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Séminaire de Géométrie Enumérative
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Année 2023 - 2024 Temps : Vendredi à 14h Lieu : Jussieu, 1516 - 413 Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université |
| Date | Orateur | Titre et résumé | Lieu |
| 22/09/2023 |
Kento Osuga,
University of Tokyo |
Refined correspondence between Hurwitz numbers and topological recursion It has been proven that one can construct generating functions of Hurwitz numbers from topological recursion on an appropriate spectral curve. In this talk, I will explore this correspondence in a refined setting. On one hand, we have enumerative invariants of maps onto possibly nonorientable surfaces, and on the other hand, we have refined topological recursion on a refined spectral curve. After reviewing both aspects, I will give a sketch of how to prove their correspondence. If time permits, I will mention how far we can extend, and also mention other applications of the refined topological recursion. This talk is partly based on work in progress with Nitin Chidambaram and Maciej Dolega. |
Jussieu, 1516 - 413 |
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20/10/2023 |
Michele Ancona, Université Côte d'Azur |
Aspects métriques et spectraux des courbes planes aléatoires Toute courbe complexe plane est munie d’une métrique riemannienne induite par la métrique ambiante de Fubini- Study du plan projectif complexe. Nous donnons des bornes inférieures probabilistes sur certaines quantités métriques et spectrales (telles que la systole ou le trou spectral) des courbes planes lorsque celles-ci sont choisies aléatoirement. Il s’agit d’un travail commun avec Damien Gayet. |
Jussieu, 1516 - 413 |
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17/11/2023 |
Francesca Carocci, Université de Genève |
Smooth compactifications of moduli of low genus curves in projective spaces —via log geometry and Gorenstein singularities Moduli spaces of stable maps in higher genus have many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus (the so called main component) does not have a modular interpretation. Constructing modular desingularisations of the main component is strictly related to understanding degenerations of canonical divisors and Gorenstein singularities. I will explain how logarithmic and tropical techniques can be used to solve the problem in genus one and two and say a few words on how the methods used can be adapted to construct modular birational models of pointed curves in low genus. This is based on a joint work with L.Battistella. |
Jussieu, 1516 - 413 |
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30/11/2023 15h |
Grigory Mikhalkin, Université de Genève |
Tropical trigonometry and wave fronts We'll take a look at geometry of angles in the tropical plane by means of the tropical wave front evolution. Resulting caustic produces a subdivision of the tropical angle to the elementary (right) angles. If both bounding rays of the angle are rational then the angle corresponds to a toric surface singularity, and the caustic subdivision corresponds to the minimal toric resolution. Otherwise, the subdivision is infinite. Joint work with Mikhail Shkolnikov. |
Jussieu, 1516 - 413 |
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08/12/2023 |
Gurvan Mével, Nantes Université |
Universal polynomials for coefficients of tropical refined invariant in genus 0 An important application of tropical geometry is Mikhalkin's correspondence theorem. It states that counting algebraic curves on toric surfaces is the same as counting tropical curves with multiplicities. Several multiplicities can be chosen. In particular, the count with the Block-Göttsche multiplicities leads to the tropical refined invariant, which is a polynomial. In this talk we will investigate the polynomial behavior of the coefficients of this invariant. |
Jussieu, 1516 - 413 |
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19/01/2024 |
Thomas Blomme, Université de Neuchâtel |
Asymptotique des invariants raffinés Il y a une dizaine d’années, I. Itenberg et G. Mikhalkin ont montré que le compte des courbes tropicales solutions de certains problèmes énumératifs avec la multiplicité de Block-Göttsche mène à de mystérieux invariants polynomiaux qui ont été au cœur de nombreux travaux depuis. Ceux-ci ont partiellement été démystifiés en les reliant à des invariants réels ou complexes. Plus récemment, E. Brugallé et A. Jaramillo-Puentes ont montré que les coefficients de co-degré fixé de ces polynômes avaient également un comportement polynomial. Des calculs explicites exigeants laissent suggérer une remarquable régularité de ces derniers et envisager une subtile « conjecture de Göttsche duale ». |
Jussieu, 1516 - 413 |
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09/02/2024 |
Sergey Finashin, Middle East Technical University |
Root eigenlattices and lines on real rational elliptic surfaces
In a joint work with V.Kharlamov, out aim is to understand the topology of sections (aka lines) of real rational elliptic surfaces. The main tool is the Mordell-Weil group (formed by automorphisms acting as a group shift in each elliptic fiber). In the complex setting, it can be identified with the lattice E_8, which acts freely and transitively on the set of lines. In the real setting, the real Mordell-Weil groups is identified with the (-1)-eigenlattice of the complex conjugation acting on E_8. It acts similarly on the set of real lines, which becomes a key to understand their topology. |
Jussieu, 1516 - 413 |
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15/03/2024 |
Shaoyun Bai, Columbia University |
Gauged linear sigma model and infinitude of Hamiltonian periodic orbits
Take an irrational rotation of the two-sphere; it only has the north and south poles as its periodic points. However, Franks proved that for any area-preserving diffeomorphism of the two-sphere, if it has more than two fixed points, then it must have infinitely many periodic points. I will discuss a generalization with Guangbo Xu of this result to all compact toric manifolds in the form of a "Betti number or infinity" dichotomy. The Floer theory package from gauged linear sigma models, also known as symplectic vortices, plays quite a surprising role. |
Jussieu, 1516 - 411 |
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22/03/2024 |
Veronica Fantini, IHES |
Invariants des Stokes en P^2 local et modularité
Le modèle local P^2 est un modèle relativement simple, utilisé dans l’étude de la symétrie miroir et de la correspondance TS/ST. En géométrie énumérative, il est naturel de considérer les invariants de Gromov-Witten du modèle P^2 local, dont les fonctions génératrices sont des fonctions quasi-modulaires pour le groupe Gamma_1(3). Dans le cadre de la correspondance TS/ST, il est possible d’introduire d'autres invariants, de nature analytique (ou plus précisément résurgente) appelés invariants de Stokes. Avec C. Rella, nous démontrons que les fonctions génératrices des invariants de Stokes pour le modèle P^2 local sont des formes modulaires quantiques holomorphes pour Gamma_1(3). Grâce à l'étude des propriétés de modularité, nous espérons pouvoi r donner une interprétation géométrique des invariants de Stokes en termes des invariants Gromov-Witten. Dans cet exposé, nous introduirons la définition des invariants de Stokes avant de décrire les propriétés modulaires de leurs fonctions génératrices. |
Jussieu, 1516 - 413 |
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29/03/2024 |
Tudor Padurariu, IMJ-PRG |
BPS invariants on C^3 via matrix factorizations
BPS invariants are virtual counts of semistable sheaves on a Calabi-Yau threefold, related to other enumerative invariants of interest such as Donaldson-Thomas (DT) or Gromov-Witten. In this talk, I will discuss refinements of BPS and DT invariants for the simplest moduli space of sheaves on a Calabi-Yau threefold, that is for moduli of points in C^3. I will first review results about a cohomological refinement and compare them with computations of the cohomology of the Hilbert scheme of points in C^2. I will then discuss results and conjectures about a categorical refinement defined using matrix factorizations. This is joint work with Yukinobu Toda. |
Jussieu, 1516 - 413 |
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05/04/2024 |
Kento Osuga, University of Tokyo |
b-Hurwitz numbers from Whittaker vectors for W-algebras
b-Hurwitz numbers count graphs embedded into possibly non-orientable surfaces weighted by the so-called measure of non-orientability. Similar to the orientable case, it can be shown that their generating function satisfies a b-deformed cut-and-join equation. In this talk, I will uncover an intriguing structure underlying b-Hurwitz numbers, namely, W-algebras of type A. More concretely, I will present how b-Hurwitz numbers and the b-deformed cut-and-join equation arise in a representation of W-algebra. This talk is based on work joint with N. Chidambaram and M. Dolega. |
Jussieu, 1516 - 413 |
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03/05/2024 |
Leo Herr, Universiteit Leiden |
Quantum K theory and log Gromov--Witten theory
Our goal is to count curves satisfying certain conditions. Gromov--Witten theorists have reduced this question to certain calculations in cohomology or Chow groups. When you perform the "same" calculations in K theory, you get different numbers! Called ``quantum K invariants,'' they admit a physics interpretation just like the usual curve counts. We present a basic toolkit before discussing connections with log geometry and surveying open, doable questions. |
Jussieu, 1516 - 413 |
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24/05/2024 14h |
Gaëtan Borot, Humboldt-Universität zu Berlin |
Symmetries of F-cohomological field theories and topological recursion
F-CohFTs are collections of cohomology classes on the moduli space of curves compatible with pinching separating cycles, unlike CohFTs which are compatible with pinching any kind of cycles. F-CohFTs appear naturally when looking at the moduli space of complex curves with compact Jacobian, and by work of Buryak and Rossi they can be used in combination with the geometry of the double ramification cycle to produce integrable hierarchies. Since there are less constraints, the world of F-CohFT is richer that those of CohFTs. I will describe a group of symmetries of F-CohFTs which enlarge the known F-Givental group of Arsie-Buryak-Rossi-Lorenzoni by a large set of linear symmetries. For CohFTs, Dunin-Barkowski, Orantin, Shadrin and Spitz have established a dictonary between (semisimple) CohFTs and topological recursion, building on Givental symmetries and Teleman reconstruction. I will describe the analogue of this dictionary in the F-world (though it is less powerful in absence of Teleman's reconstruction). This is based on a work soon to appear with Alessandro Giacchetto and Giacomo Umer. |
Jussieu, 1516 - 413 |
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24/05/2024 15h30 |
Tyler Kelly, University of Birmingham |
Exoflops for Calabi-Yau complete intersections
Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of many Calabi-Yau varieties in some sense. For example, if the Calabi-Yau is a hypersurface in a smooth projective variety Z cut out by a polynomial f, then one can take X to be the canonical bundle of Z with function W=uf, where u is the bundle coordinate—when the hypersurface is smooth, the critical locus of uf will indeed just be the hypersurface. Exoflops were introduced by Aspinwall as a way to effectively find new birational models of the quotient stack to get new geometries. They effectively create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to Z using geometric invariant theory. We will explain this technique, provide some foundational results about this, and then provide some new applications proven recently for Calabi-Yau varieties with nontrivial scaling symmetry groups. This talk contains results from a series of joint works with David Favero (UMinn), Chuck Doran (Bard/Alberta), and Aimeric Malter (Tokyo). |
Jussieu, 1516 - 413 |
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07/06/2024 |
Arthur Renaudineau, Université de Lille |
Relations entre les invariants de Welschinger de l’ellipsoïde et ceux de l’espace projectif réel de dimension 3
En suivant une idée de Kollar, Brugallé et Georgieva ont montré que les invariants de Welschinger de RP^3 s'expriment en fonction de ceux de l’hyperboloïde (RP^1xRP^1). Dans un travail en commun avec Brugallé, nous montrons que les invariants de Welschinger de l’ellipsoïde (S^2) s'expriment en fonction de ceux de RP^3 (dans le cas ou il y a au moins 2 points réels), et en fonction des invariants de Gromov-Witten de P^3 sinon. La preuve passe par les invariants relatifs et par une formule de récurrence quadratique pour ces derniers. |
Jussieu, 1516 - 101 |
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10/06/2024 17h30 |
Yizhen Zhao, Weizmann Institute |
Open r-spin and FJRW theory via the point insertion technique
Witten (91) defined an intersection theory on the moduli spaces of closed Riemann surfaces with r-spin structures. I will explain how to construct, using a technique called 'point insertion', an intersection theory for holomorphic disks with r-spin structures. This generalizes the previous construction by Buryak-Clader-Tessler and allows us to define more general open Fan-Jarvis-Ruan-Witten theories. If time permits, I will talk about the construction of the intersection theory for cylinders with r-spin structures. Based on joint works with R. Tessler. |
Jussieu, 1525 - 502 |
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18/06/2024 11h |
Xujia Chen, Harvard University |
A product operation on disk fiber bundles, and a configuration space with mouse diagrams
In this talk we will be concerned with smooth, framed fiber bundles whose fibers are the standard d-dimensional disk, trivialized along the boundary. "Kontsevich's characteristic classes" are invariants defined for these bundles: given such a bundle \pi:E \to B, we can associate to it a collection of cohomology classes in H^*(B). On the other hand, there is a "bracket operation" for these bundles defined by Sander Kupers: namely, given two such bundles \pi_1 and \pi_2 as input, we can output a "bracket bundle" [\pi_1,\pi_2]. I will talk about this bracket bundle construction and a formula relating the Kontsevich's classes of [\pi_1,\pi_2] with those of \pi_1 and \pi_2. The main input of the proof is a new but very natural configuration space generalizing the Fulton-MacPherson configuration spaces. This is a work in progress joint with Robin Koytcheff and Sander Kupers. |
Jussieu, 1525 - 502 |