Thursday, 19 April 2012

A Diophantine Dionysios.

extracted from 'In search of Riemann zeroes'  here

The following is extracted from Steve Landsburg's lyrical appreciation of Weil's achievement-

'The essence of Weil's great vision was that Diophantine problems, although they appear to concern only the ancient subject of pure arithmetic, are inextricably linked to problems in geometry and topology, many of which can be stated only in the language of twentieth century mathematics. High school seniors know that the germ of this idea goes back to Fermat's contemporary Descartes, who discovered that by "graphing'', you can translate equations into geometry. But that translation is too crude to tell you very much about Diophantine questions. You can plot a curve that represents all the solutions to an equation like x5- y3=31, but no matter how long you stare, you'll never be able to discern which points on that curve represent whole number solutions. (One solution is x=2 and y=1. How can you tell whether this is the only whole number solution? Or one of many? Or one of an infinitude?)

So it's natural to guess that if you're interested in whole numbers, geometry won't be much help. But thanks largely to Weil (and others including L.J. Mordell and Carl Ludwig Siegel), we now know that guess to be the exact opposite of the truth. Weil was able to prove that the geometric structure of a curve conveys---in ways that are highly subtle and not at all obvious---information about the arithmetic of the associated equation. From there, he articulated a grand vision of how arithmetic and geometry should be linked in far more general circumstances. This grand vision---which became known as the "Weil conjectures''---was formulated in 1948 and soon became the Holy Grail of algebraic geometry. Throughout the 1960's, a team comprising several of the world's very best mathematicians, and led by the charismatic and indefatigable Alexandre Grothendieck, developed the machinery that made it possible, in 1973, for Pierre Deligne to prove the Weil conjectures and justify the audacious courage that had allowed Weil to suggest that such an extraordinary set of statements might actually be true.

Nowadays, it would be unthinkable to work on problems in arithmetic without exploiting the power of geometry. To a large extent, it was Weil's prescience that made this development inevitable.

But that gets slightly ahead of the story. Before you can apply geometry to arithmetic, you need proper foundations for geometry. When Weil was doing his most important work in the 1940's, those foundations did not exist. For several decades, algebraic geometry had been dominated by the traditions of the "Italian school''---traditions which included a somewhat breezy attitude toward the details of proofs. There was a vast literature full of beautiful results, but it had become essentially impossible to tell which had been proven true and which had only been proven plausible.

The only remedy was to rebuild algebraic geometry from the ground up. Weil felt a particular urgency about this, because he needed a rigorous version of geometry to continue his work in arithmetic. This inspired him to write what he called "the indispensable key to my later work'', his book onFoundations of Algebraic Geometry. With the appearance of this book in 1946, the methods of the Italians were finally legitimized. In the process, Weil had to introduce new ideas and a new language, but characteristically he emphasized the continuity between his own work and the masters of the past. "Nor should one forget'', he wrote, "when discussing such subjects as algebraic geometry and in particular the work of the Italian school, that the so-called `intuition' of earlier mathematicians, reckless as their use of it may sometimes appear to us, often rested on a most painstaking study of numerous special examples, from which they gained an insight not always found among modern exponents of the axiomatic creed...Our wish and aim must be to return at the earliest possible moment to the palaces which are ours by birthright, to consolidate shaky foundations, to provide roofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone.''

Within a few decades, Weil's rebuilt palaces were no longer the foundation of geometry, but the foundation of the foundation. In the 1960's, Grothendieck and his school used the palaces themselves as the groundwork for fantastic modern skyscrapers, reworking every assumption and expanding the realm of geometry to unimaginable heights. From these heights the Weil conjectures were eventually conquered. Grothendieck's project was one of the most remarkable episodes in the history of mathematics. Weil's conjectures made that project necessary, and Weil's foundations made it possible. If Weil had never lived, I cannot imagine what modern geometry would even be about.

The Bhagvad Gita, as I argue in my book Ghalib, Gandhi & the Gita, is the dual of the education in Game Theory of the Just King.
What happens is this. First the Just King experiences 'Vishada' (Depression) being perplexed by the question of what is the duty owed to dependants and Agents. This is answered by