Workshop on moduli spaces    

• 21/06/2018.
  Jiaming Chen. Stable cohomology of Satake compactification of $\mathcal{A}_g$ and Tate extension
  Abstract. 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
  Abstract. 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
  Abstract.  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
  Abstract. (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
  Abstract. 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
  Abstract. (see below)
  
  
• 24/11/2017.
  Alexander Polishchuk (room 15.16.417) : Matrix Factorization and Cohomological Field Theories I
  Abstract. 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 (room 15.16.417): Derived Geometry and Gromov-Witten invariants.
  Abstract. 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 ( room 15-16-101 ): All genera quantum Hirzebruch-Riemann-Roch formula in permutation-equivariant quantum K-theory
  Abstract. 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
  Abstract. (see below)
  
  
• 29/09/2017.
  Alexander Polishchuk : Moduli of curves and moduli of A-infinity-structures I
  Abstract. 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
  Abstract. 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.
  Abstract. 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.
  Abstract.