(1) The coordinate functions on a free boundary minimal surface (FBMS) in the unit
ball B^n are Steklov eigenfunctions with eigenvalue 1. For many embedded FBMS in B^3 we show its first Steklov eigenspace coincides with the span of its coordinate functions, affirming a conjecture of Fraser & Li in an even stronger form. One corollary is a partial resolution of the Fraser-Schoen conjecture: the critical
catenoid is the unique embedded FBM annulus in B^3 with antipodal symmetry. This is joint work with Peter McGrath.
(2) In this talk, we discuss recently developed techniques from Khovanov homology used to exhibit pairs of exotic Seifert surfaces in the 4-ball, as well as potential applications of these techniques toward producing an infinite family of exotic slice disks bounding a common knot. This is joint work with Kyle Hayden, Seungwon Kim, Maggie Miller, and JungHwan Park.