sissa, Trieste, july 3-7 2017

**Chiu-Chu Melissa Liu,
***Gromov-Witten invariants, Fan-Jarvis-Ruan-Witten invariants,
and Mixed-Spin-P fields*

Lecture 2: Stable maps with fields

Lecture 3: Witten's top Chern class and Fan-Jarvis-Ruan-Witten invariants

Lecture 4: Landau-Ginzburg/Calabi-Yau correspondence

Lecture 5: Mixed-Spin-P Fields

Detailed abstract here.

*Anna Barbieri,**A convergence property for a deformation of Joyce generating functions*

Abstract. A generating function f for the generalized Donaldson-Thomas invariants on a (abelian) category was

introduced by Joyce in 2006. It is a continuous and holomorphic formal sum whose coefficients satisfy recursive laws, and it is known to be

well-posed and convergent only in the case of a finite abelian category. One of the open questions in Joyce's article is to study the

convergence of this function and to extend the result to a generic triangulated category.

In a joint work with J.Stoppa, we deform f into a formal power series f_s which is well-defined also in the case of triangulated categories.

We study and prove the convergence of the graded (with respect to the underlying lattice K(C)) components of a deformation of f_s.

- C.J.
Bott, Mirror
Symmetry for K3 Surfaces With Non-symplectic Automorphism

Abstract. Mirror symmetry is the phenomenon originally discovered by physicists that Calabi-Yau manifolds come in dual pairs, each of which produces the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. There are several constructions in different situations for constructing the mirror dual of a Calabi-Yau manifold. It is a natural question to ask: when two different mirror symmetry constructions apply, so they agree?

We consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry, the first inspired by the Landau-Ginzburg/Calabi-Yau correspondence, and the second more classical. In particular, for certain K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani-Boissière-Sarti and Comparin-Lyon-Priddis-Suggs show that they agree when n is prime. We will discuss new techniques needed to solve the problem when n is composite.

- Boris
Bychkov, Degrees
of the strata of Hurwitz spaces

Abstract. Let H_(0;k_1,...,k_m) be the space of meromorphic functions of degree k_1 + . . . + k_m on genus 0 algebraic curve with the numbered multiplicities of the preimages k_1, . . . , k_m of the point ∞ and the zero’s sum of the finite critical values. The closure in P\overbar{H_(0;k_1,...,k_m)} of the set of functions having prescribed ramifications forms the discriminant stratum. The degree of the stratum is the intersection index of its Poincaré dual class with the complementary degree of the first Chern class of the tautological line bundle. I will talk about the certain method of computation of the degrees of the strata of small codimension.

As a consequence we will have a closed formulae for some series of so called double Hurwitz numbers and some new relations on the generating series for integrals of ψ-classes over the moduli space of stable genus 0 curves with marked points. My talk will follow the paper arXiv:1611.00504v1

- Ritwik
Mukherjee, Counting
curves in a linear system with upto eight singular
points

Abstract. Consider a sufficiently ample line bundle L--->X over a compact complex surface X. We obtain an explicit formula for the number of curves in the linear system H^0(X, L) that pass through the appropriate number of generic points, having delta nodes and one singularity of codimension k, provided delta +k<=8.

- Alexis
Roquefeuil, Lagrangian
cone in Gromov--Witten theory

Abstract. We will introduce the notion of Lagrangian cone associated to the potential function of genus 0 Gromov--Witten invariants, as an enrichment of the Frobenius structure associated to the quantum cohomology. We will describe its geometry while relating it to properties of Gromov--Witten theory. We will then compare the cone to the quantum connection/D--module.

*Audrien Sauvaget, Tautological rings of spaces of r-spin structures with effective cycles*

Abstract. The recent developments in the study of moduli spaces of holomorphic differentials have allowed to compute the Poincaré-dual classes of loci of differentials with prescribed singularities. This classes can be expressed using the standard tautological classes of the moduli space of curves. An open conjecture by Pandharipande and Farkas gives closed formulas for these classes in terms of Chiodo's classes of moduli of r-spin structures.

We will explain how this result would allow to describe a family of subrings of the cohomology rings of the moduli spaces of r-spin structures obtained by enriching the classical tautological rings with classes of loci of effective spin structures defined by Polischuk

- Emre
Sertöz, Enumerative
geometry of theta characteristics

Abstract. Deformations of individual theta characteristics have been studied extensively. But geometric questions regarding the relationship of multiple theta characteristics require a different take on the existing moduli spaces. We define the appropriate compactification of multiple spin curves which then allow us to study problems of classical nature pertaining to pairs of theta hyperplanes.

We answer the following questions via divisor computations and degeneration arguments on these compactified moduli spaces: how many fibers of a given one-parameter family of curves admit pairs of theta hyperplanes sharing a common point of contact? If a curve admits a pair of theta hyperplanes sharing a common point of contact, does it admit others? How many points of common contact are there on this curve?

- Jason
van Zelm, Nontautological
bielliptic cycles

Abstract. Tautological classes are geometrically defined classes in the Chow ring of the moduli space of curves which are particularly well understood. The classes of many known geometrically defined loci were proven to be tautological. A bielliptic curve is a curve with a 2-to-1 map to an elliptic curve.

In this talk we will build on an idea of Graber and Pandharipande to show that the closure of the locus of bielliptic curves in the moduli space of stable curves of genus g is non-tautological when g is at least 12.