17/12/2020 |
Sergej Monavari, Universiteit Utrecht |
Donaldson-Thomas type invariants of Calabi-Yau 4-folds 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. |
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07/01/2021 |
Renata Picciotto, Columbia University |
Stable maps with fields to a projective variety 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. |
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14/01/2021 |
Marco Castronovo, Rutgers University |
Open Gromov-Witten theory and cluster mutations 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. |
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21/01/2021 |
Antoine Toussaint, IMJ-PRG |
Comparaison des orientations complexes des courbes réelles planes pseudo-holomorphes et algébriques (d’après S. Orevkov) 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. |
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25/03/2021 |
Oliver Leigh, Stockholm University |
Towards a geometric proof of Zvonkine's r-ELSV formula 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. |
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06/05/2021 |
Kirsten Wickelgren, Duke University |
An arithmetic count of rational plane curves 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. |
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02/07/2021 14h |
Andrei Gabrielov, Purdue University |
Lipschitz geometry of definable surface germs 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). |
Jussieu, 1516 - 413 |