Beijing Geometry and Physics Colloquium

Schedule

Pre-talk (for students only)

9:00am-10:00am: Bai-Ling Wang (Australia National University)
Introduction to virtual manifolds and virtual orbifolds

10:00am-10:30am: Coffee and tea break

10:30am-11:30am: Bohui Chen (Sichuan University)
Orbifold and Fredholm systems and the virtual technique

Regular talks (for all audiences)

3:00pm-4:00pm: Bohui Chen (Sichuan University)
Virtual technique for the moduli spaces of pseudo-holomorphic curves

4:00pm-4:30pm Coffee and tea break

4:30pm-5:30pm: Bai-Ling Wang (Australia National University)
Cohomological and K-theoretical Gromov-Witten invariants

Abstracts

Talk1.
Title: Introduction to virtual manifolds and virtual orbifolds
Speaker: Bai-Ling Wang
Abstract: In this first talk, we will explain the basic notion of virtual manifolds and virtual orbifolds. They arise naturally from local perturbations of moduli spaces for many non-linear elliptic partial differential equations. After we define vector bundles and virtual vector bundles, both of which play key roles in the study of moduli spaces, we will explain how to define virtual integrations for virtual manifolds and virtual orbifolds. This talk aims at graduate students with some background in differential geometry and differential topology.

Talk2.
Title: Orbifold Fredholm systems and the virtual technique
Speaker: Bohui Chen
Abstract: Solutions to a non-linear elliptic partial differential equations can be expressed as a zero set of a section of infinite dimensional vector bundle over a Banach manifold or orbifold of Sobolev type. The section is Fredholm in the sense that its linearization is a Fredholm operator between suitable Sobolev spaces. In general, the transversality of this Fredholm section can not be achieved by the so-called cokernel pertubation in the orbifold case. To resolve this, we develop the so-called virtual technique to describe the moduli space as a zero set of a virtual section S of a finite dimensional virtual vector bundle E over a finite dimensional virtual orbifold V. This talk aims at graduate students with some background in differential topology and basic functional analysis.

Talk3.
Title:Virtual technique for the moduli spaces of pseudo-holomorphic curves
Speaker: Bohui Chen
Abstract: The compactified moduli space of pseudo-holomorphic curves of fixed topological type in a symplectic manfiold is quite subtle to study in term of orbifold Fredholm system due to the non-smoothness issues and the stratifications. We shall introduce a new notion of weak orbifolds and get a weak orbifold Fredholm system for this moduli space. The main result of this talk is to show that this moduli space can still be expressed as a zero set of a virtual section S of a finite dimensional virtual vector bundle E over a finite dimensional virtual orbifold V.

Talk 4.
Title: Cohomological and K-theoretical Gromov-Witten invariants
Speaker: Bai-Ling Wang
Abstract: In this talk, we shall apply the virtual integration theory and the virtual index theory for virtual orbfiolds to define the cohomological and K-theoretical Gromov-Witten invariants for any closed symplectic manifold.