# Counting holomorphic disks using bounding chains

In 2003, Welschinger defined invariants of real symplectic

manifolds of complex dimensions 2 and 3, which are related to counts

of pseudo-holomorphic disks with boundary and interior point

constraints (Solomon, 2006). The problem of extending the definition

to higher dimensions remained open until recently (Georgieva, 2013,

and Solomon-Tukachinsky, 2016-17).

In the talk I will give some background on the problem, and describe

a generalization of Welschinger's invariants to higher dimensions,

with boundary and interior constraints, a.k.a. open Gromov-Witten

invariants. This generalization is constructed in the language of

$A_\infty$-algebras and bounding chains, where bounding chains play

the role of boundary point constraints. If time permits, we will

describe equations, a version of the open WDVV equations, which the

resulting invariants satisfy. These equations give rise to recursive

formulae that allow the computation of all invariants of

$\mathbb{C}P^n$ for odd $n$.

This is joint work with Jake Solomon.

No previous knowledge of any of the objects mentioned above will be assumed.

- Last modified: 6/06/2018