Introduction to Floer homology. Many very active areas of geometry and topology -- e.g. smooth 3- and 4-dimensional manifolds, knot theory, contact topology, symplectic geometry -- involve invariants that fall under the umbrella of "Floer theory." At its heart, Floer theory is a generalization of the fact that the topology of a smooth manifold can be understood to a significant degree by studying a single, generically chosen real-valued function on the manifold, along with its gradient flow. The course will begin with a discussion of this "classical" situation, with an eye toward generalization appropriate to the infinite-dimensional case of Floer homology. Along the way we will develop some basic geometry and topology of manifolds and introduce the appropriate analytical tools. Later we will focus on particular incarnations of Floer homology, beginning with "Lagrangian intersection Floer homology." Adaptations of this construction give rise to a number of fascinating invariants that we will also touch on: Floer homology for symplectomorphisms (whose development led to the proof of the Arnold conjecture on fixed points of certain self-maps of sympectic manifolds), Heegaard Floer homology (a very powerful invariant of 3-manifolds and of knots), and the "symplectic Khovanov homology" of Seidel and Smith. Additional possible topics include include monopole Floer homology based on the Seiberg-Witten equations, instanton Floer homology, and contact homology. Prerequisites are basic differential and algebraic topology, preferably but not necessarily including cohomology.