Pseudoholomorphic curves with boundary: Can you count them? Can you really?

Members' Seminar
Topic:Pseudoholomorphic curves with boundary: Can you count them? Can you really?
Speaker:Sara Tukachinsky
Affiliation:Member, School of Mathematics
Date:Monday, November 4
Time/Room:2:00pm - 3:00pm/Simonyi Hall 101
Video Link:https://video.ias.edu/members/2019/1104-SaraTukachinsky

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with various constraints on boundary and interior marked points. The presence of boundary poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions.The construction is rather abstract. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called open WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation. For the projective space, this PDE generates recursion relations that allow the computation of all invariants. Moreover, the open WDVV can be reinterpreted as an associativity of a suitable version of a quantum product. No prior knowledge of any of the above notions will be assumed.