Launch of a new PeakMath space
On this very day, May 6th, exactly 5! years have passed since the birth of André Weil.
One of the truly great mathematical minds of the 20th century, Weil proposed his famous Weil Conjectures on the structural properties of local zeta functions. They were eventually resolved by himself in the one-dimensional case, and in the subsequent decades the general case was the driving force behind the revolution of algebraic geometry carried out by Grothendieck and his school, through the languages of schemes, motives, topoi and new cohomology theories.
In spite of all their resounding success, Weil, Grothendieck and all of their contemporaries left us with perhaps the most profound mathematical challenge ever known to humankind. Can the methods which resolved the Weil Conjectures (including the "local Riemann hypothesis") be transferred in some form to the world of global L-functions? Can some of the underlying ideas be rephrased and reimagined, in a form that would allow a serious attack on the classical Riemann Hypothesis and other immortal problems, unyielding in the world of primes and number fields, like the BSD conjecture, the abc conjecture, and the mysterious connections with quantum field theory that seem to appear for reasons no-one truly understands?
These fundamental mysteries stand as an invitation to a great adventure. It is my hope that this PeakMath project can become a small part of your own voyage into the wilderness of the mathematical landscape, and that some of the insights you may find here will serve you well, as you expand your own horizons and move to take on seemingly unconquerable mathematical foes.
We shall explore together many different aspects of the dream which sometimes is referred to as a "geometry over the field with one element". This is precisely the conjectural and still highly speculative transfer of insights from Weil's and Grothendieck's generations to the great open problems of contemporary number theory. Sometimes the discussion will be held on a rather elementary and accessible level, while at other times we will dive into current research articles and recent breakthroughs. Welcome to join with fellow travellers on an epic journey! Paraphrasing Yuri Manin, we shall seek to conquer the unfathomable abyss.
Still round the corner there may wait
A new road or a secret gate
And though I oft have passed them by
A day will come at last when I
Shall take the hidden paths that run
West of the Moon and East of the Sun
(Frodo, on his last journey to the Grey Havens)
Recommended reading
For the human story of the Weil Conjectures and the remarkable lives of André Weil and his sister Simone, I want to recommend the book The Weil Conjectures, by Karen Olsson. I meant to show this book in Episode 11 of the RH Saga, but forgot to bring it along to the recording studio! It is beautifully written, although it doesn't go much into the mathematical details of the conjectures themselves. Brian Hayes has written a review of the book.
Simone and André exchanged many letters. From a mathematical point of view, the most important one may be the 1940 letter from André, in which he describes his view of analogy in mathematics. It can be found in English translation in the AMS Notices.
As a special bonus, I would also recommend Weil's 1947 essay, L'avenir des mathématiques, available in French in his Oevres Scientifiques (Volume I, pp. 359) and in English translation on JSTOR. (In case you don't have library access to JSTOR, you can still create a free account and read up to 100 articles per month). On the third and fourth page of the essay, Weil speaks about the dream connecting the Riemann Hypothesis, general L-functions, and the ideas that later led to the Langlands program.
These are some of the directions which can and must be followed up […]; it is not impossible that we are here close to principles of extraordinary fertility and that, once the first decisive step on this road has been taken, we shall gain access to vast domains whose existence is hardly suspected.