Eigenvalue problems have played a pivotal role in the study of minimal surface theory. In this talk, I will provide an overview, starting with basic concepts and progressing to recent development. Specifically, I will explain free boundary minimal surfaces in the unit ball and their connection to eigenvalue problems. Key conjectures in this area will be highlighted, along with a discussion of my recent contribution.

We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstnerič and Kalaj, we prove the boundary Schwarz lemma for such minimal disks.

Can large language models improve feedback on undergraduate proof writing without lowering standards? I present a practical way to guide an LLM so its comments are focused, consistent, and aligned with course aims—while academic judgement stays with the instructor. In a pilot with first-year work, this approach surfaced common misconceptions early and shortened feedback cycles. I’ll outline the idea, note what helped and what failed, and discuss where AI can responsibly sit alongside human marking.