Instanton and Heegaard Floer homologies of surgeries on torus knots

A Floer homology is an invariant of a closed, oriented 3-manifold $Y$ that arises as the homology of a chain complex whose generators are either the set of solutions to a differential equation or the intersection points between Lagrangian manifold, and its differential arises as the count of solutions of a differential equation on $Y \times \mathbb{R}$. The Instanton Floer chain complex is generated by flat connections on a principal $SU(2)$-bundle, and the differential counts solutions to the Yang-Mills equation (known as instantons). The Heegaard Floer chain complex is generated by the intersection points of curves in a Heegaard diagram for $Y$ and its differential counts solutions to the Cauchy-Riemann equation (known as pseudoholomorphic Whitney discs). In the talk I will show that these invariants are the same when the 3-manifold is integral surgery on $S^3$ along a torus knot. This is joint work with Tye Lidman and Christopher Scaduto.