MA562 Introduction to Differential Geometry and Topology
Fall 2023, Prof. Manuel Rivera, Purdue University
Course description: This course will be an introduction to the geometry and topology of smooth manifolds. We will begin by defining smooth manifolds and smooth maps, derivatives and tangent spaces, and discuss the inverse and implicit function theorems. We will continue by discussing the basics of transversality and intersection theory on manifolds. Then we will discuss vector fields, differential forms, and integration on manifolds. Some of the highlights of the course include the Whitney embedding theorem, Poincaré-Hopf index theorem, Stokes’ theorem, and Gauss-Bonnet theorem.
Instructor: Prof. Manuel Rivera (manuelr at purdue dot edu), Office: Mathematics Building 708
Textbook: We will follow Differential Topology (Guillemin and Pollack). Other good references are Introduction to Smooth Manifolds (Lee), Topology from the Differentiable Viewpoint (Milnor), and Differential Forms (Guillemin and Haine – freely available here.)
Course schedule: Tuesdays and Thursdays 10:30-11:45am at Helen Schlemann Hall (Recitation Building) 123.
Office hours: TBD
Homework: A homework problem set will be posted every two weeks. Each problem set will contain some exercises from Guillemin and Pollack’s book as well as exercises not in the book. Solutions to the exercises can probably be found online. Do not look them up; try each problem as hard as you can and if you get stuck discuss with me or your colleagues. Write clear and complete solutions to those problems you are able to solve. Collaboration is encouraged but you must write up your own solutions and indicate who you worked with.
Here are some “Comments on style” written by J. Munkres and here are some “Guidelines for good mathematical writing” written by Francis Su. Please read these during the first week of the course and keep them in mind while writing your homework and exam solutions.
Grader: Milana Golich
Exams: We will have a midterm and a final exam. The midterm will be a take home exam and the final will be in class. You will have three days to work on the midterm. Consulting sources other than the books listed above will not be permitted. The final exam will be in real time and the format will be similar to that of a qualifying exam. The dates of the midterm and final will be determined soon.
Grading scale: Homework 50 %, midterm 20%, final exam 30%.
Schedule: We will follow Guillemin and Pollack’s book almost linearly.
Special accommodations: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.
In this mathematics course accommodations are managed between the instructor, the student and DRC Testing Center. If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php
Homework problems – updated every two weeks. Please send your solutions through email to both the instructor and the grader.
riveramanuel.com 