MATH 9161, Fall 2017 : Differential Geometry

Tentative Schedule: Wednesday and Friday, 13h00 - 14h30 in MC 107. Note that this may still change in case of time conflicts with other courses or seminars.

First Meeting: Friday September 8 in MC 107.

Prerequisites: Calculus on manifolds and basic algebraic topology.

Textbook: We will follow Peter Michor's excellent book "Topics in Differential Geometry" published by the AMS. This book is available on P. Michor's website. Please have a look to get a better idea of the course. You can also find a review here.

Course outline:

The aim of this course is to provide an introduction to differential geometry with an emphasis on various structures on manifolds. We will start with a review of foundational material (smooth manifolds and maps, derivatives, flows, the Inverse function theorem and its corollaries, Frobenius theorem, etc). Then, we will cover vector bundles and tensor calculus from both a local and a global point of view. This will allow us to study geometric structures (Riemannian, complex, symplectic, etc) in a unified framework. Throughout the course, we will present many examples. If time permits, specific topics may be covered to accommodate students interests and goals. This may include Chern-Weil theory of characteristic classes, more about Lie groups, symmetric spaces, etc.

Evaluation: Homeworks 30%, Midterm 30%, Written project 25%, Oral presentation 15%.

The complete pdf outline will be available here later.

Other References: I encourage each student to review the material with the help of some standard text. Here is a list of some of my favorite references:

  • "Introduction to Smooth Manifolds" by John Lee.
    Covers the basics of manifolds with many examples. A good reference.
  • "Manifolds and Differential Geometry" by Jeffrey Lee.
    Similar to Jeffrey Lee's book with additional topics like connections and Riemannian Geometry. A very good reference for this course.
  • "Foundations of Differential Manifolds and Lie Groups" by F. Warner.
    More formal than the above references. Gives very few concrete examples, but covers sheaf cohomology.
  • "Semi-Riemannian Geometry" by B. O'Neil.
    Covers (Semi)-Riemannian Geometry with applications to General Relativity.
  • "Riemannian Geometry and Geometric Analysis'' by J. Jost.
    Very good selection of topics. Nice discussion of some important differential operators.
  • "Curvature and Characteristic Classes" by J. Dupont.
    Introduction to classical Chern-Weil theory of characteristic classes.
  • "Characteristic Classes" by J. Milnor and J. Stasheff.
    A classic on characteristic classes on manifolds.
  • "Differential and Riemannian Manifolds", by Serge Lang
    More advanced, it is one of the few textbooks that covers infinite dimensional manifolds.

Announcements

  • The first meeting is Friday September 8, 2017 in MC 107.


Policies & Regulations:

Medical Accommodation: If homework is missed and sufficient documentation is provided, the homework can be handed in later. If an exam is missed and sufficient documentation is provided, a make-up exam will be offered.

Failure to follow these rules may result in a grade of zero.

Scholastic offences: Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at the following Web site:http://www.uwo.ca/univsec/handbook/appeals/scholastic_discipline_grad.pdf

Accessibility: Please contact the course instructor if you require material in an alternate format or if you require any other arrangements to make this course more accessible to you. You may also wish to contact Services for Students with Disabilities (SSD) at 519-661-2111 x82147 for any specific question regarding an accommodation.

Support Services: Learning-skills counsellors at the Student Development Centre are ready to help you improve your learning skills. Students who are in emotional/mental distress should refer to Mental Health@Western for a complete list of options about how to obtain help. Additional student-run support services are offered by the USC. The website for Registrarial Services is http://www.registrar.uwo.ca.

Last Updated: 2017-09-04