My course begins

I just put up lecture notes for the first lecture from my course Algebraic Geometry II, a course on the complex approach to algebraic geometry, loosely taught out of Claire Voisin’s book.

The mathematical content of my opening lecture is something I have often considered as a blog topic: Seven ways of computing the cohomology of S^1. I think a lot of you will like it.

I am going to have my students take turns preparing electronic notes, which I will edit and post on the course website. Come read along!

Notes to date:
Differential forms
Sheaves
Sheaf cohomology
Sheaf cohomology II, de Rham cohomology
Complex differential forms
Dolbeault’s Lemma
Cohomology Vanishes on Polydiscs
deRham computation of topological cohomology of affine varieties — preview
Cartan’s Lemma
Gluing resolutions

12 thoughts on “My course begins”

  1. The algebraic de Rham computation really cleared something up for me! That property you mention of Laurent polynomials shows up in an algebraic proof of the Lagrange inversion formula, but I didn’t know of a good way to conceptualize it until now.

  2. Helpful notes! A minor gripe from a Frenchman: I don’t think the spelling “DeRham cohomology” (vs. “de Rham”) has become standard even in English, and I hope it never does!

  7. Typos are fixed, thank you! Regarding other questions:

    I plan to post notes by just adding links to the website. I’m told that RSS is “Really Simple”, is it simple enough to deal with that?

    Alternatively, if there were a lot of demand, I could set up a blogspot blog and just post “The new notes are up” every time that they are. I could also do that here, but I’m not sure how my co-bloggers would feel about me flooding the blog with these posts.

    Regarding turning the notes into a wiki-book: I don’t want to do this for two reasons. First, I don’t want to take responsibility for maintaining it, and I don’t necessarily trust someone else to do so. I am going to read and edit all the LaTeX before I post it; which should hopefully limit the errors to only my own mistakes. Doing that in a second format means more work for me; turning that responsibility over to others means risking the errors they create. (This may be somewhat irrational of me — similar to a person who prefers driving to flying because he feels more in control — but I would prefer to maintain control of the record of my course.)

    Second, I have stated that students retain copyright to their work, except for giving me permission to edit it and post it online, with attribution to them. I don’t expect most students will care about this, but I thought I should set them a good role model of how they should want publishers to behave. (I also was unable to find a clear statement in the Faculty Handbook as to what was permitted in this regard, so it struck me as safest to only the request the minimal rights I would need.) It seems to me that incorporating their work into a publicly editable wiki is questionable under those terms.

  8. David, jekyll and similar “static blog generators” could help in this situation. You could put a blog-like structure on your home page for the course (including a feed or even comments via disqus etc).

    It’s easy to set up, even easier to run and I’d gladly help set it up etc (in person or otherwise).

    Btw, have you considered posting the notes under cc-licenses?

  10. You could just edit this very post with a single note at the end like: “Last updated mm/dd/yy, last topic blah”

    The comments section of this post could be a central place to suggest corrections to future errors.

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