Séminaires : Géométrie énumérative

b-Hurwitz numbers count graphs embedded into possibly non-orientable surfaces. weighted by the so-called measure of non-orient ability. 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.

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.

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 pouvoir 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. 

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.

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.

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 ». 

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.

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.

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 binational models of pointed curves in low genus. This is based on a joint work with L.Battistella.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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).

In 1981, Tetsuji Shioda proved that, for each integer m > 0 prime to 6, the 3m^2 lines contained in the Fermat surface \Phi_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 \Phi_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.

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.

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.

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.

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.

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).

Recently, there has been increasing interest in solving q-difference equations associated to quantum invariants of 3-manifolds as q-series. I will discuss an algorithmic approach to constructing such solutions, studied for example by Dreyfus, and relate such solutions to state integrals of Andersen-Kashaev. Along with exact computations of monodromy, this proves quantum modularity of such solutions in examples and, in particular, the quantum modularity of the q-Borel resummation of the coloured Jones polynomial. This is based on joint work with Garoufalidis, Gu and Mariño.

Rational Gromov-Witten and Welschinger invariants of the plane (or another del Pezo surface) count complex and real rational curves passing through appropriate configuration of points in general position and having nodes as their only singularities. One can count complex curves with non-nodal singularities (cusps etc.) which, under some regularity conditions, always leads to enumerative invariants. In 2006, Welschinger noticed that the count of real plane rational unicuspidal curves does depend on the point constraint, and he suggested a real enumerative invariant counting together the unicuspidal curves, reducible nodal curves, and irreducible curves matching some point and tangency conditions. We address the following question: Does there exist real enumerative invariants counting only rational curves with a given collection of non-nodal singularities and with weights depending only on the topology of the real point set?

Theorem 1. In degree 4, such invariants exist only for curves having either one singular point A_5, or one singular point D_4, or one singular point E_6, or three ordinary cusps, provided that, in the latter case, the constraint consists of four pairs of complex conjugate points. All these invariants are positive.

Theorem 2. For each degree d>4, there exists an enumerative invariant that counts real rational curves with one singularity of order d-1, combined of the transversal local branches of odd orders.

Joint work with O. Bojan.

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 one-parameter deformation of Hurwitz numbers, the ``b-deformed Hurwitz numbers''. The Goulden-Jackson b-conjecture from 1996 (and variants) asserts that these numbers are well defined (positive) and have to do with the enumeration of maps on non-oriented surfaces. I will talk about recent progress towards the conjecture, and other developments related to b-deformed "monotone" Hurwitz numbers and \beta-ensembles of random matrices.

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 r-th root of L. More precisely, for a positive integer r we define the stack of r-th roots of L, which is a finite flat cover of S of degree r^{2g}. It carries a universal r-th root of L (as a log line bundle), and the locus where this r-th root is (logarithmically) trivial defines a lift of DR(L) to the stack of r-th roots. Pixton's formula for DR(L) admits a fairly straightforward lift to this setting. 

After giving an overview of known results about stable rationality of hypersurfaces, I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder. One technique to construct interesting degenerations is the use of tropical geometry; I will demonstrate this technique by proving the stable irrationality of the quartic fivefold. 

Un modèle de Landau-Ginzburg twisté est un couple (X,s), où X est un schéma (sur une base S) et s est une sectionne globale d'un fibré en droites L sur X.
Dans cet exposé, nous allons étudier la réalisation motivique (et l-adique) de la catégorie de singularités attachées à un modèle de Landau-Ginzburg twisté. Pour faire ça, on devra introduire un formalisme de cycles évanescents approprié. Tous ça, ainsi qu'un théorème du a D.Orlov et à J.Burke-M.Walker, nous permettra de calculer la réalisation l-adique de la catégorie des singularités de la fibre spécial d'un schéma régulier sur un anneau noetherien, local régulier de dimension n. Cette formule généralise un résultat du à A.Blanc-M.Robalo-B.Toën-G.Vezzosi, qui a fortement inspiré ce travail.

I will survey a rather intruiging approach to some problems in geometric topology that start by reformulating them as problems in intersection theory. I will start by explaining, on a specific problem, biased pairing theory, which studies the way that the Hodge-Riemann bilinear relation degenerates on an ideal, and review how this limits for instance the complexity of simplicial complex embeddable in a fixed manifold. I will then discuss a conjecture of Singer concerning the vanishing of l^2 cohomology on non-positively curved manifolds, and use biased pairing theory to relate it to Hodge theory on a Hilbert space that arises as the limit of Chow rings of certain complex varieties.

Motivated from Mirror Symmetry near large complex structure limit, a dgBV algebra will be constructed associated to a possibly degenerate Calabi-Yau variety equipped with local thickening data. Using this, we prove unobstructedness of smoothing of degenerated Log CY satisfying Hodge-deRham degeneracy property.

I will present here a correspondence between well-chosen quantum K-theoretical Gromov-Witten invariants of different flag varieties. I will also discuss how this correspondence implies some finiteness properties of the big quantum K-ring of flag varieties.

In a joint work with V. Kharlamov, we suggest a short proof of O. Benoist and O. Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface X of dimension ≥2 the real lines on X generate the whole group H_1(X(ℝ);ℤ/2).

I will present here a correspondence between well-chosen quantum K-theoretical Gromov-Witten invariants of different flag varieties. I will also discuss how this correspondence implies some finiteness properties of the big quantum K-ring of flag varieties.

The celebrated ELSV formula expresses Hurwitz numbers in terms of intersection theory of the moduli space of stable curves. Hurwitz numbers enumerate branched covers of the Riemann sphere with prescribed ramification profiles. 

Since the original ELSV was found, many more ELSV-type formulae appeared in the literature, especially in connection with Eynard-Orantin topological recursion theory. They connect different conditions on the ramification profiles of the Hurwitz problem with the integration of different cohomological classes which have been studied independently. 

We will go through this interplay, focusing on a conjecture proposed by Zvonkine and a conjecture of Goulden, Jackson, and Vakil. In both these conjectures, classes introduced by Chiodo play a key role.

Kummer and Shaw have introduced the separating semigroup Sep(S) of a real curve S. The semigroup is made of topological multidegrees of totally real algebraic maps from S to the Riemann sphere and can be considered in the context of a 1-dimensional version of Hilbert's 16th problem. We'll explore this point of view and classify Sep(S) for curves of genera up to four.

In this talk, I will introduce two enumerative theories coming from a variation of GIT stability condition. One of them is the Gromov-Witten theory of a Calabi-Yau complete intersection; the other one is a theory of a family of isolated singularities fibered over a projective line, which is developed by Fan, Jarvis, and Ruan recently. I will show these two theories are equivalent after analytic continuation. For Calabi-Yau complete intersections of two cubics, I will show that this equivalence is directly related - via Chern character - to the equivalences between the derived category of coherent sheaves and that of matrix factorizations of the singularities. This generalizes Chiodo-Iritani-Ruan's theorem matching Orlov's equivalences and quantum LG/CY correspondence for hypersurfaces. 

We compute the mod 2 cohomology of the real Lagrangians in Calabi--Yau threefolds, using a long exact sequence linking it to the cohomology of the Calabi--Yau. We will describe this sequence explicitly, and as an application will illustrate this computation for the quintic threefold. This is joint work with Thomas Prince and with Bernd Siebert.

Shustin introduced an invariant of del Pezzo surfaces to count real curves of positive genera. By considering the properties of these invariants under Morse transformation, we found a genus decreasing phenomenon for these invariants.  In this talk, we will present a genus decreasing formula for these invariants and discuss possible generalization of it.