Khovanov showed, more than 20 years ago, that Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. In these lectures, I will describe two solutions to this problem, which originate in string theory, and which are related by a version of (homological) mirror symmetry. The first approach was recently proven by Ben Webster to be equivalent to his purely algebraic approach to categorification using KRLW algebras. The second approach is completely new and much simpler. The resulting theory describes a precise way in which Chern-Simons (or quantum group) knot invariants derive from a more fundamental theory. It makes many predictions both for algebraic and symplectic geometry, two areas of mathematics connected by mirror symmetry, and for knot theory.
I will describe this story in the course of several lectures, pointing out open problems along the way. The lectures are based on:
M. Aganagic, “Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves,’’ arXiv:2004.14518
M. Aganagic, “Knot Categorification from Mirror Symmetry, Part II: Lagrangians,’’ arXiv:2105.06039
While the subject brings together substantial amount of mathematics and physics, the end result is accessible to graduate students. Correspondingly, the lectures should get more accessible as we go along.