Through the years I’ve written several series of notes on topics in differential geometry. Some have been accompaniment for courses I had been teaching, and some have been for lecture series I’ve been invited to give. A few people have told me that making them available could be useful, particularly to graduate students.
These notes were developed to support the first three weeks of a course on Geometric Analysis I taught at Penn in 2015. Later in the course, we went through selected parts of Aubin’s “Some Nonlinear Problems in Riemannian Geometry.”
Penn has a course on differential manifold theory that sits between the undergraduate and graduate levels; this is Math 501/465. I taught this in Spring 2014. The material moves from the theory of curves and surfaces, to Eulerian and Gaussian curvature and the Theorema Egregium, and makes a push toward intrinsic Riemannian geometry, ending with Christoffel symbols and the Riemann tensor.
ICMS Workshop: Summer School on Ricci Curvature — Edinburgh, UK, 2013
In early July 2013, I gave 4 lectures at the ICMS summer school on Ricci curvature, the focus of which was ideas in geometric analysis revolving around epsilon-regularity.
At Penn in 2012-2013 I taught two semesters of Lie algebra theory to graduate students. Partly for my own sake, I wrote up these lecture notes. The first semester was largely based on the book “Introduction to Lie Algebras and Representation Theory” by James E. Humphreys. The second semester drew partly from that source, but from numerous other sources as well.
USTC, Hefei, China 2011
I spent a few weeks on the USTC campus, in Hefei, China, in the summer of 2011, where I was invited to give some lectures in the annual summer school there. I gave 10 lectures on the topics related to F-structures and collapsing Riemannian manifolds.
In 2010 I taught a class at Stony Brook University on F-structures and collapsing. The reason for these notes was that at the time there were very limited resources that collected the research necessary for a class like this, and certainly no textbooks.