  In these notes, I will sketch a new approach to Khovanov homology of knots
and links based on counting the solutions of certain elliptic partial
differential equations in four and five dimensions. The equations are
formulated on four and five-dimensional manifolds with boundary, with a rather
subtle boundary condition that encodes the knots and links. The construction is
formally analogous to Floer and Donaldson theory in three and four dimensions.
It was discovered using quantum field theory arguments but can be described and
understood purely in terms of classical gauge theory. (Based on a lecture at
the conference Low-Dimensional Manifolds and High-Dimensional Categories,
University of California at Berkeley, June 2011).
