In the AoPS Olympiad Problem Solving class today, one of the students brought to my attention the existence of a branch of mathematics called differential Galois theory. It is essentially Galois theory used on differential equations. As a result, it is possible to prove that certain functions do not possess elementary antiderivatives. I had always wondered how they did that. Now I need to find a good book on the topic. Has anyone read one?

I managed to be excessively stupid today. I dropped one of my fountain pens, and as a result, the nib got bent slightly. I have more or less repaired it now, but I’ll still probably do a bit of fiddling with it for a few days.

Most of the CDs I ordered last week arrived yesterday. I now have a recording of the Art of Fugue played by Gustav Leonhardt, a recording of the Well-Tempered Clavier book 1 played by Leonhardt, and a recording of the complete Well-Tempered Clavier played by Sviatoslav Richter. Now I’m just waiting for the Mass in B Minor.

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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.

I also want to know about differential Galois theory, so let me know if you find out anything. Although I might find out thirdhand, as I’m pretty sure I heard about it from the same person (a math person I know from back home, who some of our mutual LJ acquaintances know as well).

Yes, it was indeed someone from upstate New York (who also lived in the Bay Area for a year, so I met him in person as well).

Oh, right. I think he said he’d met you. (BTW, I wish I’d had him when I was class assisting Olympiad Problem Solving last semester. Hope that class is going well.)

Okay here is an article on it (in postscript).

Differential Algebra

Hello, just a passer-by looking at other math lover’s livejournals. Anyway, here’s a classic text on differential algebra (free online text! http://www.ams.org/online_bks/coll33/). Differential Galois theory is of course, a subbranch of the field. The subject seems more or less dead for some reason. My school (Rutgers Newark) has two faculty members interested in it, but I’ve only found I think two other schools with faculty having interests in the field (Bucknell and some well known Texas school, maybe Uni of Tex. Austin?)

For combinations of ODE/PDE and algebra, it seems that ‘symmetry analysis of ODEs/PDEs’ is alive. I think this is somewhat different, utilizing more of a lie group to ODEs/PDEs approach rather treating ODEs/PDEs as abstract algebraic equations and using tools like ‘modified’ Galois Theory and so on. There are people doing this scattered around. Applying noncommutative harmonic analysis and group representation theory to PDEs seems alive also, though I’m not sure if this is the same.

Here’s two (respective) books on both:

http://www.amazon.com/exec/obidos/tg/detail/-/0387950001/102-8255686-5862566?v=glance

http://www.amazon.com/exec/obidos/ASIN/0821815237/qid%3D1110037378/sr%3D11-1/ref%3Dsr%5F11%5F1/102-8255686-5862566

Historical note: Sophus Lie actually invented Lie Groups over 100 years ago in attempts to create a theory of equations for ODEs like Galois did for polynomials.

Anyway contact me if you wish: jbi2@njit.edu

Josh

One More Thing

Oh yes, as you may (or may not) know, there are lots of beautiful combinations of algebra and analysis, in ways that are much more nontrivial than just using linear algebra (ie basic functional analysis.) Two such are group representation theory (which is in general the study of homomorphisms of groups into groups of bounded operators in Hilbert Spaces) and operator algebras (which is the study of, you guessed it heh, operators in Hilbert Spaces that form an algebra.) The latter amazingly enough, is applicable to a WIDE range of fields: differential geometry (noncommutative geometry), geometric topology, representation theory, dynamical systems, quantum physics. Hell, the Jones Polynomial, the famous ‘knot invariant’ was discovered via Von Neumann Algebras, which are special cases of operator algebras!!!