It came as an enormous (and pleasant) surprise when Prof. Raghunathan asked me, nearly a year ago, whether I would be willing to give a lecture or two on Indian classical music appreciation at the ICM. The idea would be to present some aspects of Indian culture to the participants, specially those from outside India, and to prepare them to some extent for the planned live concert. Accordingly I gave two lectures, one on Sunday August 22 and the second on Tuesday August 24. The vocal concert by Ustad Rashid Khan took place on August 25.

During my first lecture the sound and video (files embedded in my powerpoint presentation) worked well, and speaking in Hall 2 was a thrill since it was there that Vishwanathan Anand, a couple of days later, played simultaneous chess against 40 participants (apparently unmoved by the gratuitous questions about his Indian-ness or lack thereof).

The second lecture was held in the infinitely larger Hall 4, and like everything conducted there, was videotaped. Hall 4 was even more of a thrill given that I was on the stage where the President of India and the Fields medallists had stood a few days earlier, but for me the thrill quickly evaporated when the sound failed to work and some time was wasted getting things in order. The video of this session can be viewed by going to this page and selecting Part 3 under "24th Aug 10 Time 15:00 – 18:00 /Hall4" or you can download this flv file. Unfortunately due to the sound problem, by the end of an hour I was only 45 minutes into the talk. Since the video was programmed for an hour, it failed to capture the last 20 minutes (in which incidentally Pandit Kumar Gandharva features twice).

Now about the content of the talks. I was asked by the press unit there (= R. Ramachandran, better known as "Bajji") to send a writeup for the ICM daily newsletter, so I might as well reproduce that here.

Titled "A Mathematician's Guide to Hindustani Classical Music", this pair of talks on the musical tradition of North India has been put together specially for the ICM. The first talk presented a brief history of Indian music, which has its roots in religious chanting from Vedic times around 5000 BCE. The textbook "Natya Shastra " by Bharata, the basis for the Bharata Natyam dance form presented at the ICM on Friday, has some reference to this music, and more details including an embryonic concept of raga appear in Matanga's Brihaddeshi in the 8th century. By around the 11th century Persian and Arabic influences started to enrich the music and around this time the North and South Indian streams of music began to diverge. The present lectures focus exclusively on the North Indian or "Hindustani" tradition, which will be presented at the ICM in a live concert by Ustad Rashid Khan on Wednesday.

The nature of Hindustani music evolved during the 12th to 18th centuries, partly in response to the Bhakti movement in Hinduism, in which participatory and devotional love for the divine being (rather than formal worship of God as an idealised entity) became the principal theme. Another contributing factor was the patronage of the Mughal emperors. By the 18th century the "khayal" form of music was established. It remains an oral tradition even today, despite many books and treatises on the subject, some of which have established a rudimentary notation.

In the first of these talks, the notion of raga is introduced by playing short clips of pairs of performances, by different musicians, of the same raga. The common features between the members of a pair serve to illuminate the concept of the raga, even to a complete novice. A definition can then be built up through a series of successive approximations. In its barest form, a raga is a set of notes selected from the 12 notes of the musical scale. But then these notes must be combined into patterns following certain rules. One can emulate the definition of a topological space in mathematics by saying that a raga R={S,U,T} is a subset S of notes from the musical scale together with a collection U of subsets of S and a set T of rules for combining elements of U! But art is not mathematics, so we need to add an aesthetics clause: the rules for combination must give rise to desirable results and create an appropriate mood. It is this mood that lies at the heart of a raga, which some authors consider to be a "living entity" rather than a mere combination of proportions and form. Parallel to raga, the concept of tala (rhythm) is briefly developed.

In the second talk the notion of "gharanas" or schools of music is briefly introduced (parallels with mathematics are quite strong!) and video clips used to illustrate some of the instruments and show how they are played. This is followed by a description of the structure of a typical performance, the different types of movements (introductory, slow and fast) and the complementary role of compositions and variations. The bulk of the talk consists of audio and video clips of performances by some of the leading musicians of India (many of them sadly no more) illustrating different segments and features of a performance. In selected cases the lyrics and their significance are highlighted. The association of ragas with times of day and seasons is also briefly discussed. The talk closes with a short outline of the "lighter" forms: thumri, tappa and bhajan that are usually performed towards the end of a concert.

## Tuesday, August 31, 2010

## 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.

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.

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