My heavy workload seems to have died down for now, so I can just enjoy learning on my own for a while and try not to fall too far behind on my next wave of applications. In particular, geometric group theory and local class field theory are fun.

I went to two seminars this past week that I really liked. On Tuesday Chuck Akemann talked about multiplicative linear functionals on $\ell^\infty(\mathbb{N})$, which involves ultrafilters and the Stone-Čech compactification of $\mathbb{N}$. I felt I finally understood this stuff properly.

On Wednesday, Jon McCammond gave a talk on 2 and 3. The point is that they are the only primes dividing the orders of all nonabelian finite simple groups, there’s a very high-density sphere packing arrangement in $\mathbb{R}^{24}$, and in the explicit formula for the partition function, you have to sum over 24th roots of unity. Apparently, all these are closely related, and they have to do with the fact that the only p-torsions in $GL_2(\mathbb{Z})$ are 2-torsion and 3-torsion, and in particular, $GL_2(\mathbb{Z})$ has an index 24 free subgroup. Jon McCammond always seems to find such interesting stuff to work on, and it doesn’t hurt that he’s a really amazing lecturer. (I can’t immediately think of anyone else nearly as good as he is.) A few weeks ago, there was a group theory problem that some friends and I were working on but couldn’t solve, and then a few days later Jon McCammond showed me how to do it by drawing a bunch of pictures. I didn’t even know that one could solve group theory problems by drawing pictures before that!


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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5 Responses to

  1. deviantq says:

    Yeah, group theory via pictures is definitely awesome when it works. I don’t really have much skill in the area but me and another student were playing with using pictures to represent group actions, and that was pretty helpful. I’d like to see a textbook or webpage or just a “popular science” book that does group theory via pictures.
    Actually, I’d like to see a “popular science” book about group theory in general. I would really, really benefit from something that was about giving me an intuitive grasp of all the concepts, instead of focusing on formal definitions and proofs (which are, of course, what math is all about—but it helps to _intuitively understand_ these formal objects too). Do you know of any such book?

    • Simon says:

      In this case he was using pictures to represent generators and relations. You draw different sorts of line segments to represent different generators, so you end up with a graph in which all cycles are relations.
      I haven’t read any popular books on group theory (although I remember hearing about one on the quintic which I think was called The Equation That Couldn’t Be Solved). You might try looking at some books on combinatorial group theory though. They cover different topics from standard group theory courses, but they might still be helpful.

  2. grumf says:

    I have him for math 111a! He used pictures to prove that in symmetric groups any element can be rewritten as either an even or an odd number of 2 cycles but not both! Much better than the nasty inductive proof in the textbook!

    • Simon says:

      How do you prove that with pictures?

      • grumf says:

        basically he drew:
        some rule for 5 elements like this:
        1 2 3 4 5
        | | \ / |
        | | \ / |
        | | /\ |
        | | / \ |
        1 2 3 4 5
        so this is (34).. but he drew something a lot more complicated maybe
        (24531) and he made sure that no more than 2 lines passed through any point. then he just ordered looked at each intersection from top to bottom and looked at was the result of the intersection (just a 2 cycle).
        so he showed us that we can break any cycle into a 2 cycles.
        for the identity we concluded that there must be an even number of intersections.
        and similarily he showed that if there was either an even or an odd number of intersections if you redraw the lines but preserve the top and bottom points then you’re still going to end up having either even or an odd number of intersections (the same as you had before).
        best drawn not explained like this =/
        also i dont know if this is exactly “proof”.
        My physics buddy always says that if you draw pictures you are “showing” or “arguing” but not “proving” =~(

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