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MTH557

Basics

Title: MTH557 - Topics in Differential Geometry

Instructor: Mike Gage

Email: gage@math.rochester.edu

Office: Hylan 1010

Office hours: TBA

Time: 9:30am -- 10:45am Tuesday and Thursday (We may have to change the time if there are too many conflicts. Please e-mail or contact the instructor if you are interested in the course.)

Room: Hylan 1104

Content:

• Comparison theorems in differential equations (fairly easy -- basic geometry and ODE)
• Comparison theorems in geometry (application of the above principals to differential geometry -- in particular how does curvature determine geometry and topology). This subject is less elementary and students benefit by knowing the basic facts about Riemannian Geometry (covariant derivatives, parallel transport) -- such as that contained in the book Riemannian Manifolds by John Lee. Completion of MTH 255 should also be sufficient to benefit from the material.

  I will try to approach this topic so that students without this background, but willing to take certain facts on faith, will learn the basic framework for differential geometry comparison theorems.

• Other topics (to be determined by interests of the class and time) Possibilities are:
  • Applications of comparison theorems to geometric PDE (curve shortening show, possibly the Ricci flow results, depending on student interests and backgrounds)
  • Convex geometry and Finsler geometry -- Extensions of comparison theorems in the context of Finsler geometry.

Classroom details:

• There will be no tests. Some exercises will be suggested and the results either discussed in class or reviewed by the instructor.
• Some of the topics may be run in seminar fashion with volunteers from the class presenting some or all of the topic.
• Auditors are welcome.
• Those taking the course for credit, particularly undergraduates, will need to attend most lectures and actively participate in the class. The kind of participation expected will be adapted to the background of the students.

Texts

There will be no single text for the course. Some reference texts which will be on one day reserve in the library will include:

Hubbard and West -- Differential equations: a dynamical systems approach

V. I. Arnol'd: Ordinary Differential Equations

John Lee: Riemannian Manifolds

Hermann Karcher: Riemannian Comparison Constructions -- article in Global Differential Geometry, edited by Chern