Knot homologies and applications to low-dimensional topology
Roughly speaking, low dimensional topology is the study of 3-dimensional manifolds and the 4-dimensional cobordisms between them. Low dimensional topologists also like to study knots, i.e., smoothly imbedded circles in 3-manifolds (up to ambient isotopy), because of a classical theorem of Lickorish-Wallace: Every closed, connected, oriented (c.c.o.) 3-manifold can be obtained from the three-sphere (the simplest c.c.o. 3-maniofld) by doing surgery on a finite collection of knots. A corollary of the Lickorish-Wallace theorem is that any c.c.o. manifold is the boundary of some 4-manifold.
Yet knots are remarkably tricky to study directly; it is difficult to tell, just by staring at pictures of two knots, whether they are the same or different. We confront this problem through the use of knot invariants, algebraic objects associated to knots that do not depend upon how the knots are drawn. I will discuss a couple of these: Khovanov homology and knot Floer homology, both inspired by ideas in physics. In the less than ten years since their introduction, they have generated a flurry of activity and a stunning array of applications. There are also intriguing connections between the two theories that have yet to be fully understood.