Math 132: Differential Topology (Spring 2026)
Meeting time and location:
Tue Thu 9:00 AM - 10:15 AM
at
Science Center TBD.
Instructor:
Sunghyuk Park
Email: sunghyukpark at math dot harvard dot edu
Office hours:
TBD at
Science Center 231
Course goals:
This course is an introduction to the topology of smooth manifolds and smooth maps, from the perspective of oriented intersection theory.
Here are the topics I am planning to talk about:
- Smooth manifolds and smooth maps
- Transversality, mod 2 intersection theory, winding numbers, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem
- Oriented intersection theory, Lefschetz fixed point theorem, vector fields, index and degree, Poincare-Hopf theorem, Euler characteristic
- Differential forms, integration, de Rham cohomology, Stokes theorem, Gauss-Bonnet theorem
Textbook:
Guillemin & Pollack - Differential Topology [
pdf]
See also Dan Freed's
Lecture Notes
Prerequisites:
Mathematics 22a,b, 23a,b or 25a,b or 55a,b or 112; or an equivalent background in mathematics (vector calculus, introductory real analysis, and basic topology).
Grading:
There will be weekly* homework assignments (20%*), as well as in-person midterm and final exams (80%*). (* subject to change)
Make-Up Exam Policy: There will be no make-up exams. If you miss an exam for a valid reason (e.g. documented illness or family emergency), your score on the other exam score will replace the missed exam.
AI policy:
While there is no specific restriction on the use of generative AI tools in this course, the in-person exams will carry much greater weight than the assignments to ensure a fair and accurate evaluation of your understanding.
Harvard College Honor Code:
"Members of the Harvard College community commit themselves to producing academic work of integrity –
that is, work that adheres to the scholarly and intellectual standards of accurate attribution of sources, appropriate collection and use of data, and transparent acknowledgement of the contribution of others to their ideas, discoveries, interpretations, and conclusions.
Cheating on exams or problem sets, plagiarizing or misrepresenting the ideas or language of someone else as one's own, falsifying data, or any other instance of academic dishonesty violates the standards of our community, as well as the standards of the wider world of learning and affairs."