Knots, quivers, and combinatorics

I will present a recently found surprising relation between knot invariants and quiver representation theory, that we refer to as the knots-quivers correspondence. Consequences of this relation include the proof of one of the famous Labastida-Marino-Ooguri-Vafa integrality conjecture of BPS invariants, explicit (and unknown before) formulas for colored HOMFLY-PT polynomials for various knots, new viewpoint on knot homologies, a novel type of categorification, new dualities between quivers, solutions to certain combinatorial problems, and many others