Abstracts
- pdf version
Courses:
Chiu-Chu Melissa Liu,
Gromov-Witten invariants, Fan-Jarvis-Ruan-Witten invariants,
and Mixed-Spin-P fields
Lecture 1: Moduli of stable maps and Gromov-Witten
invariants
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.
Cristina Manolache,
Boundary contributions to enumerative invariants
Contributed
talks:
- 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.