Knots-quivers correspondence

I will present a surprising relation between knot invariants and quiver representation theory, motivated
by various string theory constructions involving BPS states. Consequences of this relation include the
proof of the famous Labastida-Marino-Ooguri-Vafa conjecture (at least for symmetric representations),
explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new
viewpoint on knot homologies, a novel type of categorification, new dualities between quivers,
and many others.