I was a bit envious at first that Conway was giving a talk at Berkeley yesterday, but we had a famous mathematician giving a talk here as well: Sir Michael Atiyah. He was talking about quaternions and what happens when you try to do analysis with them. At first he was going slowly enough that I could understand what he said, but my knowledge became insufficient when he started getting into subtleties of sheaf cohomology. (I deserve to be punished for not knowing enough about sheaf cohomology given that we spent some time on the subject in complex analysis last year.) But Atiyah managed to convince me somehow that I almost understood what was going on until I really started to think about it more. So even though I didn’t really understand it, I really enjoyed the talk.

I really don’t know what to take next quarter. I talked to my advisor yesterday, and he told me that I should get some of my requirements out of the way because a few years ago a student saved them for the end, and when he gave a good answer to a question on an exam or paper or something, the grader thought he had cheated(!) because he wasn’t used to people giving good answers, and the student had his graduation delayed as a result. So maybe I’ll do that. I have to decide between continuing differential geometry and continuing probability theory next quarter since they conflict. Probability theory is certainly a lot easier, but that isn’t a good reason to continue one over the other.

The problems for the noncommutative rings class are very hard this time. It took me all morning and afternoon on Sunday to solve a single problem, but since then things have gone a little better.

And here’s a problem that I don’t know how to solve. Find a group that is isomorphic to a direct product of three copies of itself but not of two. I can’t imagine it should be that hard though.

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## 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.

an equally annoying problem is the following i once had for a group thy course:

show that the groups GL4(Z/2Z) and A8 are isomorphic…

the idea is actually quite simple but carrying out the machinery for that problem was not so fun…

i dunno about your group thy problem of finding a group that is isomorphic to a d. p of 3 copies of itself but not 2.

then again, i once had a prof. intentionally assign wrong problems on occasion to trip us up…it could be the same with you.

it depends on the prof’s personality, i guess.

It’s not a problem assigned in a class. It’s just a problem that a professor told a graduate student, who proceeded to tell it to me. The professor is on sabbatical this quarter and is thus not available to answer questions about it. The graduate student doesn’t know how to solve it either.

is conway on a national tour or something? he was just giving a talk at harvard a week or two ago… weird.

Quaternions are pretty…

So what _does_ happen when you try to do analysis with them? I think they’re neat and would be interested to hear more about them besides as a novelty.

I suppose anticommutativity probably causes some issues, no?

Re: Quaternions are pretty…

Well, you don’t have any entire (“analytic” everywhere on H) functions except constants, but there are meromorphic functions. I think Atiyah’s point was that you can do better by saying that Cauchy-Riemann-like equations only have to hold in some directions. But noncommutativity seems not to cause too much of a problem. The fact that you have too many complex rays is more problematical. I should try to find a book on quaternionic analysis so that I can go through the topics Atiyah talked about at a more leisurely pace.