Monday, August 30, 2010

The Great Mathematics Bazaar

I haven't blogged in exactly two months, the longest hiatus since I started. Not sure why. Now (under pressure from my brother!) I'm giving it another shot.

I just got back from the International Congress of Mathematicians at Hyderabad. That should give me plenty to write about! It's a long conference, all of 4+4 days with a day's break in between. And it's an enormous conference -- somewhere between 3000-4000 participants. It takes place once in four years and covers "all" areas of mathematics.

First the academics. Four young (under-40 is the rule) mathematicians received the prestigious Fields Medal. It was quite a thrill to be present as the medals were announced, and the awardees came on stage to receive them from the diminutive President of India, Her Excellency Smt. Pratibha Patil, shimmering in an exquisite silk sari. With each medal, a tug-of-war ensued as the recipient tried to take it from her hands, but she grimly held on to it and gestured with her head that they should face the camera! Only after the photo-op did she allow herself an impish smile and relinquish the medal to the winner.

There were other awards including the Chern, Gauss and Nevanlinna prizes.

The awards ceremony was followed by laudatory talks, by (who else) laudators, specially chosen, each of whom lectured on the work of an awardee. Unfortunately for the most part they were marred by (i) poor transparencies in microscopic fonts, (ii) halting and uncertain accounts of the work, (iii) super-technical accounts lacking in the big picture, (iv) all of the above. Later, however, the medallists themselves gave talks on their own work (every day after lunch) and these were by and large superb. Vietnamese Ngo Bau-Chau (I'm missing half a dozen accent marks that belong to his name) and Israeli Elon Lindenstrauss won the award for work that was "purely mathematical" in nature. But the work of the other two: Stas Smirnov, a Russian working in France, and Frenchman Cedric Villani -- the latter wearing fashionably long hair and what appeared to be a bouquet of red silk ribbons on the front of his shirt -- was motivated by rather straightforward problems in physics.

To get to the latter first: Villani studied the rate of increase of entropy and the approach to equilibrium predicted by the Boltzmann equation. He also illuminated the phenomenon of Landau damping and studied optimal transport theory. The last one is described as follows in the nicely written work profile that you can find here: "Suppose you have a bunch of mines and a bunch of factories, in different locations, with varying costs to move the ore from each particular mine to each particular factory. What is the cheapest way to transport the ore?" I find it wonderful that the highest level of mathematics today still deals with problems that are relatively simple to state.

Stas Smirnov proved the existence of the continuum limit of certain lattice models (models of systems like magnets where microscopic spins sit at each site of a discrete lattice and interact with their neighbours). Physicists of course use such a limit all the time without having any proof that it is rigorously defined. Smirnov spoke very engagingly about it and I felt I understood very clearly (at the time) what it was he had done, though not in any detail how exactly he had done it.

The work of Lindenstrauss too seemed fairly accessible at least in its motivations. Ergodic theory, the study of how dynamical systems do - or don't - go everywhere eventually, originates in celestial mechanics. Number theory deals, among other things, with how many integer solutions there are to a given polynomial equation or inequality. In finding a connection between the two branches of mathematics, he made major progress on Littlewood's conjecture: on how a pair of irrational numbers can be approximated by fractions in a correlated way.

That leaves only Ngo, whose work on the Fundamental Lemma of Langlands remained rather obscure to me despite his valiant attempts. I can only say here that the Langlands programme attempts to relate automorphic forms and number theory among a wide canvas of interconnections, and that Ngo proved what Langlands had thought would be a simple result (hence the name "Lemma") that turned out to defy attempts for decades.

Since this is getting rather long I shall end this instalment here. To make the next one interesting let me mention that I gave two invited talks at the ICM (and I have an "Invited Speaker" badge to prove it!). But they were not about mathematics, nor even physics. More on that soon.


Rahul Basu said...

The "bouquet of red silk ribbons" is called a lavallière

Gaurav Goswami said...

Dear Prof. Mukhi,

You'll be surprised (I hope) to know that so many of us missed your blogging for the last two months! Please blog regularly!

Sunil Mukhi said...

Gaurav: Yes I'm greatly surprised, and happy, to know this! Thanks!

Kaushik said...

I encountered Ergodic theory first (& only )in statistical physics. Please do tell us more about its origins in celestial mechanics.

Sunil Mukhi said...

Kaushik: Oops, I think I propagated a slight misconception.

What I had in mind was a statement in this article about the work of Lindenstrauss, where it describes ergodic theory as "a field of mathematics initially developed to understand celestial mechanics".

Now of course it was Boltzmann who formulated the ergodic hypothesis (and gave it its name) in the context of statistical mechanics, but as I understand, Birkhoff, von Neumann and Kolmogorov (in that order) converted this to a branch of mathematics which studied dynamical systems using measures. Birkhoff, who came before the others, must surely have been motivated by his own seminal work on the 3-body problem. So in that sense celestial mechanics could be called one of the origins of the branch of mathematics called ergodic theory (but even then, statistical mechanics probably ought to be cited first).

Sukratu said...

Felt nice to read this piece on the Congress of Mathematicians. After a long time, we got to read an article here on academic events (actually, after a long time we got to read A NEW article someone mentions above!)
The point about mathematics still addressing problems that look easy
to state....I would say we have seen
this many times till today. Says a lot about our own perception from outside the