NMAK21002U Cancelled Topics in Geometry

We will generally be concerned with various advanced topics from modern and classical differential geometry, geometric analysis, Riemannian geometry and topology - close to active areas of research - such as: Minimal surfaces, prescribed curvature problems, variational problems in geometry, curvature flows, geometric analytical methods, with the precise contents varying each year and depending on the interests of the participants.

Fall 2023: A tentative plan for this year is to give an introduction to the modern theory of minimal 2-surfaces in 3-manifolds, via excerpts from Colding-Minicozzi's book A Course in Minimal Surfaces (AMS, 2011) plus other relevant notes/articles. Time permitting we will also include some applications to other areas of geometry and topology.

Contents in past years:
Fall 2021: Themed "Interactions Between Topology & Geometric Analysis", the main topics we covered were:
(1) Proof of Hopf's Theorem on uniqueness of the round sphere as the only genus 0 constant mean curvature 2-dimensional closed surface in R³, i.e. "(sometimes) soap bubbles are round" (via the Poincaré-Hopf index of vector fields combined with complex analysis, among other things).
(2) Proof of existence of non-trivial closed geodesic curves on any smooth closed n-dimensional manifold, via so-called min-max variational methods, which make use of some basic algebraic topology (homotopy groups) combined with L²-theory for differential (Euler-Lagrange) equations.

Fall 2020: This year, the course ran informally, and we discussed advanced topics in Mean Curvature Flow, such as gluing problems and uniqueness questions for singularity models in the flow.