Expository notes:

• Minimal surfaces and comparison geometry

  Submitted to Proceedings of the ICM (2026). [arXiv:2510.04481]

• Introduction to minimal surfaces (Winter 2025)

  Notes from a course (Math 286) given at Stanford. Last updated September 15, 2025.

• The Morse index of a minimal surface (Winter 2024)

  Notes from a minicourse given at the 5th Taiwan International Conference on Geometry at NTU. Last updated July 31, 2025

• The Bernstein problem, generalizations, and applications (Summer 2023)

  Notes from a minicourse given at UCL. Last updated July 14, 2023.

• Stable minimal surfaces and positive scalar curvature (Fall 2021)

  Notes from a course (Math 258) given at Stanford. Last updated December 5, 2021.

• Recent results concerning topological obstructions to positive scalar curvature (with Chao Li)

  Perspectives in Scalar Curvature, Ch. 6, World Scientific Publishing Company, vol. 2, pp. 215–230 (2023). [journal] [MR 4577915]

• The Morse index of a minimal surface (with Davi Maximo)

  Notices Amer. Math. Soc. vol. 68, no. 6, pp. 892–898 (2021) [MR 4270401]

• Some new generic regularity results for minimal surfaces and mean curvature flows (Spring 2021)

  Notes for talks given in the 2nd geometric analysis festival. Last updated May 2021.

• Geometric features of the Allen–Cahn equation (Summer 2019)

  Notes from a course I gave at a Princeton RTG summer school. Last updated June 29, 2023. See also an earlier version from a 2018 UConn summer school.

• Minimal surfaces and scalar curvature (Summer 2019)

  Notes from a mini-course I gave at a 2019 CIMAT (in Guanajuato, Mexico) conference on scalar curvature and relativity. Last updated February 19, 2020.

• Topics in mean curvature flow (Taught by Brian White, Spring 2015)

  Notes from Brian White's course on mean curvature flow. Last updated (typos corrected): February 18, 2022.

• Ricci flow (Taught by Richard Bamler, Winter 2014)

  Notes from Richard Bamler's course on Ricci flow. These were a joint effort with Christos Mantoulidis. Probably in a final state (missing the last two lectures). Last updated: June 8, 2015.

• Topics in minimal surfaces (Taught by Brian White, Spring 2013)

  Notes from Brian White's course on minimal surfaces. These were a joint effort with Christos Mantoulidis. Last updated (typos corrected): August 14, 2023.

• Notes on linearization stability (Winter 2013)

  Notes for several (expository) talks I gave in Rafe Mazzeo's course "Classics in Analysis." The notes are on linearization stability of scalar curvature and the Einstein equations in the sense of Fischer–Marsden. Last updated: March 8, 2013.

• Geometric measure theory (Taught by Brian White, Spring 2012)

  Notes from Brian White's course on flat chains. Last updated: November 6, 2014.