Category Archives: opinions

How to succeed

My new kitten is a highly successful cat — at least he’s successful against me. To find the silver lining in my defeat, I’ve been cataloging the secrets of his success. They turn out, with minimal translation, to be valid advice for mathematicians or, really, for anyone.

  1. If you’re being dragged away, concentrate on what you can take with you.
  2. Don’t worry for a second about how failing makes you look.
  3. If the direct approach is being guarded, try cozying up to your target gradually.
  4. Always have two projects on the go: if one is being guarded, the other may not be.
  5. Be ready to eat anything.
  6. Keep to a schedule.
  7. Patrol regularly for opportunities.

The most powerful secret is not available to the rest of us.

  1. Be charming in the way only a cat can be.

And some probably aren’t good advice.

  1. Always attack people wiping something with a paper towel.

Photo: Mackey Functor at five months, 6 lbs 3 oz.

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March for Science

The March for Science will be on Earth Day, Saturday, April 22, 2017, in 605 cities across the U.S. and beyond. Come out and march for truth against lies! Check out the web site for details of the march in your city. I’ll be marching in Houston.

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Defend freedom

The ACLU (American Civil Liberties Union) is more important than ever now. If you can afford to, support the ACLU’s efforts by donating and/or joining.

Several tech executives are offering to match donations, so this can be a way to multiply the effect of yours. 5 Feb update: Here is a compiled list that appears to be up to date.

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Why I like the spin group / New paper: Essential dimension of the spin groups in characteristic 2

9df3daf31bf116de8c6d6bdddff66a7fjpg Mathematics is about rich objects as well as big theories. This post is about one of my favorite rich objects, the spin group, inspired by my new paper Essential dimension of the spin groups in characteristic 2. What I mean by “rich” is being simple enough to be tractable yet complicated enough to exhibit interesting behavior and retaining this characteristic when viewed from many different theoretical angles.

Other objects in mathematics are rich in this way. In algebraic geometry, K3 surfaces come to mind, and rich objects live at various levels of sophistication: the Leech lattice, the symmetric groups, E8, the complex projective plane,…. I’d guess other people have other favorites.

Back to spin. The orthogonal group is a fundamental example in mathematics: much of Euclidean geometry amounts to studying the orthogonal group O(3) of linear isometries of R3, or its connected component, the rotation group SO(3). The 19th century revealed the striking new phenomenon that the group SO(n) has a double covering space which is also a connected group, the spin group Spin(n). That story probably started with Hamilton’s discovery of quaternions (where Spin(3) is the group S3 of unit quaternions), followed by Clifford’s construction of Clifford algebras. (A vivid illustration of this double covering is the Balinese cup trick.)

In the 20th century, the spin groups became central to quantum mechanics and the properties of elementary particles. In this post, though, I want to focus on the spin groups in algebra and topology. In terms of the general classification of Lie groups or algebraic groups, the spin groups seem straightforward: they are the simply connected groups of type B and D, just as the groups SL(n) are the simply connected groups of type A. In many ways, however, the spin groups are more complex and mysterious.

One basic reason for the richness of the spin groups is that their smallest faithful representations are very high dimensional. Namely, whereas SO(n) has a faithful representation of dimension n, the smallest faithful representation of its double cover Spin(n) is the spin representation, of dimension about 2n/2. As a result, it can be hard to get a clear view of the spin groups.

For example, to understand a group G (and the corresponding principal G-bundles), topologists want to compute the cohomology of the classifying space BG. Quillen computed the mod 2 cohomology ring of the classifying space BSpin(n) for all n. These rings become more and more complicated as n increases, and the complete answer was an impressive achievement. For other cohomology theories such as complex cobordism MU, MU*BSpin(n) is known only for n at most 10, by Kono and Yagita.

In the theory of algebraic groups, it is especially important to study principal G-bundles over fields. One measure of the complexity of such bundles is the essential dimension of G. For the spin groups, a remarkable discovery by Brosnan, Reichstein, and Vistoli was that the essential dimension of Spin(n) is reasonably small for n at most 14 but then increases exponentially in n. Later, Chernousov and Merkurjev computed the essential dimension of Spin(n) exactly for all n, over a field of characteristic zero.

Even after those results, there are still mysteries about how the spin groups are changing around n = 15. Merkurjev has suggested the possible explanation that the quotient of a vector space by a generically free action of Spin(n) is a rational variety for small n, but not for n at least 15. Karpenko’s paper gives some evidence for this view, but it remains a fascinating open question. The spin groups are far from yielding up all their secrets.

Image is a still from The Aristocats (Disney, 1970). Recommended soundtrack: Cowcube’s Ye Olde Skool.

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Book review: The Serre–Tate correspondence

This is not Serre's catFor the past month I’ve been punctuating my life by reading the correspondence between Jean-Pierre Serre and John Tate, recently published in two volumes. Anyone interested in the development of number theory and algebraic geometry will find something to enjoy here.

The book was presumably suggested by the success of the Grothendieck–Serre correspondence, published by the Société Mathématique de France in 2001 and in English translation by the American Mathematical Society in 2003. The Grothendieck–Serre correspondence, beyond its outstanding mathematical interest, has the additional personal fascination of Grothendieck’s story. At first a complete outsider to algebraic geometry, he becomes the master builder of the subject in the 1960s, before rejecting mathematics and, by the end, the rest of humanity.

By comparison, Serre and Tate are reasonable men. The attraction of their correspondence lies in the mathematical ideas that they gradually develop, over the years from 1956 to 2009. Some of the key topics are Galois cohomology (essentially created by Serre and Tate), Tate’s notion of rigid analytic spaces, the Tate conjecture on algebraic cycles, Tate’s invention of p-adic Hodge theory, and Serre’s work on the image of Galois representations, for example for elliptic curves.

Serre usually writes in French, and Tate in English; but both writers make occasional use of the other language for the fun of it.

One running theme is Tate’s reluctance to write up or publish some of his best work. Serre encourages Tate and edits Tate’s papers, but sometimes has to concede defeat. Mazur and Serre started to prepare the publication of Tate’s Collected Papers in about 1990, which would include letters and unpublished work; sadly, nothing has appeared. Serre reports that the AMS has revived the project, and concludes: “I cross my fingers.”

A major topic of the correspondence starting in the 1970s is the relation between modular forms and Galois representations. Deligne and Serre showed in 1974 that a modular form of weight 1 determines a Galois representation with image a finite subgroup of PGL(2,C). At that time, however, it was a serious computational problem to give any example at all of a modular form of weight 1 for which the image is an “interesting” subgroup (that is, A4, S4, or A5, not a cyclic or dihedral group). Tate and a group of students found the first example on June 21, 1974. Soon Tate becomes fascinated with the HP25 programmable calculator as a way to experiment in number theory.

Both Serre and Tate are strongly averse to abstract theories unmoored to explicit examples, especially in number theory. This is a very attractive attitude, but it had one unfortunate effect. One of Serre’s best conjectures, saying that odd Galois representations into GL(2) of a finite field come from modular forms, was formulated in letters to Tate in 1973. But for lack of numerical evidence, Serre ended up delaying publication until 1987. The conjecture played a significant role in the lines of ideas leading to Wiles’s proof of Fermat’s last theorem. Serre’s Conjecture was finally proved by Khare and Wintenberger.

Finally, the correspondence has its share of mathematical gossip. One memorable incident is the Fields Medals of 1974. Tate is on the Fields Medal committee, and Serre suggests “Manin-Mumford-Arnold” as not a bad list, with Arnold as the strongest candidate outside number theory and algebraic geometry. In the event, the award went only to two people, Bombieri and Mumford. At least in the case of Arnold, it seems clear (compare this MathOverflow question) that this was a disastrous result of official anti-Semitism in the USSR, with the Soviet representative to the International Mathematical Union, Pontryagin, refusing to allow the medal to go to Arnold.

I hope that some mathematical readers will go on from the Serre–Tate correspondence to Serre’s Collected Papers. Serre took the idea of cohomology from topology into algebraic geometry and then into number theory. He is one of the finest writers of mathematics. I recommend his papers without reservation.

Correspondance Serre–Tate, 2 volumes. Editée par Pierre Colmez et Jean-Pierre Serre. Société Mathématique de France (2015).

Photo was from the Cambridge branch of Cats Protection, but a different cat is now featured.

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Favorite Christmas movies

Nora and Nick Charles in The Thin Man

This week Town Topics asked Princetonians on the street to name their favorite Christmas movies. The results were infuriatingly dull. As a public service, here is a list of the favorite Christmas movies in my household.

Metropolitan
The Thin Man
Remember the Night
Christmas in July
Enemy of the State
The Royal Tenenbaums
Auntie Mame
The Miracle of Morgan’s Creek
My Night at Maud’s

Nick Smith in Metropolitan

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Exceptional Lie Groups, Ranked

5. E7

4. E6

3. G2

2. F4

1. E8

Update (28 July 2016): The rankings of E7 and E6 should be switched. E6 is the worst least good exceptional Lie group.

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