Refined Gromov-Witten invariants

Gromov-Witten theory is a framework for enumerating holomorphic curves in a Kähler manifold X. The case where X is Calabi-Yau of complex dimension three is particularly rich and features connections to seemingly unrelated areas in mathematics and mathematical physics, for instance knot invariants. I will introduce a refinement of Gromov-Witten invariants of Calabi-Yau threefolds as proposed in joint work with A. Brini. I will explain how our proposal formalises certain ideas in mathematical physics and mention several sanity checks our proposal passes. I will discuss applications and relations between our refinement and other developments in enumerative geometry. Special emphasis will be on refined knot invariants where I will comment on current obstructions and possible gains.