Le groupe de travail se tient en général le vendredi matin de 10h à 12h en Salle 15-16-413 (site Jussieu).
21/06/2018. Jiaming Chen : Stable cohomology of Satake compactification of $\mathcal{A}_g$ and Tate extension
Résumé. In 1983, Charney and Lee showed that the rational cohomology of the Satake-Baily-Borel
compactification $\mathcal{A}_g^{bb}$ of $A_g$ stabilizes as g tends to infinity and they computed this stable
cohomology as a Hopf algebra. Their methods are purely topological, which involving Hermitian K-theory and the relation with Borel-Serre compactifiaction. We give a relatively simple algebrogeometric proof
of their theorem and show that this stable cohomology comes with a mixed Hodge
structure of which we determine the Hodge numbers. We find that the mixed Hodge
structure on the primitive cohomology in degrees $4r + 2$ with $r > 1$ is an extension
of $\mathbb{Q}(-2r-1)$ by $\mathbb{Q}(0)$; in particular, it is not pure. (This is a joint work with Eduard Looijenga.)
15/06/2018. Omid Amini : Stacky limit linear series
Résumé. After motivating the question of understanding the limits of linear series in families of curves, I will report on a joint work still in progress with Eduardo Esteves in which we take into account the full combinatorics of the dual graphs in order to construct a space of limit linear series for general families of curves.
I will explain how graphs lead to combinatorially interesting geometric tilings of Euclidean spaces, which in turn lead to an arrangement of toric varieties that we may call toric tilings (and which can be embedded into a somehow exotic infinite dimensional “algebraic variety”). The moduli problem is then described in terms of these toric tilings. Time permitting I will treat the particular case of degenerations of canonical sheaves where other combinatorially interesting features of the approach show up.
08/06/2018. Alexandru Oancea : Théorie de Floer et structures A-infini sur les cônes
Résumé. La première partie de l’exposé sera une discussion générale de la construction de structures A-infini sur des complexes de Floer. Cette construction est sous-jacente à la définition de la catégorie de Fukaya. La deuxième partie de l’exposé expliquera une construction de Abouzaid et Seidel, qui définissent des structures A-infini sur les cônes de certains morphismes de continuation en utilisant des espaces de modules enrichis de façon “opéradique”. Cette construction est sous-jacente à la définition de la catégorie de Fukaya enroulée pour des variétés non-compactes.
01/06/2018. Penka Georgieva : The local real Gromov-Witten theory of curves
Résumé. The local Gromov-Witten theory of curves studied by Bryan and Pandharipande revealed strong structural results for the local GW invariants, which were later used by Ionel and Parker in the proof of the Gopakumar-Vafa conjecture. In this talk I will report on a joint work with Eleny Ionel on the extension of these results to the real setting. Similarly to the classical case, we obtain a complete solution in terms of representation theoretic data using the formalism of an extended Klein TQFT. The local real version of the Gopakumar-Vafa formula is obtained as a corollary.
25/05/2018. Anton Zorich : Masur--Veech volumes, Siegel--Veech constants and intersection numbers of moduli spaces
Résumé. (with V. Delecroix, E. Goujard and P. Zograf) The cotangent space to the moduli space $\cM_{g,n}$ of complex curve of genus $g$ and $n$ marked points can be identified to $\cQ_{g,n}$ the moduli space of pairs $(C,q)$, where $C$ is a complex curve of genus $g$, and $q$ is a meromorphic quadratic differential on $C$ with $n$ simple poles and no other pole. This cotangent space comes with a natural symplectic form and associated volume form called the Masur--Veech volume form. We provide a formula for the volume of the level hypersurface of quadratic differentials of area $1$. We also provide a formula of similar nature for the so called Siegel--Veech constant of $\cQ_{g,n}$.
Both the volume and the Siegel--Veech constant are expressed as polynomials in the intersection numbers of $\psi$-classes supported on the boundary components of the Deligne-Mumford compactification $\overline{\cM}_{g,n}$. The formula obtained in this article are derived from lattice point counting involving the Kontsevich volume polynomials $N_{g,n}(b_1^2, \ldots, b_n^2)$ that also appear in Mirzakhani topological recursion for the Weil--Petersson volumes of the moduli space $\cM_{g,n}$.
18/05/2018. Adrien Sauvaget : A construction of Witten’s r-spin class via log-geometry
Résumé. (This is joint work with Q. Chen, F. Janda, Y. Ruan and D. Zvonkine) We give one more construction of Witten's r-spin class. This construction uses the theory of log Gromov-Witten invariants as developped by Gross-Siebert and Ambramovich-Chen. The main advantage of this construction is that it makes Witten's class available for torus-localization computations.
11/06/2018. Dimitri Zvonkine : A cohomological field theory with nontautological classes
Résumé. We construct the first known example of a cohomological field theory whose values are not limited to tautological classes on the moduli space of curves.
01/12/2017. Alexander Polishchuk : Matrix Factorization and Cohomological Field Theories II
Résumé. (voir en-dessous)
24/11/2017. Alexander Polishchuk (en salle 15.16.417) : Matrix Factorization and Cohomological Field Theories I
Résumé. I will describe the construction (joint with Vaintrob) of the Witten’s virtual class
on the moduli spaces of generalized spin curves, associated with a nondegenerate
quasihomogeneous polynomial. The key feature of the construction is the use of categories of
matrix factorizations, so I will spend a lot of time on the topics related to them.
17/11/2017. Marco Robalo (en salle 15.16.417): Derived Geometry and Gromov-Witten invariants.
Résumé. In this talk I will explain a joint work with Etienne Mann where we use the framework of derived algebraic geometry to understand and extract the axioms of Gromov-Witten theory in the algebraic setting.
10/11/2017. Alexander Givental ( EN SALLE 15-16-101 ): All genera quantum Hirzebruch-Riemann-Roch formula in permutation-equivariant quantum K-theory
Résumé. K-theoretic Gromov-Witten invariants compute holomorphic Euler characteristics of interesting
vector bundles over moduli spaces of stable maps -- pretty much the same way as cohomological
GW-invariants compute intersection numbers between characteristic classes of such bundles.
In the first hour of the talk, I will discuss a construction of K-theoretic GW-invariants storing not only
Euler charactristics of sheaf cohomology, but also the action induced on the sheaf cohomology by renumbering
of marked points. In the second hour, I will outline the adelic quantum mechanical formula which expresses
such permutation-equivariant quantum K-theory in terms of cohomological GW-invariants.
06/10/2017. Alexander Polishchuk : Moduli of curves and moduli of A-infinity-structures II
Résumé. (voir en-dessous)
29/09/2017. Alexander Polishchuk : Moduli of curves and moduli of A-infinity-structures I
Résumé. In these talks I will discuss the connection between the moduli spaces in the title. Considering families of A-infinity algebra structures on a given finite-dimensional graded associative algebra
is a natural moduli problem in the world of noncommutative geometry. I will present a general criterion for the existence of a nice moduli space parametrizing A-infinity structures (assuming vanishing of certain Hochschild cohomology spaces). I will then focus on two classes of examples of such moduli spaces. In the first type of examples the moduli spaces turn out to be isomorphic to certain moduli spaces of (possibly singular) curves. I will sketch an application to the arithmetic homological mirror symmetry for punctured tori. The second example is related to solutions of the associative Yang-Baxter equation and cannot be described purely in terms of moduli of commutative geometric objects. Rather, we get the moduli space of curves equipped with certain noncommutative orders.
13/10/2017. Adrien Sauvaget : Localization by cosection
Résumé. Given a perfect obstruction theory on a DM stack and a section
from the obstruction sheaf to the structure, Kiem and Li construct a cycle
called the localized virtual cycle. Chang, Li, Li and Liu used this
formalism to give a new construction of Witten's top Chern
class of a Landau-Ginzburg model. This construction is natural but
not effective for computations. We will explain how to modify this
formalism to obtain explicit computations.
22/09/2017. Dimitri Zvonkine : Plugging r=0 into the space of r-th roots.
Résumé. Consider a complex curve C endowed with line bundle L whose r-th tensor power is the trivial or the canonical line bundle. The moduli space of pairs (C,L) is a ramified covering of the moduli space Mbar_{g,n} of algebraic curves. It carries several natural cohomology classes whose projections to Mbar_{g,n} turn out to be polynomial in r. We will state several theorems and conjectures that relate the constant term of this polynomial (obtained by plugging r=0) to the Poincaré dual cohomology classes of important geometric loci in Mbar_{g,n}. This is joint work with F. Janda, R. Pandharipande, and A. Pixton.
15/09/2017. Ilia Itenberg : Courbes algébriques réelles finies.
Résumé.
Année 2016-2017
Le groupe de travail pour 2016-2017 porte sur les espaces des modules de structures r-spin et sur la théorie des cycles de double
ramification (cycles DR). L'objectif du groupe de travail est à la fois d'étudier les récents développements dans le domaine et
d'explorer plusieurs conjectures ouvertes: formalisme des mixed P-fields et construction de la classe de Witten par cosection (travaux
de Huai-Liang Chang, Jun Li, Wei-Ping Li and Chiu-Chu Melissa Liu), la formule pour les cycles DR (Janda, Pandharipande, Pixton, Zvonkine)
et les conjecturales généralisations de cette formule pour les espaces des modules de différentielles.
10/03/2017. Paolo Rossi: Hiérarchies de ramification double
Résumé. Dans cet exposé je présenterai les derniers développements dans notre programme de recherche sur les hiérarchies de ramification doubles (DR), en collaboration avec A. Buryak, B. Dubrovin et J. Guéré. Il s'agit d'une nouvelle construction de systèmes intégrables quantiques associés aux théories cohomologiques des champs (CohFT). J'expliquerai la construction et je présenterai quelque application. On conjecture que, dans la limite classique, ces hiérarchies DR soient équivalentes, à changement de coordonnées près, aux hiérarchies de Dubrovin-Zhang, intervenant dans la conjecture de Witten et ses généralisations. Cette conjecture est équivalente à une certaine famille de relations dans les sous-anneau tautologique de la cohomologie de l'espace de modules des courbes stables et on l'a vérifié pour plusieurs COhFT et en genre bas pour toutes les CohFT.
27/05/2016. Adrien Sauvaget et Dimitri Zvonkine: Généralisation des cycles DR pour les espaces de différentielles.
Résumé. Il s'agit d'une introduction générale aux problématiques que l'on souhaite aborder.
Voici le calendrier du groupe de travail que nous avons commencé en 2015 sur la théorie de Gromov-Witten et espaces des modules.
27/05/2016. Dimitri Zvonkine: Polynomiality in r of Witten's r-spin class.
Résumé. Fix n nonnegative integers a_1, ..., a_n whose sum is
equal to 2g-2. Denote by Hol(a_1, ..., a_n) the locus of curves
(C, x_1, ..., x_n) in Mbar_{g,n} such that there exists a
holomorphic differential with zeros at the marke points x_i of
orders exactly a_i. The goal of the talk is to discuss the following
conjecture. The projection of Witten's r-spin class W_{g,n}(a_1,
..., a_n) to the moduli space Mbar_{g,n} is a polynomial in r and
the constant term of this polynomial is Poincaré dual to Hol(a_1,
..., a_n). In an attempt to prove this conjecture we arrived at a
conjectural expression of Witten's class that should allow one to
compute higher genus Gromov-Witten invariants of the quintic.
20/05/2016. Ming Zhang: Tautological relations of moduli space of curves via quasi-modular forms.
The tautological relation of moduli space of curves is a classical and important topic in algebraic geometry. Recently, Pandharipande, Pixton and Zvonkine proved a very general class of tautological relations by studying the shifted Witten class near the discriminant. Pixton further conjectured that these are all the relations. In this talk I will introduce a new approach to obtaining tautological relations on $\overline{M}_{g,n}$ by using the (quasi)modularity of the shifted FJRW CohFT of $x^3+y^3+z^3$. Whether these tautological relations are different from Pixton’s relations is unknown. This is based on my work in progress.
27/11/2015. Penka Georgieva: Mirror symmetry at genus 1 for the quintic 3-fold.
I will describe what is involved in the proof following the works of A. Zinger confirming the BCOV predictions in genus 1.
20/11/2015. Dimitri Zvonkine: Teleman's claffication theorem of semi-simple cohomological field theories.
06/11/2015. Dimitri Zvonkine: Counting real polynomials with fixed ramification profiles.
There are n^{n−3} (properly normalized) complex degree n polynomials with n−1 fixed critical values. This can be found by establishing a one-to-one correspondence between these polynomials and marked trees, which are enumerated by the Cayley formula. The number of (properly normalized) real degree n polynomials with n − 1 fixed real critical values is equal to the n-th Euler-Bernoulli number. This can be found by establishing a one-to-one correspondence between these polynomials and alternating permutations. The problem above can be generalized by allowing multiple critical values and fixing their ramification profiles. In the complex case this problem is solved; in the real case, however, the answer depends on the order of the critical values on the real line. Thus the question arises whether it is possible to attribute a sign to every real polynomial in such a way that the number of polynomials counted with signs is invariant under permutations of critical values. We construct a sign with this property and study the invariant thus obtained.
23/10/2015.
Inonut Ciocan-Fontaine : Quasi-maps III.
16/10/2015. Inout Ciocan-Fontaine : Quasi-maps II.
09/10/2015 Inout Ciocan-Fontaine : Quasi-maps I.
Résumé. In a series a 3 talks I will introudce this different compactification of spaces of curves.
25/09/2015. Andreas Gerstenberger, Symplectic hypersurfaces and transversality.
Résumé. In this talk I will give a short overview of a construction technique for a Gromov-Witten pseudocycle as introduced in symplectic geometry by Cieliebak-Mohnke. The moduli spaces considered consist of holomorphic curves defined on surfaces with additional marked points, which are constrained to be mapped to an almost complex hypersurface which is Poincare dual to a large multiple of the symplectic form (which exists by a result of Donaldson). This moduli space can be shown to have a compactification which is comprised of holomorphic curves defined on stable surfaces, which allows for application of standard geometric transversality methods to obtain a well defined pseudocycle.
18/09/2015. Valentin Tonita : Quantum K theory.
Résumé. For a project manifold X, let X_{0,n,d} be the moduli spaces of (genus 0, degree d ) stable maps to X.
I will define both "ordinary" and permutation equivariant K-theoretic Gromov-Witten invariants (recently
introduced by A. Givental) as holomorphic Euler characteristics on X_{0,n,d}, respectively X_{0,n,d}/S_n .
I will characterize their generating series in terms of the cohomological Gromov-Witten theory of X. Time permitting I will explain recent progress in the permutation-equivariant theory: use of localization for
K -theoretic I functions of toric manifolds, twisted K-theoretic
Gromov-Witten invariants etc.
11/09/2015. Adrien Sauvaget. Classes de cohomologie de strates dans l'espace des différentielles.
Résumé. Soit g un entier. Le fibré de Hodge H_g est l'espace de module des paires (courbe de genre g, différentielle holomorphe). C'est un fibré au dessus de l'espace des Modules de courbes M_g. Cette espace est stratifié par des sous-espaces H_g(m_1,...,m_n) où les m_i vérifient m_1+...+m_n=2g-2, des paires (courbe, différentielleavec des zéros d'ordre m_i). L'exposé portera sur la compactification de ces strates et leur classe de cohomologie.
Année 2014-2015.
19/02/2015. Adrien Sauvaget, introduction générale à la théorie des espaces des
modules sous divers points de vue et au formalisme des classes tautologiques (Notes de l'exposé pdf).
26/02/2015. Mattia Galeotti, courbes de r-spin, morphisme naturel de revêtement,
classification des singularités (Notes de l'exposé pdf).
12/03/2015. Ivan Bazhov, comptage des courbes dans les surfaces K3.
26/03/2015. Adrien Sauvaget, nombres d'Hurwitz, Formule ELSV, Toda Hierarchy.
09/04/2015. Malick Camara, relations FZ, courbes de r-spin, théorie des champs cohomologique.
Adrien SAUVAGET
adrien.sauvaget@math.cnrs.fr
Laboratoire AGM, bureau 502
Université de Cergy Pontoise
2 avenue Adolphe Chauvin,
95302 Cergy-Pontoise Cedex (France)