Tag Archives: deligne

In which I attend a conference on fundamental groups in arithmetic geometry from the comfort of my own home

I don’t watch videotaped lectures — in general I’ve found the difficulty of seeing the board and hearing the lecturer makes it impossible for me to maintain enough focus to engage with the mathematics and take good notes.  In fact, I think the only online video lecture I’ve ever viewed all the way through was one of my own, because I somehow lost the notes I’d used and needed to generate a new set so I could give the talk again.

But I was really sorry not to be able to make last week’s introductory workshop for the Newton Institute’s special semester on non-abelian fundamental groups in arithmetic geometry — so sorry that I decided to try watching the recorded lectures on my laptop.  And they’re great!  Crisp sound and visuals, appropriately timed close-ups on the board, and even a camera pointed at the audience so you can see the people asking questions.  And you can download the talks to your iPod!   Three cheers for the A/V team at the Newton Institute.

As of tonight just the Monday and Tuesday talks are up, which is already plenty to keep me busy.  I just watched Deligne talk about counting l-adic local systems on curves over finite fields; highly recommended.

When I was first giving public lectures, someone gave me the hoary advice that I should quell nervousness by imagining the members of the audience in their underwear.  Strange to think that, in this new broadband world, most of them actually are.

One-second precis of Deligne’s talk:  starting with Drinfel’d in the early 80s, you can count the number of l-adic local systems on a curve over F_q by applying whatever version of the Langlands correspondence you have available and then using an appropriate trace formula to count automorphic forms.  It turns out that the number of rank-d l-adic local systems “defined over F_{q^n}” seems to behave as if it were governed by a Lefschetz fixed point formula, i.e. as if it were the number of F_{q^n}-rational points on some variety.  But what variety?  Not the moduli space of rank-d vector bundles with connection on the curve; that has dimension twice as large as the dimension of the purported variety suggested by the result of the counting problem.  But one still may hope — bolstered to some extent by recent work of Arinkin and Flicker — that the point count is reasonably legible and has something to do with the hyperkahler geometry of that moduli space.  I don’t think that summary made tons of sense — so watch the video!

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“Le Groupe Fondamental de la Droite Projective Moins Trois Points” is now online

The three papers that influenced me the most at the beginning of my mathematical career were “Rational Isogenies of Prime Degree,” by my advisor, Barry Mazur; Serre’s “Sur les représentations modulaires de degré 2 de \text {Gal}({\overline {\Bbb Q}}/{\Bbb Q});” and Deligne’s 200-page monograph on the fundamental group of the projective line minus three points.  The year after I got my Ph.D. I used to carry around a battered Xerox of this paper wherever I went, together with a notebook in which I recorded my confusions, questions, and insights about what I was reading.  This was the paper where I learned what a motive was, or at least some of the things a motive should be; where I first encountered the idea of a Tannakian category; where I first learned the definition of a Hodge structure, and what was meant by “periods.” Most importantly, I learned Deligne’s philosophy about the fundamental group:  that the grand questions proposed by Grothendieck in the “Esquisse d’un Programme” regarding the action of Gal(Q) on the etale fundamental group \pi := \pi_1^{et}(\mathbf{P}^1/\overline{\mathbf{Q}} - 0,1,\infty) were simply beyond our current reach, but that the nilpotent completion of \pi — which seems like only a tiny, tentative step into the non-abelian world! — nonetheless contains a huge amount of arithmetic information.  My favorite contemporary manifestation of this philosophy is Minhyong Kim’s remarkable work on non-abelian Chabauty.

Anyway:  Deligne’s article appears in the MSRI volume Galois Groups over Q, which is long out of print; I bought a copy at MSRI in 1999 and I don’t know anyone who’s gotten their hands on one since.  Kirsten Wickelgren, a young master of the nilpotent fundamental group, asked me the obvious-in-retrospect question of whether it was possible to get Deligne’s article back in print.  I talked to MSRI about this and it turns out that, since Springer owns the copyright, the book can’t be reprinted; but Deligne himself is allowed to make a scan of the article available on his personal web page.  Deligne graciously agreed:  and now, here it is, a publicly available .pdf scan of “Le Groupe Fondamental de la Droite Projective Moins Trois Points.”

Enjoy!

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Mathematicians, a coffee table book from Princeton University Press

Just got in the mail a coffee-table book from PUP, which will appeal to you if you like looking at big photographic portraits of mathematicians while you drink your coffee.  I do!  The pictures, by Mariana Cook, are agreeable, but what really sells the book for me are the short essays that accompany the photos.

At least two of these would make good openings for novels. Pictured here,  Ed Nelson:

I had the great good fortune to be the youngest of four sons with a seven-year gap between my brothers and me, born into a warm and loving family.  This was in Georgia, in the depths of the Depression, where my father organized interracial conferences.  He was the sixth Methodist minister in lineal descent.  While driving he would amuse himself by mentally representing the license plate numbers of cars as the sum of four squares.

And Kate Okikiolu:

My mother is British, from a family with a trade-union background and a central interest in class struggle; she met my father, who is Nigerian, while both were students of mathematics in London.  My father was a very talented mathematician, and after my parents married, he went on to a position in the mathematics department of the University of East Anglia.  While I was growing up, the elementary school I attended was extremely ethnically homogeneous.  I was unable to escape from heavy issues concerning race, which my mother always explained in a political context.  My parents separated after my father resigned his university position to focus on his inventions, and my mother then finished her education and became a school mathematics teacher.

Less novelistic but very keenly observed is this, from the Vicomte Deligne, on the role of intuition in geometry:

You have more than one picture for each mathematical object.  Each of them is wrong but we know how each is wrong.  That helps us determine what should be true.

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