## Tag Archives: grothendieck

• There’s a new biography of Grothendieck, this one in French.  Any chance it’ll be translated?
• Let felons vote and let them carry guns — the ultimate left-right compromise reform?  Why not?  Everybody believes there’s some core of constitutional rights an American doesn’t give up, no matter what they do.  Felon or no felon, you have the right to free speech and the right to a trial by jury.  I think voting belongs in that inner circle.  I don’t really feel that way about gun ownership, but I get that a lot of people do.  And — purely as a practical matter — the typical felon who’s served his time is surely more correct in feeling he needs a firearm to protect himself than, say, I do.
• “Pinch my cheeks and call me gorgeous — it’s Raven!”  This panel has been floating in my memory for about thirty years.  CJ really likes the Teen Titans show that’s on Cartoon Network now, and watching him watching it inspired me to see if I could actually find an image.  Thanks, tumblr.
• Indietracks Compilation 2015.  As always, a great collection of songs.
• At some point I will try to find time to think more seriously about the claim by Josh Miller and Adam Sarjurjo that the famous Gilovich-Vallone-Tversky study finding no evidence for the hot hand in basketball actually found strong evidence for the hot hand in basketball.  The whole thing comes down to screwy endpoint problems when you average results of a bunch of short trials.  It has some relation to the perils of averaging ratios.
• Pretty sure this cartoon calculus book is the very one that was sitting on the shelf in Mrs. Levin’s 6th grade classroom, which I became absolutely obsessed with.
•  Do you think the most Shazammed songs are the most popular songs, or songs that best combine popularity with being a song no one knows the name of?  I like that you can see the country-by-country charts:  here’s Thailand, where they love Meghan Trainor, or don’t know her name.
• Good-looking conference at the Newton Institute about large graphs.

## Grothendieck-Winnicott update

One good feature of meeting Adam Phillips was that I got to ask him about Grothendieck’s use of the phrase “the capacity to be alone,” generally associated with the psychoanalyst D.W. Winnicott.  Winnicott was Phillips’s analyst’s analyst, and Phillips has written extensively on him, so I thought I’d run the quote by him.  Phillips told me:

• Grothendieck’s conception of the capacity to be alone as “a basic capacity in all of us from the day of our birth” is certainly not that of Winnicott, who was talking about a capacity that’s acquired later via the developing relationship between infant and mother.
• Familiarity with psychoanalytic terminology was fairly common in France at the time, and doesn’t necessarily mean Grothendieck was psychoanalyzed or had any particular interest in analytic theory; in particular, the French analyst Francoise Dolto had a radio show in the 1970s which helped popularize Winnicott’s ideas in France.

## It’s Grothenday!

The G-Man turns 83 today.

Suggested Grothenday activity;  locate the hackiest, most awkward argument in a paper you’re working on, and replace it with an elegant proof that follows effortlessly once the correct definitions are set down.

Bonus points if this extends the original statement to the relative case over an arbitrary base scheme.

## Illusie’s reminiscences of Grothendieck

More Grothendieckiana:  Illusie chats with Bloch and Drinfel’d about Grothendieck, Chicago, January 2007.  Via Andy Putman.

## The capacity to be alone

I’d never encountered this exquisitely characterizing passage from Grothendieck’s memoir before.  I think even non-mathematicians will find it of interest.

In those critical years I learned how to be alone.[But even]this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation[1945-1948],when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is  in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

I’ll add just one remark:  “The capacity to be alone” is a phrase made famous by the psychoanalyst D.W. Winnicott, who understood the development of this capacity to be a crucial phase in the maturation of the child.  Winnicott’s sense of the term is quite specific: “the basis of the capacity to be alone is a paradox; it is the experience of being alone while someone else is present.”  I don’t know whether Grothendieck was quoting Winnicott here (is it known whether he was analyzed, or familiar with the psychoanalytic literature at all?) but his sense of the phrase is much the same.  The challenge is not to do mathematics in isolation, but to preserve a necessary circle of isolation and autonomy around oneself even while part of a mathematical community.

I should say that this is totally foreign to my own mode of mathematical work, which involves near-constant communication with collaborators and other colleagues and a close attention to the “notions of the consensus,” which I find are usually quite useful.

Also, Grothendieck’s distinction between himself and the less profound mathematicians who were quick studies and winners of competitions should give John Tierney something to think about.

## “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!

## Happy Grothenday!

Alexander Grothendieck is 80 today. It’s truly surprising that his strange and marvelous biography — to put it all in one sentence, he rewrote much of the foundation of number theory and geometry in an immense burst of energy in the 1960s, then, over time, came to feel that the mathematical establishment had betrayed him and his ideas, and moved to the Pyrenees to be alone and herd sheep — is not better known.

Read the excellent biographical article by Allyn Jackson, “Comme Appelé du Néant — As If Summoned From the Void”: Part I, Part II. And if that doesn’t sate you, skim through the mass of scanned manuscripts, appreciations both technical and non-, and photographs at The Grothendieck Circle.

My own story: in my last year of grad school I came across Grothendieck’s famous late article, “Esquisse d’un programme” (actually an unsuccessful grant application.) My advisor saw me reading it, and, aware of its seductive effect, told me, “I forbid you from reading the Esquisse until your thesis is finished!”

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