Rigidity of cotangent fibers of Riemann surfaces

I will discuss joint work with Georgios Dimitroglou Rizell which studies symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our work is motivated by analogies to Arnold’s celebrated Nearby Lagrangian Conjecture and by work of Ekholm and Smith in higher dimensions. We crucially use several special properties of pseudo-holomorphic curves in dimension 4 as well as some fairly recent developments in the analysis of punctured pseudo-holomorphic curves. I will review these developments, highlighting in particular why certain pseudo-holomorphic curve techniques are particularly effective in dimension 4.