MATH 8302: Differential Topology — Fall 2026
MATH 8302: Differential Topology
Time
10:10 AM–11:00 AM MWF
Location
Vincent Hall 6
Instructor
Erkao Bao
Office
Vincent Hall 365
Zoom Meeting ID
Office Hours
TBA
Course Summary
The course will cover smooth manifolds, tangent spaces, embeddings and immersions, Sard’s theorem, the Frobenius theorem, differential forms, integration, curvature, and the Gauss-Bonnet theorem.
Time permitting, the course will also discuss de Rham theory and duality in manifolds.
Prerequisites
Knowledge of basic point-set topology and multivariate calculus will be essential for this course.
Course Goals and Objectives
- Develop facility with smooth manifolds and tangent spaces.
- Study embeddings and immersions and apply Sard’s theorem.
- Understand the Frobenius theorem, differential forms, and integration on manifolds.
- Study curvature and the Gauss-Bonnet theorem.
- If time permits, become familiar with de Rham theory and duality in manifolds.
Course Format and Important Dates
The course meets in person on Mondays, Wednesdays, and Fridays. A detailed day-by-day plan will be posted on Canvas. The order and pace of topics may be adjusted as the course develops. No weekly schedule is included in this syllabus.
- First class meeting: Wednesday, September 9, 2026
- Last class meeting: Wednesday, December 16, 2026
- Midterm exam window: Tuesday, October 27–Saturday, October 31, 2026
- Final exam window: Monday, December 21–Wednesday, December 23, 2026
Exact examination release and submission times will be announced on Canvas. The final examination timing will comply with the University’s Fall 2026 final-examination schedule.
Course Links
- Canvas
- University of Minnesota Fall 2026 academic calendar
- Fall 2026 final exam schedule
- University course catalog entry
Textbook
The course will very loosely follow:
- John M. Lee, Introduction to Smooth Manifolds.
The textbook is optional.
Homework
Homework will normally be assigned weekly on Canvas. Collaboration on homework is encouraged.
Exams
There will be one midterm exam and one final exam. Both examinations are take-home examinations. Collaboration is not allowed. Books and notes may be used, but internet access is not allowed during the examinations.
- Midterm exam: October 27–31, 2026
- Final exam: December 21–23, 2026
Grading Policy
The final grade will be based on the following components:
- Homework: 60%
- Exams: 40%
Course Policies
Attendance
Regular attendance is expected. Attendance is not a separate graded component, but students are responsible for announcements and material covered in class and on Canvas.
Collaboration and Academic Integrity
Collaboration on homework is encouraged. Collaboration is not allowed on either examination. Books and notes may be used on examinations, but internet access is not allowed. All work must comply with the University Student Conduct Code and scholastic-integrity policies.
Make-up Exams and Late Work
Students who need an accommodation because of a compelling circumstance should contact the instructor as soon as possible. Any revised arrangements will be made in accordance with University policy.
Incompletes
An incomplete will be considered only in circumstances permitted by University policy and with a written plan for completing the remaining work.
Course Schedule
A detailed day-by-day plan will be maintained on Canvas. No weekly schedule is included in this syllabus.
Required University Policy Statements
The University of Minnesota Recommended Policy Statements for Syllabi are incorporated by reference. They address the Student Conduct Code, scholastic dishonesty, excused absences and makeup work, appropriate use of class notes and course materials, University grading scales, sexual misconduct, equity and equal opportunity, disability accommodations, mental health and stress management, and academic freedom and responsibility.
Review the complete, current statements in the University’s Recommended Policy Statements for Syllabi.
Students seeking disability-related accommodations should contact the Disability Resource Center and share a current accommodation letter with the instructor as early as possible.