Holomorphic disks and low-dimensional topology

I will discuss recent work with Zoltan Szabo, in which we use techniques from symplectic geometry -- holomorphic disks, and Lagrangian Floer homology -- to construct topological invariants for three- and four-manifolds. These invariants yield many of the results which have been proved using their gauge-theoretic predecessors (Donaldson-Floer and Seiberg-Witten theory), though the new invariants are constructed using more topological and combinatorial input, rendering them easier to calculate. Moreover, they also have applications to classical topological questions which have not been addressed by gauge theory, including the problem of representing lens spaces as surgeries on knots in the three-sphere.