Northern California Symplectic Geometry Seminar: Orientations of moduli spaces for Legendrian contact homology/Real Gromov-Witten theory in all genera

Seminar | April 3 | 2:30-5:10 p.m. | 939 Evans Hall

 Cecilia Karlsson/Penka Georgieva, Stanford/Pierre et Marie Curie

 Department of Mathematics

2:30 PM Karlsson) : I will give an introduction to Legendrian contact homology, which is an invariant of Legendrian submanifolds that is defined by using pseudo-holomorphic disk techniques. In particular, I will explain how one can define this homology with integer coefficients by orienting the moduli space of the pseudo-holomorphic disks. I will also discuss how one can make this invariant easier to compute by replacing the pseudo-holomorphic disks with gradient flow trees, and how the moduli space of these trees can be oriented in a computable way.

3:30-4:00 PM: Tea Break in 1015 Evans Hall

4:10 PM (Georgieva): We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces. (A symmetric surface is a surface with an orientation reversing involution, and a real bundle pair is a complex bundle with a conjugation lifting the involution on the base.) This allows us to endow the uncompactified moduli spaces of real J-holomorphic maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces. In reasonably regular cases, these invariants can be used to obtain lower bounds for counts of real curves of arbitrary genus. This is joint work with A. Zinger.

There will be a dinner at 6:00 PM

D. Auroux, Y. Eliashberg, D. Fuchs, V. Ginzburg, M. Hutchings, E. Ionel, R. Montgomery, K. Wehrheim, A. Weinstein