Instructor
Professor Shelly Harvey
Herman Brown 446
shelly@rice.edu

Course Information
Class meets: TR 1-2:15pm in HB 423 (the first two weeks of class will be on Zoom. See Canvas for the Zoom link.
Office Hours: Mondays 3-4pm, Fridays 1:30-2:30pm, directly after class, or by appointment.
Webpage: http://math.rice.edu/~shelly/543s22/

Required textbook
Characteristic Classes by Milnor and Stasheff

Prerequisites
You should have taken Math 444/539. Specifically, knowledge of smooth manifolds and tangent spaces will be mostly assumed (though there will be a brief recap at the beginning of the semester). Milnor’s Topology from a Differentiable Viewpoint is a good reference for this material. You should have taken or be concurrently enrolled in Math 445/540.

Course objectives and learning outcomes
This class is an important foundational course in bundles and characteristic classes. Any graduate student who is interested in topology, geometry, algebraic geometry, or number theory should take it. I would encourage first year students to take the course as it is only offered every other year.

In the smooth and other categories, much of the fine structure of manifolds and varieties is encoded in the tangent bundle or other associated bundles. In this course, we will cover the following topics.

1. A brief summary of smooth manifolds and tangent spaces.

2. Basics of vector (and fibre) bundles, building towards characteristic classes (Steifel–Whitney, Euler, Chern, Pontryagin).

3. Existence of exotic smooth structures on S7 (time-permitting).

By the end of the course, the student will have a thorough understanding of the material presented in class.

Homework and in class participation
I will assign homework problems on occasion. You are encouraged to do the problems that you can and turn them in; this is the best way to learn a subject. If you turn them in, I will look at them and give you feedback. In addition, we will have in class activities. You are required to participate in these as much as you can with your mathematical background (there may some homework problems or activities that first year students may not have the background to do). There will be no exams.

Exams
There will be no exams.

Grades
Students are required to attend every lecture and participate in the in class activities (unless they have permission from the instructor to miss). Your grade is completely determined by attendance. In order to get an A, you must miss fewer than 6 lectures. If you miss 7-10 lectures, you get a B. If you miss 11-14 lectures, you get a C. If you miss 15-18 lectures, you get a D. If you miss more than 18 lectures, you get an F.

Statement of Conduct
The Department of Mathematics supports an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. We expect all participants in mathematics courses (students and faculty alike) to treat each other with courtesy and respect, and to adhere to the mathematics department standards of collegiality, respect, and sensitivity: mathweb.rice.edu/department-statement-collegiality-respect-and-sensitivity as well as the Rice Student Code of Conduct. If you think you have experienced or witnessed unprofessional or antagonistic behavior, then the matter should be brought to the attention of the instructor and/or department chair. The Ombudsperson is also available as an intermediate, informal option, and contacting them will not necessarily trigger a formal inquiry. See the above website for details on how to contact the Ombudsperson.

Disability Support
The Department of Mathematics supports an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. We expect all participants in mathematics courses (students and faculty alike) to treat each other with courtesy and respect, and to adhere to the mathematics department standards of collegiality, respect, and sensitivity: mathweb.rice.edu/department-statement-collegiality-respect-and-sensitivity as well as the Rice Student Code of Conduct. If you think you have experienced or witnessed unprofessional or antagonistic behavior, then the matter should be brought to the attention of the instructor and/or department chair. The Ombudsperson is also available as an intermediate, informal option, and contacting them will not necessarily trigger a formal inquiry. See the above website for details on how to contact the Ombudsperson.

Title IX Responsible Employee Notification
Rice University cares about your wellbeing and safety. Rice encourages any student who has experienced an incident of harassment, pregnancy discrimination or gender discrimination or relationship, sexual, or other forms interpersonal violence to seek support through The SAFE Office. Students should be aware when seeking support on campus that most employees, including myself, as the instructor/TA, are required by Title IX to disclose all incidents of non-consensual interpersonal behaviors to Title IX professionals on campus who can act to support that student and meet their needs. For more information, please visit safe.rice.edu or email titleixsupport@rice.edu.