'Knot theory' is the study of closed, embedded curves in
three-dimensional space. Classically, knots can be studied via a
various computable polynomial invariants, such as the Alexander
polynomial. In this first talk, I will recall the basics of knot
theory and the Alexander polynomial, and then move on to a more modern
knot invariant, 'knot Floer homology', a knot invariant with more
algebraic structure associated to a knot. I will describe applications
of knot Floer homology to traditional questions in knot theory, and
sketch its definition. This knot invariant was originally defined in
2003 in joint work with Zoltan Szabo, and independently by Jake
Rasmussen. A combinatorial formulation was given in joint work with
Ciprian Manolescu and Sucharit Sarkar in 2006.