Séminaire de Géométrie Enumérative


Contacts :
Penka Georgieva, Elba Garcia-Failde
Ilia Itenberg, Alessandro Chiodo


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
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
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
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
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
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
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
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
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
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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