This course will build on the work done in Math 58J (Lecture notes below) and discuss more advanced topics in differential forms. In the first half of the course the instructor will cover the homotopy invariance properties of deRham cohomology and homological intersection theory of submanifolds using differential forms. In the second half the students will give presentations on the topics listed below.
Lecture notes
- As of 27/04/2023
Student presentations
- Tolga: Morse-deRham comparison
- Ali: Cech-deRham comparison
- Yalim: Formality for Kahler manifolds
- Mert-Ibrahim: Algebraic deRham theory
Plan
- Until 27.03, regular lectures
- 27.03 and 30.03 meetings for presentation prep.
- Next two weeks, the first presentations
- Spring break
- Next two weeks, the second presentations
- Tie up the loose ends/bonus topics
Resources
- Introduction to differential topology - J. W. Robbin, D. A. Salamon - link
- Differential forms - V. Guillemin, P. Haine - link
- Differential forms in algebraic topology - R. Bott, L. W. Tu - link
- These notes from a differential topology class I had taught
- M. Usher's lecture notes - link (Moser argument)
- Elliptic operators, topology and asymptotic methods - J. Roe - link (Hodge theory)
- Complex geometry - D. Huybrechts
- Hodge Theory and Complex Algebraic Geometry - C. Voisin, L. Schneps
- On the deRham cohomology of algebraic varieties - A. Grothendieck - link
- Supersymmetry and Morse theory - E. Witten - link
- Finite volume flows and Morse theory - F. Harvey, H. Lawson - link
- Characteristic classes - J. Milnor, J. Stasheff - link
- Real homotopy theory of Kähler manifolds - P. Deligne et. al. - link
In this course we will discuss singular cohomology, focusing on the case of manifolds, deRham cohomology of smooth manifolds, and the comparison between the two that goes by the name of deRham's Theorem.
Lecture notes
- As of 28/05/2022
Guidelines for writing notes in LaTeX
- Use both your notes from the actual lecture and the notes I will send you after the lecture. The goal is to write something that represents what happened in the class, while being clear and accurate.
- Send me the new figures and the updated tex file by email when you are done.
- I will read what you wrote, perhaps change some things, update the shared Dropbox file and upload the pdf here.
- When texing use the created environments for theorems, lemmas, exercises etc. Display all the long equations. If you have to refer to something label them and use the ref command. In general keep consistent with the style of the document.
Guidelines for homework
- Your homework consists of the exercises that I will assign during the lectures. These will also be recorded in the lecture notes.
- Keep your solutions in a computer file altogether. You will email them to me at the end of the semester.
- You can keep updating your solutions from previous weeks. One restriction is that you can only use what was done at the time of assignment of the exercise in your solutions.
- There will be an oral exam at the end of the semester where I will ask you about the solutions that you submitted.
Preliminary reading
- Fundamental group of a topological space. We will not cover this very important notion in the course, which would typically be covered before homology/cohomology in an introductory algebraic topology course. It is very important that you learn it yourself. This will be a good preparation for the first part of the course as well.
- Definitions of topological and smooth manifolds. We will cover these in the course, but you might want to do some reading by yourself. You can read this historical account for example.
- Exterior algebra of a finite dimensional vector space. The linear algebra that is used in the theory of differential forms can feel a bit quick, so you might want to just start thinking about it by reading this from Wikipedia etc.
- Here is also a historical account for differential forms.
On reading historical accounts: these are generally quite difficult to understand, but for a good reason. I advise you to suffer through them as much as you can.
Resources
- A concise course in algebraic topology - J. P. May - link
- Algebraic topology - A. Hatcher - link
- Differential forms - V. Guillemin, P. Haine - link
- Introduction to smooth manifolds - J. W. Lee - link
- Introduction to differential topology - J. W. Robbin, D. A. Salamon - link
- Differential forms in algebraic topology - R. Bott, L. W. Tu - link