Dancing Numbers

June 15, 2007

I can’t resist one last post before I go off into the webless wilderness of the Atlantic coast. I wasn’t sure what to expect with this blogging idea but I was more than taken aback by how fast the blog’s existence seems to have spread. It’s exciting.

Over the years I’ve read articles about Twyla Tharp, the american dancer and coreographer. Somehow, I always found great affinity to her ideas. A few months ago, in a New York Times article, she described how she sometimes went for days without dealing with numbers (hid all the clocks in her house, for example) to give the intuitive side of her brain more prominence. Despite being a number theorist, I found the thought strangely appealing rather than heretic.

This morning at the Harvard Coop I found she just wrote a book “The creative habit” (published by Simon and Schuster) that I hadn’t seen before. I just had to buy it. Here’s a representative bit:

“I will keep stressing the point about creativity being augmented by routine and habit. Get used to it. In these pages a philosopical tug of war will periodically rear its head. It is the perennial debate, born in the Romantic era, between the beliefs that all creative acts are born of (a) some transcendent, inexplicable Dyonisian act of inspiration, a kiss of God on you brow that allows you to give the world the Magic Flute, or (b) hard work.

If it isn’t obvious already, I come down on the side of hard work. That’s why this book is called ‘The Creative Habit’. Creativity is a habit, and the best creativity is a result of good work habits. That’s it in a nutshell.”

Other ideas in the book remind me of L. Pasteur’s quote: “Chance favors the prepared mind”. To students I have often compared doing mathematics to fishing. You mostly sit around waiting for ideas to bite but when a big one does you better be ready.

Then a strange thing happened. I was sitting on the lawn of Harvard Yard reading the book when a guy comes over to ask me if he could borrow it. He wanted to get a picture sitting on the lawn of Harvard Yard reading a book. “Isn’t that what people do at Harvard?”, he said. When he finished posing for the photo and was ready to leave he asked me “And who the hell is Twyla Tharp?”


Symmetric Functions 0

June 14, 2007

I’ll be gone on vacation until the end of the month and won’t be posting. When I come back I plan to write more about symmetric functions, starting from scratch and hopefully reaching eventually something about Macdonald polynomials. I will be teaching a graduate class on representation theory of finite groups this Fall and I see this project as preparing the lectures for the second half of the course.

The standard reference (“the scriptures” as Schiffmann put it in his talk at AIM) is Ian G. Macdonald’s book: Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University
Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2. (The second edition is more that just a cosmetic change of the first.)

This book is truly amazing. There’s absolutely no fat in it. This, however, can make it a bit hard to get into. In my case, the motivation to read it came from needing his description of the irreducible characters of GL_n (k), with k a finite field (chapter IV), which is in turn based on the original description by Green. I was hooked.

Symmetric functions are at the very least a fantastic computational tool. They can be used to package quite a bit of information in a very concise form. Typically this information is combinatorial or related to representations of certain finite groups (the symmetric group, obviosuly, but also, as mentioned above, GL_n(k) and so on). This is already evident in the description by Frobenius of the irreducible characters of the symmetric groups. In a situation that is quite typical the values of the characters (the character table) appear as the transition matrix (i.e. a change of basis matrix) between two natural bases of the ring \Lambda of symmetric functions in infinitely many variables: the power sums and the Schur functions. (I can’t refrain from quoting Macdonald to the effect that elements of \Lambda, itself a limit of polynomial rings, are neither polynomials nor functions but we have to call them something! See his paper “A new class of symmetric functions”, Publ. IRMA Strasbourg, 1988 372/S-20, Actes 20e Seminaire Lotharingien, p. 131-171, were he introduces the Macdonald polynomials. )

We also know from the work of Haiman that symmetric functions are relevant to the geometry of Hilbert schemes. My experience with our work with Hausel and Letellier is that they are also crucial to understanding the cohomology of character varieties. I am convinced that tough we use them mostly as a computational device the connection goes well beyond that.

Non-sequitur: Farshid Hajir pointed out during my talk in Amherst last Fall that Frobenius appeared in it in three different forms: 1) the Frobenius substitution, the absolute bread and butter of Number Theory, 2) the trace formula expressing the number of points of a character variety over a finite field in terms of irreducible characters of the target group and 3) Frobenius algebras, the building block of topological field theories. Frobenius was just amazing! I highly recommend his Collected Works; I would bet money that there’s still a lot to be dug up from them.

For the history on how Frobenius developed the theory of characters (a pretty convoluted one and hard to fathom from modern accounts) I recommend Representations of Finite Groups: A Hundred Years, Part I by T. Y. Lam.

Communicating mathematics today

June 10, 2007

A few thoughts about how we can use current technology to communicate mathematics. Disclaimer: I have no vested interest in any of the sites or products I mention and I am sure there are plenty of alternatives. Neither can I guarantee their success, naturally. They just happen to be things I know and have used myself. I’d be quite interested in hearing what other people are using.

A good place to start is Skype. It is a great internet phone system which works quite well. It’s free from computer to computer (for now anyway) and fairly cheap from computer to land-line. A useful feature for collaborations with more than two people is the possibility to make conference calls. There are Windows, Mac and Linux versions of Skype.

What I haven’t found a good way to do yet is to also be able to write mathematics for others to read while on Skype. I haven’t tried the beta version of Skype with camera. During the AIM workshop I heard that some people use iChat successfully. I also heard of some version of tablet PC’s used for this purpose.

— I did a little searching since writing the above paragraph and found something that could be quite useful. It’s skrbl (read it aloud). You create a whiteboard where several people can write or draw on. It doesn’t require any downloads; it works directly on the assigned webpage. It worked for me on Firefox when I tried it. The mouse is a bit cumbersome to use to write formulas but perhaps one should use instead some kind of pointer device. It looks like this in combination with Skype could be a good setup for collaborating in mathematics.

For talks I have occasionally used a digital voice recorder, something like this: Olympus WS-100 . They are pretty inexpensive and work well. It’s easy to record and download the sound file directly to computer afterwards. A colleague of mine listens to talks while driving some times.

I understand that a consortium of 5 UK universities will start this year a program of teaching graduate courses to students in all of them through the web. I’m quite interested to see how this goes. I’ve also seen joint seminars (UBC and SF in Vancouver) done with video conferencing. Investing in this kind of technology seems to be a pretty good idea to me.

Finally, for all my classes I no longer use the blackboard. Instead, I write in regular white paper with a thick black marker. This gets projected by a Document Camera to a screen for the audience to read. (You can find one description of the camera here .) After class I scan the notes and put them on the web. For this I use Igal, a series of perl scripts to organize fotos on a webpage. You can find some examples in my website.

I find the fact that I keep a record of exactly what I said in class (including asides, tangents, answers to questions, etc.) very useful. The students typically do too and, no, it does not seem to translate into them feeling less inclined to come to class.

Here it is

June 10, 2007

As was suggested at the end of the recent AIM workshop Arithmetic harmonic analysis on character and quiver varieties I am starting this blog as a way to continue the discussions and interactions that took place there.

You can use Latex code in your comments.
Check the details.