## The Road Ahead

I think that’s about enough of the representation theory of the symmetric groups for now. There’s a lot more to say, but we’ve been at this for months and I’m itching for something new.

I’m thinking next up is some differential geometry, although I’ll have to take a little side trip to cover some basic differential equations before too long. I am still looking for a good reference that gets straight to the basic existence and uniqueness theorems of differential equations. If anyone has any suggestions, I’m all ears.

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That would be cool, maybe towards getting people towards able to understand the basics of GR? I suspect, sigh, that I’d fall off the wagon pretty soon, as usual, since what I really think I ought to learn about ATM is strong monads ….

Comment by Avery Andrews | February 19, 2011 |

My vote is for something about differential geometry.

Comment by Donal Henry | February 19, 2011 |

I recommend Hurewitz’s little book “Lectures on Ordinary Differential Equations”, which begins with a beautiful treatment of the basic existence and uniqueness theorems (with a couple of different proofs, plus some things that lots of people don’t know, like that you can prove existence [but not uniqueness] without needing a Lipschitz condition).

Comment by Andy P. | February 19, 2011 |

Sounds good, Andy. Unfortunately it doesn’t seem to be in print, and I no longer have easy access to an academic library.

What book would you pick to teach an upper-level undergrad ODEs class?

Comment by John Armstrong | February 19, 2011 |

I would be interested in seeing your approach. I studied ODEs, curves and surfaces last term and I am currently starting manifolds.

My professor didn’t use a book, he made his own approach to the existence theorems via functional analysis techniques.

Comment by gvaerg | February 19, 2011 |

I’ve really enjoyed the Strogatz book (non-linear dynamics and chaos) — easy read.

Comment by Andrew | February 19, 2011 |

No comments but I just wanted to say that I am interested to hear your take on differential geometry. I am currently trying to learn a little myself.

Comment by Jake | February 19, 2011 |

You might check out Arnol’d:

http://www.amazon.com/Ordinary-Differential-Equations-Universitext-Vladimir/dp/3540345639/ref=ntt_at_ep_dpt_2

The approach is geometric. After an initial chapter of examples, he gets down to the first existence and uniquenss theorem on page 36 (of my 1992 edition). Lots of diagrams, lots of applications to mechanics, which you may or may not be interested in.

Comment by Santo | February 21, 2011 |

Thanks, Santo. I do generally like Arnol’d, and geometry besides.

Just to make sure: his e/u theorems are in real vector spaces (though with one time dimension for derivatives, of course), correct?

Comment by John Armstrong | February 21, 2011 |

John, yes, Arnol’d deals with real vector spaces, and almost entirely with just two-dimensions. Since his book is loaded with mechancis applications, the two dimensions are usually one space dimension and one time dimension.

I devoted quite some time to studying the representation theory of symmetric groups when working on my thesis, so your series of posts on this topic (excellent, as always) was a nostalgic trip down memory lane for me. Thanks!

Comment by Santo | February 21, 2011 |

Two dimensions is actually the important one. If you can prove it in two dimensions you can prove it in dimensions. The proofs for one dimension don’t always clearly generalize, since matrices are just numbers.

Comment by John Armstrong | February 21, 2011 |

An online favorite, Chapter 6 of Loomis and Sternberg:

Comment by Kevin | February 22, 2011 |

Another suggestion is

http://www.amazon.com/Ordinary-Differential-Equations-Graduate-Mathematics/dp/0387984593

from page 60 on.

Comment by Landau | February 24, 2011 |

Were you planning on covering any topics on Differential Manifolds? Just wondering… Also, I just discovered your blog and I have really enjoyed it –especially, the representation theory posts.

Comment by bryguy | February 18, 2013 |

I was bryguy, but having a real job and another major hobby has put a major dent in my ability to maintain this one…

Comment by John Armstrong | February 18, 2013 |