Mathematics 621, Spring 2018

Differential Geometry

Wednesdays and Fridays, 3:05-4:20, Physics 205

Office hours: Mondays 2:00-3:00, Thursdays 11:00-12:00

I will need to be away several times during the semester. To make up for missed classes, we will be holding class on occasional Mondays 3:05-4:20. Here's the current list of canceled and make-up classes:

Homework assignments

Problem sets will be posted here as they are assigned.

Course notes part 1, 2018 edition; course notes part 2, 2018 edition

for last year's course notes)

Course syllabus (please note: contrary to what's stated there, homework sets will probably be generally due on Fridays)

Course synopsis:

This course is a graduate-level introduction to foundational material in differential geometry. Differential geometry underlies modern treatments of many areas of mathematics and physics, including geometric analysis, topology, gauge theory, general relativity, and string theory. The main topics of study will be organized into two overall sections, differential topology (differential manifolds, vector fields, tensors, differential forms, and vector bundles) and Riemannian geometry (Riemannian metrics, connections, geodesics, curvature, and topological curvature theorems). Additional advanced topics will be considered if time permits.

The textbook for this course is Riemannian Geometry by Manfredo Perdigao do Carmo. As a supplementary source, some of the material covered in the class can be found in Riemannian Geometry by Gallot, Hulin, and Lafontaine, and Smooth Manifolds by Lee.

The grading for this class is based on weekly homework sets and a take-home final exam.

The formal prerequisite for this course is Math 532, which we will mainly use for the Inverse and Implicit Function Theorems. If you haven't taken Math 532, please consult with me to see if Math 621 is appropriate for you. Our course is fast-paced and is likely to be extremely difficult if you haven't at least taken 500-level courses in the past. For some indication of the level of our course, you may want to take a look at the textbook and at my lecture notes from last year's course (the link is here).