S2 Topics in Differential Geometry and Geometric Analysis

As a new option in 2019/2020, we shall offer an additional sequence of geometry topics, focused on differential geometry and geometric analysis.  The lectures will take place at UCL on Thursday from 10-12 in Physics D3 (1st floor), so they will not clash with the Friday Geometry Topics at Imperial College.

The first-year cohort will be consulted at the end of November 2019 to gauge interest in this option and to finalize the topics to be covered.  A provisional list of topics (in no particular order) is as follows:

  • Principles of PDE on manifolds
  • Einstein manifolds
  • Geometric flows
  • Minimal surfaces
  • Geometric measure theory
  • Comparison theorems in Riemannian geometry
  • Isoperimetric and Sobolev inequalities

There will not be enough time to cover these all, but depending on interest and student background, some of these could be studied in the junior geometry seminar.

The pattern will follow that of the successful Topics in Geometry Course, and so for each of the 5 topics covered, we shall need an owner (one of the students on the course) and a wrap-up leader (a senior student or post-doc).

Most relevant S1 topics are: Forms and currents, connections and curvature, the Kaehler condition, and Hodge Theory - these will be assumed as prerequisites for Topics in DGGA.

The propaganda for the course is similar to that of Geometry Topics.  In addition to giving you a broad overview of current research topics, we hope this course will emphasis some of the common themes and techniques in this field, while also giving an introduction to the different available techniques.

One unifying idea which we shall try to bring out through the course is the relation between linear and nonlinear phemonena: many geometric problems lead to nonlinear PDE.  Some of these PDE can be very successfully as perturbations of a linear problem: in others nonlinear effects are to the fore and completely different techniques are required.