Mathematical Miscellany

I’ve spent much of the past week trying to learn a bit of algebraic topology. Homology is good stuff — homology groups are not difficult to calculate, but they are powerful topological invariants. I was looking forward to cohomology more in fact, but I keep alternating between feeling that cohomology is a trivial dualization of homology and too difficult for me to grasp at the moment, but I think both of those are in fact false. Of course, cohomology has more structure than homology with the cup product and all that, but I’m not supposed to know about cup products yet. I’ve managed to come to terms with Hatcher’s book somewhat — I used to feel that he went to great lengths to explain all the geometric stuff that I could figure out on my own anyway while neglecting formalism with categories and functors and all that. Well, I still do, but now I don’t mind it so much for some reason.

I thought it was fine time for a break today, so I got myself a book on umbral calculus, a topic I have been peripherally interested in since high school. I didn’t get too far yet, but it looks like fun.

Ars Mathematica posted a link to the proof of the Feit-Thompson Theorem today. I have been looking for it for some time without success, so I am pleased by this.


About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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3 Responses to Mathematical Miscellany

  1. Anonymous says:

    peter may has his book on alg. topology online as well..
    he introduces categories and functors almost immediately..the treatment is somewhat “high tech” for me i guess…
    and you’re right, hatcher doesnt introduce categories until the very last possible moment..i suppose there’s a fairly straightforward reason for that…you can sometimes lose intuition quickly if you don’t *see* what’s going on a more concrete level..
    speaking from personal experience, i remember a lecture where a professor put up the statement of the yoneda lemma and asked us what familiar theorem from group theory it was generalizing..(cayley’s thm)…(i sat there dumbfounded)…and it was only then that i realized that i really wasn’t appreciating what’s going on in category theory at all even though i had played around with it quite a bit…
    i guess the cool part of it is that once you get the hang of it, there’s a nice mix of the highly generalized approach with specific problems that you’re able to have with category theory…
    and i take a dimmer view of hatcher than you i guess..i think it’s an abomination..bleh.

    • Simon says:

      Yeah, I’ve read a bit of May’s book — I even referenced it in an earlier LiveJournal post in December I think. I’m glad I’m not the only one who has issues with Hatcher’s book — most people I talk to seem to be crazy about it. I think I only read it because it’s the only algebraic topology book I have a paper copy of.
      P.S. Who are you?

      • Anonymous says:

        same here. i only read hatcher because there doesnt seem to be anything else out there.
        and who am i? im a math grad student.

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