Math 7590: Characteristic Classes

Course Content:

The following is a collection of topics that I hope to cover in this course.

  • Intro to vector and fiber bundles
  • Classifying vector bundles
  • Intro to Stiefel-Whitney classes
  • The algebraic topology of Grassmannians
  • Existence of Stiefel-Whitney classes
  • Chern classes
  • The Euler class
  • Pontryagin classes
  • Chern and Pontryagin numbers
  • Connections, curvature and Chern-Weil theory

Text: We won’t follow one book exclusively, but Milnor and Stasheff’s Characteristic Classes is the best single resource for this course. Other fantastic resources include Hatcher’s Vector Bundles and K-Theory, Bott and Tu’s Differential Forms in Algebraic Topology and Johan Dupont’s Fibre Bundles and Chern-Weil Theory.

Logistics:

Classroom:  Lockett 130
Time:  9 – 10:20
Office Hours:  Tuesdays 3 – 4, Thursdays  12 – 1

Homework and Grades:

Homework Policy: Homework is to be turned in to my office or mailbox by 4:00 PM on Fridays.  If you know in advance you will be unable to turn in homework when it’s due, you should plan to turn it in ahead of time. Religious holidays are not an excuse for late homework! If you know that there is an impending religious holiday, it is your responsibility to make arrangements to submit homework early. Since homework is only due once a week, this should be manageable. I will drop your lowest homework score to allow for missed assignments or for assignments that pose special difficulty.

Homework must be neat, well-organized, and legible. Ragged margins, multi-column and otherwise over-dense formats, and unstapled sheets are unacceptable. If your handwriting is difficult to read, type your homework. If you tend to scratch out or erase incorrect parts of solutions, do a rough draft or type your homework. Write in paragraphs, sentences, and English words. Use punctuation and conjunctions to indicate your flow of thought rather than arrows or telepathy. Shoot for lucidity rather than terseness.

Final Course Grades: Your grade will be determined by your performance on homework assignments and your overall classroom engagement:

  • Homework 80%
  • Participation 20%

Academic Dishonesty: While copying your written work from somebody else or from any other source is considered cheating, you are strongly encouraged to talk to others about the material you are learning — this includes fellow students, me and anyone else in the department.  The more engaged you are with the material the better. That said, the work you hand in must be your own. The best way to avoid any suspicion of copying is to write out your homework solutions on your own, not whilst working with a friend. Cheating on exams is unacceptable. Any cheating during midterms or finals will result in you failing the exam.

Disability Support: Students who may need accommodations because of a documented disability should meet with me privately within the first week of classes. In addition, students with disabilities should also contact the Office of Disability Services.