Involutive Floer Theory and Surfaces in Stabilized Four-Manifolds

Given a four-manifold with non-vanishing Seiberg-Witten invariants, the adjunction inequality provides a lower bound on the genus of any smoothly embedded surface representing a fixed homology class. Stabilization (that is, taking a connected sum with a product of two 2-spheres) always kills the Seiberg-Witten invariants of a four-manifold, so the traditional adjunction inequality no longer applies. In this talk, I will discuss an invariant for closed, spin four-manifolds arising from involutive Heegaard and monopole Floer theory which can recover information about the genus of embedded surfaces in a wider class of four-manifolds, including certain stabilized four-manifolds. This work is advised by Ciprian Manolescu.