HN Books @HNBooksMonth

"I Want to be a Mathematician: An Automathography" -- http://www.amazon.com/Want-Mathematician-An-Automathography/...

This was a very influential book for me in just thinking about what it would be like to do math for a career. And I'm not a mathematician, so I think it worked. :-)

The article by William P. Thurston (a Fields medalist) called "On Progress and Proof in Mathematics

http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-...

(which I learned about from a comment here on HN, thanks) does a good job of demonstrating how a mathematician who makes new discoveries has to invent a new language for describing those discoveries. Then the mathematician has to relentlessly practice communicating those results first to other professional mathematicians, helping them to see the connections between their research and the new research results. Mathematicians who work hard at communicating with other mathematicians, Thurston says, can help greatly with the progress of mathematical research.

After edit: Paul Halmos, quite a well regarded mathematician who by his own self-evaluation was not in the same league as Fields medalists, wrote in his "automathography"

http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...

that he learned a lot of mathematics as he continued his career after his Ph.D. degree by "reading the first ten pages of a lot of mathematics books." Sometimes he could only get ten pages into a book by another mathematician before he was lost, but by reading dozens and dozens of books, ten pages each, he gained more conceptual foundations in more areas of mathematical research and could gradually apply what he self-learned to advance his own research. I strongly encourage students I know to follow that same strategy of reading at least the introductory portion of many books on subjects they desire to know. Don't just read what your professor assigns you to read. Go to the library and read widely. Read as far as you can before you get stuck, and then find another book and start reading it from the beginning until you get stuck again. Eventually, you will find that you can read harder books, and go farther before getting stuck.

⬐ kenjackson

tokenadult, you're the only other person I know who read that book. I found it dusty in a university library years ago while I was pursuing my PhD and it was an eye opening read. I'd love to buy a copy ,but it is so expensive. But thanks for reminding me of it.

⬐ gjm11

It's $30 from Amazon US; that's not so bad, surely. (And yes, it's a very nice book; Halmos was an outstanding writer.)

⬐ kenjackson

You're right. Will order. Where did all these versions come from? I had looked years back and it was $100+. Weird.

⬐ lliiffee

About Halmos, that is quite scary. I happened to be reading "Finite-Dimensional Vector Spaces" the last few days and while the material is pretty elementary, Halmos is obviously very good, and the idea that there are people leagues ahead of him is amazing...

⬐ NY_USA_Hacker

Well, Halmos wrote the first version of 'Finite Dimensional Vector Spaces' while he was working as an assistant to von Neumann at the Institute for Advanced Study. So the book is doing finite dimensional linear algebra over the real and complex numbers using the techniques of Hilbert space theory, a von Neumann speciality. Once von Neumann had to explain what he meant by 'Hilbert space' to Hilbert!

But that Halmos was a good writer is no joke: He was one of the best of the 20th century.

⬐ lliiffee

This is slightly off-topic, but I noticed that you did work on stochastic optimal control. Do you have any books you would recommend on the subject (either optimal control or stochastic optimal control)? Ideally as good as Halmos. :)

I come from a physics/cs background, and find that standard treatments of control are very much intended for EE folks. But this seems like a historical accident-- the techniques would seem to be very broadly useful in other fields.

⬐ NY_USA_Hacker

If you want something written as well as Halmos, then start writing, and good luck!

"Historical accident": Well, sure, in part nearly all the pure math departments pushed out any such topics!

You are correct: Optimal control has been mostly in advanced parts of electrical engineering.

So, optimal control in EE was an example doing math outside math departments. Of math done outside math departments, control theory is relatively good mathematically.

And, yes, there should be applications elsewhere.

For physics, yes, the deterministic theory of optimal control has been seen as replacing the older calculus of variations which goes back to Newton.

My references are old.

More recent work on stochastic optimal control has been by R. T. Rockafellar at University of Washington.

For

Stuart E. Dreyfus and Averill M. Law, 'The Art and Theory of Dynamic Programming', ISBN 0-12-221860-4, Academic Press, New York.

'dynamic programming' is a big part of the discrete time versions of optimal control. It can be stochastic or deterministic. The linear-quadratic-Gaussian (linear 'plant' or system, quadratic cost to be minimized, and Gaussian exogenous random variables) has 'deterministic equivalence' -- nice -- and this book treats it. The book is a good, elementary start. Apparently Dreyfus was a R. Bellman student.

For

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, 'The Mathematical Theory of Optimal Processes', ISBN 0-470-69381-9, Interscience Publishers, John Wiley & Sons, New York.

it was, of course, the main source of the Pontryagin maximum principle, and, thus, a swift kick in the back side for parts of US aerospace in the 1960s.

For

Michael Athans and Peter L. Falb, 'Optimal Control: An Introduction to the Theory and Its Applications', McGraw-Hill Book Company, New York.

Athans was long in EE at MIT and did some military work, e.g., on parts of the C5A airplane. Falb was at Brown's Division of Applied Mathematics.

One Athans story was the 'control' for least time to climb to, say, 100,000 feet for an F-4: Go up to a few thousand feet, go into a dive, get supersonic, get the lower drag of a few hundred knots above Mach 1, and then with the lower drag continue supersonic to the final altitude. I was not able to know if that 'control' idea was just intuitive or directly from computation and the Pontryagin maximum principle which is, after all, just a necessary condition, i.e., local optimality.

Also in that division at Brown, see the papers of Harold Kushner. As I recall, he wrote out a stochastic version of the Pontryagin maximum principle.

For

E. B. Dynkin and A. A. Yushkevich, 'Controlled Markov Processes', ISBN 0-387-90387-9, Springer-Verlag, Berlin.

there are connections with economic planning.

For

Dimitri P. Bertsekas and Steven E. Shreve, 'Stochastic Optimal Control: The Discrete Time Case', ISBN 0-12-093260-1, Academic Press, New York.

the math is done carefully. So there is a lot of attention to measurability. Part of the reason is the issue of 'measurable selection': This can be a deep subject, but often a relatively simple way out is via regular conditional probabilities as in

Leo Breiman, 'Probability', ISBN 0-89871-296-3, SIAM, Philadelphia.

For

David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York.

this has likely the easiest mathematical treatment of deterministic optimal control and also Kalman filtering and also the math needed for 'least action' in physics.

For

E. B. Lee and L. Markus, 'Foundations of Optimal Control Theory', ISBN 0471-52263-5, John Wiley & Sons, New York.

this tried to be clean mathematically when it was written. At the beginning, should know some relatively advanced results in ordinary differential equations, e.g., as in

Earl A. Coddington and Norman Levinson, 'Theory of Ordinary Differential Equations', McGraw-Hill, New York.

For more, you can consider non-linear filtering and connections with mathematical finance.

The above is just from my bookshelf. Likely now a better bibliography could be assembled via the Internet which actually is at least a little larger than my bookshelf!

⬐ lliiffee

Thank you thank you thank you!

⬐ scott_s

Some brilliant people are truly, honestly humble. It's possible he's better than he gives himself credit for.

Great suggestions!

Two more must read books: 1. "The Mathematical Experience" by Davis and Hersh (http://en.wikipedia.org/wiki/The_Mathematical_Experience#cit...) 2. "I Want To Be a Mathematician" (http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...).

Probably the key thing for me in helping my math ability was to actively try to prove theorems. Before reading a proof, I always try to solve the theorem myself first. And then after reading the actual proof... try to prove it again. You'll be surprised how many times you can't prove a theorem for which you just read the proof!!!

But this will help you get better at doing proofs, and understanding math. And it will also help you appreciate good proofs, because you would have already tried to solve it. You'll say, "Ahh... I didn't even think to try that, but that was exactly the step I would have needed".

Lastly, as someone else mentioned -- the proofs you read in texts are polished proofs. Often those theorems proved have been attempted by famous mathematicians who failed to prove it in their lifetimes. Take your time, be rigorous, and thoughtful. If you do that, you come out ahead regardless.

This quote is from Halmos' "automathography", I Want To Be A Mathematician. I loved reading this book (though I'm no mathematician).

http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...