The question of which symplectic 4-manifolds are complex projective surfaces reduces in principle (via branched covering constructions) to the question of which symplectic curves in the complex projective plane CP^2 are isotopic to algebraic curves - the latter is known as the symplectic isotopy problem. 



The longstanding symplectic isotopy conjecture posits that every smooth symplectic curve in CP^2 is isotopic to an algebraic curve. In this talk, I will describe a new algebraic theory I have developed on the braid groups in order to prove that all degree three symplectic curves in CP^2 with only A_n-singularities (an A_n-singularity is locally modelled by w^2 = z^n and includes nodes and cusps) are isotopic to algebraic curves. The proof is independent of Gromov's theory of pseudoholomorphic curves, and the theory also addresses the symplectic isotopy conjecture in full generality in upcoming work.

I will review the necessary background from scratch, and along the way, we will discuss beautiful ideas from algebraic geometry, symplectic geometry, monodromy theory and geometric group theory and how they unite in the study of plane curves and 4-manifolds.