Title: Extending the Landau-Ginzburg/Calabi-Yau correspondence to non-Calabi-Yau hypersurfaces in weighted projective space
Abstract: In the early days of mirror symmetry, physicists noticed a remarkable relation between the Calabi-Yau geometry of a hypersurface in projective space defined by a homogenous polynomial W and the singularity theory of the Landau-Ginzburg model with superpotential W. This relation came to be known as the Landau-Ginzburg/Calabi-Yau correspondence. In this talk, I will explain how this correspondence can be extended to non-Calabi-Yau hypersurfaces in weighted projective space using the recently introduced Fan-Jarvis-Ruan-Witten theory as the mathematical formalism behind Landau-Ginzburg models.
Title: Inertia Eigenvalues and Donaldson-Thomas theory
Abstract: I will explain an approach to a conceptual understanding of some of Joyce's work on generalized Donaldson-Thomas invariants. This approach is based on studying the eigenvalue spectrum of the inertia operator on the Grothendieck group of stacks, and is based on ideas of Tom Bridgeland. This is joint work with Pooya Ronagh.
Title: The Donaldson-Thomas theory of K3xE via motivic and toric methods.
Abstract: Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods
Title: Non-concave invariants of Calabi-Yau LG potentials in genus 0 and in genus 1
Abstract: I will present some techniques to determine non-concave invariants in the case of Calabi-Yau Landau-Ginzburg potentials in terms of r-spin invariants. In genus zero the computation is due to Jérémy Guéré, it is systematic and leads to a generating function via the Givental's formalism. In genus one, we illustrate some progress and concrete computations in the case of the Fermat quintic -- this is work in collaboration with Yongbin Ruan.
Title: All genus mirror symmetry for toric Calabi-Yau 3-orbifolds
Abstract: The remodeling conjecture proposed by Bouchard-Klemm-Marino-Pasquetti relates Gromov-Witten invariants of a toric Calabi-Yau 3-manifold/3-orbifold to Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. It can be viewed as a version of all genus open-closed mirror symmetry. In this talk, I will describe results on this conjecture based on joint work with Bohan Fang and Zhengyu Zong.
Title: Quantum cohomology of Hilb(K3,n) and the Igusa cusp form
Abstract: I will present recent results and conjectures on the quantum cohomology of the Hilbert schemes of points of a K3 surface. They relate the counting of rational curves on the Hilbert scheme to Jacobi forms and a Siegel modular form, the Igusa cusp form. This has applications to a GW/Hilb correspondence for K3 surfaces and the Gromov-Witten theory of K3 x E, where E is an elliptic curve.
Tite: Virtual fundamental cycles and implicit atlases
Abstract: I will talk about a method of constructing virtual fundamental cycles on moduli spaces of J-holomorphic maps. The construction uses quite a bit of homological algebra, in particular homotopy colimits and homotopy sheaves, and most of the action happens "at the chain level". I will also mention some applications to existing and conjectural enumerative invariants in symplectic and contact geometry.