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Séminaire de Géométrie Enumérative
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Année 2022 - 2023 Temps : Vendredi à 14h Lieu : Jussieu, 1516 - 413 Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université |
| Date | Orateur | Titre et résumé | Lieu |
| 23/09/2022 |
Xiaohan Yan,
IMJ-PRG |
Quantum K-theory of flag varieties via non-abelian localization 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). |
Jussieu, 1516 - 413 |
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26/09/2022 14h |
Kendric Schefers, UT Austin |
Microlocal perspective on homology 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. |
Jussieu, 1516 - 413 |
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30/09/2022 |
Ellena Moskovsky, Monash University |
Generalising Narayana polynomials using topological recursion 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. |
Jussieu, 1516 - 413 |
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14/10/2022 |
Maria Yakerson, IMJ-PRG |
On the cohomology of Quot schemes of infinite affine space 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. |
Jussieu, 1516 - 413 |
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21/10/2022 |
Tyler Kelly, University of Birmingham |
Open Mirror Symmetry for Landau-Ginzburg models 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. |
Jussieu, 1516 - 413 |
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07/11/2022 11h |
Alex Degtyarev, Bilkent University |
Lines generate the Picard group of a Fermat surface In 1981, Tetsuji Shioda proved that, for each integer m>0 prime to 6, the 3m^2 lines contained in the Fermat surface Φ_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 Φ_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. |
Jussieu, 1516 - 413 |
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18/11/2022 |
Gabriele Rembado, University of Bonn |
Local wild mapping class groups 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). |
Jussieu, 1525 - 101 |
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25/11/2022 |
Grigory Mikhalkin, Université de Genève |
Enumeration of curves in ellipsoid cobordisms 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. |
Jussieu, 1516 - 413 |
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02/12/2022 |
Adrien Sauvaget, CNRS et Université de Cergy-Pontoise |
On spin GW/Hurwitz correspondence 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. |
Jussieu, 1516 - 413 |
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09/12/2022 |
Denis Auroux, Harvard University |
Fonctions analytiques et homologie de Floer pour les surfaces de Riemann et leurs miroirs 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). |
Jussieu, 1516 - 413 |
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12/12/2022 11h |
Pierre Descombes, LPTHE, Sorbonne Université |
Donaldson-Thomas theory of local Calabi-Yau threefolds Donaldson-Thomas (DT) theory is a modern branch of enumerative and algebraic geometry, taking inspiration from string theory, aiming at counting sheaves on calabi-yau threefolds. I will expose some recent progress on DT theory on crepant resolutions of Calabi-Yau singularities. I will present a toric localization for DT invariants of toric singularities. I will also speak about a recursive procedure called 'attractor flow tree formula' in order to compute DT invariants in terms of initial data called 'attractor DT invariants'. I will in particular present a joint work with Pierrick Bousseau, Bruno Le Floch and Boris Pioline studying this procedure for the case of the local projective plane, and an ongoing work aiming to describe the attractor DT invariants for all crepant resolutions. |
Jussieu, 1516 - 413 |
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13/01/2023 15h |
John Alexander Cruz Morales, Universidad Nacional de Colombia |
An approach to Dubrovin conjecture from GLSM point of view In this talk I will review part of an ongoing project aiming to understand Dubrovin conjecture from the perspective of Gauged Linear Sigma Models (GLSM). One important part of the Dubrovin conjecture relates the geometry of the (bounded) derived category of coherent sheaves of a Fano manifold X and the asymptotic behaviour of its quantum differential equation around infinity. I will explain the role of the hemisphere partition function introduced by Hori and Romo in this story by analysing one simple (but non trivial) case, namely: the CP^{k-1}-model. If time permits, I will also discuss the CP^{k-1}-model with twisted masses (equivariant model). This is a joint work with Jin Chen and Mauricio Romo. |
Jussieu, 1516 - 413 |
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20/01/2023 |
Paolo Rossi, Università degli Studi di Padova |
Meromorphic differentials and integrable hierarchies On the complex projective line, for any configuration of n distinct marked points and n integers whose sum is -2, there is a meromorphic differential, unique up to rescaling, whose zeros and poles coincide with those marked points and have order prescribed by the integers. Working up to the complex 3-dimensional group of projective transformations, this means that the number of meromorphic differentials with prescribed order of zeros and poles is finite if and only if n=3. Since the sum of the orders needs two be -2, only two cases are left: two zeros and one pole or two poles and one zero. These numbers of differentials can be enriched by allowing extra poles, but with the constraint that their residue vanishes. Problem: compute these two families of integer numbers. In joint works with A. Buryak and D. Zvonkine we provide the answer to this problem and a certain higher genus generalization, together with and intriguing relation to integrable systems. |
Jussieu, 1516 - 413 |
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27/01/2023 |
Thomas Blomme, Université de Genève |
Invariants raffinés et polynomialité Dans le plan projectif complexe, il est possible de compter les courbes de degré et genre fixé passant par un nombre de points convenable pour obtenir un nombre qui s’avère ne pas dépendre du choix des points. En géométrie tropicale, I. Itenberg et G. Mikhalkin ont montré qu’il est possible de compter les courbes tropicales solutions du problème analogue avec des multiplicités polynomiales de sorte que le compte polynomial obtenu soit également invariant. Dans cet exposé on s’intéressera à des généralisations de ces invariants polynomiaux dans d’autre surfaces, leur calcul ainsi les propriétés de régularité qui en découlent. |
Jussieu, 1516 - 413 |
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17/02/2023 |
Ailsa Keating, University of Cambridge |
An infinitely generated symplectic mapping class group This talk will focus on a recent joint result with Ivan Smith: there are Stein 3-manifolds whose symplectic mapping class groups cannot be finitely generated. The key example is the so-called conifold smoothing, i.e. the Stein manifold which is the complement of a smooth conic in T* S3; we will carefully describe relevant symplectic features, along with ideas from mirror symmetry used in the proof. Time allowing, we will sketch some consequences on the mirror side. |
Jussieu, 1516 - 413 |
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10/03/2023 |
Paolo Gregori, Institut de Physique Théorique |
Resurgence and large genus asymptotics of intersection numbers In this talk, I will present a new approach to the computation of the large genus asymptotics of intersection numbers of psi-classes, Theta-classes, and of r-spin intersection numbers. This technique is based on a resurgent analysis of the generating functions of such intersection numbers, which are computed via determinantal formulae, and relies heavily on the presence of an underlying first order differential system for each of the problems taken into consideration. With this approach we are able to extend the results of Aggarwal (2021) with the computation of subleading corrections, and to obtain completely new results on r-spin and Theta-class intersection numbers. |
Jussieu, 1516 - 411 |
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17/03/2023 |
Nathan Priddis, Brigham Young University |
BHK mirror symmetry Almost 30 years ago, Berglund and Hübsch proposed a version of Mirror Symmetry for quasihomogeneous potentials, which was later completed by Krawitz. To a Landau-Ginzburg pair (W,G) of a potential W and a group of symmetries G, we can relate its BHK mirror (W‘,G‘) by a simple rule. In this presentation, we will discuss BHK mirror symmetry, its relation with other forms of mirror symmetry, and an extension of BHK mirror symmetry to nonabelian groups. |
Jussieu, 1516 - 413 |
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24/03/2023 |
Amanda Hirschi, University of Cambridge |
Global Kuranishi charts for moduli spaces in symplectic GW theory I will explain the construction of a global Kuranishi chart for moduli spaces of pseudoholomorphic stable maps and show how this allows for a straightforward definition of symplectic GW invariants. Afterwards, I will outline how global Kuranishi charts can be used to prove a product formula for GW invariants and, if time permits, describe a virtual localisation result in the vein of the analogous result by Graber-Pandharipande. |
Jussieu, 1516 - 413 |
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31/03/2023 |
Gleb Koshevoy, IITP RAS |
Maximal green sequences for triangle products The existence of maximal green sequences is an important property of a cluster algebra. We construct explicit maximal green sequences for triangle products of an acylic quiver with a Dynkin quiver. As an application we deduce from the work of Gross-Hacking-Keel-Kontsevich the full Fock-Goncharov conjecture for big double Bruhat cells for simply-connected, connected, semisimple groups of simply-laced type. Maximal green sequences are also useful for computing Donaldson-Thomas invariants and transformations. We compute normalized DT-invariants for triangle products of Dynkin quivers. For products of the Dynkin quivers, the DT-transformations are related to the Robinson-Schensted-Knuth bijection. The talk is based on joint work with V. Genz and work in progress with T. Scrimshaw. |
Jussieu, 1516 - 413 |
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07/04/2023 |
Charles Arnal, INRIA, Sophia Antipolis |
Recursively patchworking real algebraic hypersurfaces with asymptotically large Betti numbers I will present a new technique that builds on previous work by O. Viro and I. Itenberg and allows one to effortlessly define families of real projective algebraic hypersurfaces using already-defined families in lower dimensions as building blocks. The asymptotic (in the degree) Betti numbers of the real parts of the resulting families can then be recovered from the asymptotic Betti numbers of the real parts of the building blocks. Using this technique, I will explain how families of real algebraic hypersurfaces whose real parts have asymptotically large Betti numbers can be constructed in any dimension. The results presented in this talk can also be found in https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/topo.12251. |
Jussieu, 1516 - 413 |
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14/04/2023 |
Renzo Cavalieri, Colorado State University |
A log/tropical take on Hurwitz numbers I will present some recent joint work with Hannah Markwig and Dhruv Ranganathan, in which we interpret double Hurwitz numbers as intersection numbers of the double ramification cycle with a logarithmic boundary class on the moduli space of curves. This approach removes the "need" for a branch morphism and therefore allows the generalization to related enumerative problems on moduli spaces of pluricanonical divisors - which have a natural combinatorial structure coming from their tropical interpretation. |
Jussieu, 1516 - 413 |
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21/04/2023 |
Boris Bychkov, University of Haifa |
Topological recursion for generalized double Hurwitz numbers Topological recursion is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani's hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. In the talk I will define the topological recursion, spectral curves and their invariants, and illustrate it with examples; I will introduce the Fock space formalism which proved to be very efficient for computing TR-invariants for the various classes of Hurwitz-type numbers and I will describe our results on explicit closed algebraic formulas for generating functions of generalized double Hurwitz numbers, and how this allows to prove topological recursion for a wide class of problems. If time permits I'll talk about the implications for the so-called ELSV-type formulas (relating Hurwitz-type numbers to intersection numbers on the moduli spaces of algebraic curves). The talk is based on the series of joint works with P. Dunin-Barkowski, M. Kazarian and S. Shadrin. |
Jussieu, 1516 - 413 |
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16/06/2023 15h |
Guangbo Xu, Texas A&M University |
Arnold conjecture over Z Arnold's conjecture on the numbers of fixed points of Hamiltonian diffeomorphisms on symplectic manifolds has motivated numerous important developments in geometry and topology, most notably the invention of Floer homology. In this talk I will present the recent proof of the integral version of the Arnold conjecture: on any closed symplectic manifold, the number of fixed points of a nondegenerate Hamiltonian diffeomorphism is bounded from below by a version of total Betti number over Z, which takes accounts of torsions of all characteristics. This result strengthens the rational version proved by Fukaya-Ono and Liu-Tian as well as the finite field version proved recently by Abouzaid-Blumberg. The most crucial inputs of the proof are 1) Fukaya-Ono's idea of integral counting of holomorphic curves, the detail of which has been worked out recently by a previous joint work with Shaoyun Bai, and 2) the construction of global Kuranishi charts for the Hamiltonian Floer flow category which generalizes the recent work of Abouzaid-McLean-Smith. This talk is based on the joint work with Shaoyun Bai (arxiv: 2209.08599). |
Jussieu, 1516 - 413 |