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.
Category Archives: opinions
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.
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.
For 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.
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.
Update (28 July 2016): The rankings of E7 and E6 should be switched. E6 is the
worst least good exceptional Lie group.
Tom Graber recently asked me why people believe the Hodge conjecture, given the sparse evidence for its truth. I didn’t have time then to answer fully (I was giving a talk), but it’s a question that deserves a full answer. So I’ve sketched below what I feel are the reasons for believing the Hodge conjecture.
The Hodge conjecture is perhaps the most famous problem in algebraic geometry. But progress on the Hodge conjecture is slow, and a lot of algebraic geometry goes in different directions from the Hodge conjecture. Why should we believe the Hodge conjecture? How important will it be to solve the problem?
The Hodge conjecture is about the relation between topology and algebraic geometry. The cohomology with complex coefficients of a smooth complex projective variety splits as a direct sum of linear subspaces, the Hodge decomposition H i(X,C) = Σj = 0i H j,i-j(X). The cohomology class of a complex subvariety of codimension p lies in the middle piece H p,p(X) of H 2p(X,C). The Hodge conjecture asserts that any element of H 2p(X,Q) which lies in the middle piece of the Hodge decomposition is the class of an algebraic cycle, meaning a Q‑linear combination of complex subvarieties.
The main evidence for the Hodge conjecture is the Lefschetz (1,1)-theorem, which implies the Hodge conjecture for codimension-1 cycles. Together with the hard Lefschetz theorem, this also implies the Hodge conjecture of cycles of dimension 1. These results are part of algebraic geometers’ good understanding of line bundles and codimension-one subvarieties.
Not much is known about the Hodge conjecture in other cases, starting with 2‑cycles on 4‑folds. For example, it holds for uniruled 4‑folds (Conte-Murre, 1978). That includes 4‑fold hypersurfaces of degree at most 5, but the Hodge conjecture remains unknown for smooth 4‑fold hypersurfaces of degree at least 6. Why should we believe the conjecture?
One reason to believe the Hodge conjecture is that it suggests a close relation between Hodge theory and algebraic cycles, and this hope has led to a long series of discoveries about algebraic cycles. For example, Griffiths used Hodge theory to show that homological and algebraic equivalence for algebraic cycles can be different. (That is, an algebraic cycle with rational coefficients can represent zero in cohomology without being connected to zero through a continuous family of algebraic cycles.) Mumford used Hodge theory to show that the Chow group of zero-cycles modulo rational equivalence can be infinite-dimensional. There are many more discoveries in the same spirit, many of them summarized in Voisin’s book Hodge Theory and Complex Algebraic Geometry.
Another reason for hope about the Hodge conjecture is that it is part of a wide family of conjectures about algebraic cycles. These conjectures add conviction to each other, and some of them have been proved, or checked for satisfying families of examples.
The closest analog is the Tate conjecture, which describes the image of algebraic cycles in etale cohomology for a smooth projective variety over a finitely generated field, as the space of cohomology classes fixed by the Galois group. The Tate conjecture is not known even for codimension-1 cycles. But Tate proved the Tate conjecture for codimension-1 cycles on abelian varieties over finite fields. Faltings proved the Tate conjecture for codimension-1 cycles on abelian varieties over number fields by a deep argument, part of his proof of the Mordell conjecture. An important piece of evidence for the Hodge conjecture is Deligne’s theorem that Hodge cycles on abelian varieties are “absolute Hodge”, meaning that they satisfy the arithmetic properties (Galois invariance) that algebraic cycles would satisfy. This means that the Hodge and Tate conjectures for abelian varieties are closely related.
The Tate conjecture belongs to a broad family of conjectures about algebraic cycles in an arithmetic context. These include the Birch–Swinnerton-Dyer conjecture, on the arithmetic of elliptic curves, and a vast generalization, the Bloch–Kato conjecture on special values of zeta functions. One relation among these conjectures is that the Birch–Swinnerton-Dyer conjecture for elliptic curves over global fields of positive characteristic is equivalent to the Tate conjecture for elliptic surfaces, by Tate. Some of the main advances in number theory over the past 30 years, by Kolyvagin and others, have proved the Birch–Swinnerton-Dyer conjecture for elliptic curves over the rationals of analytic rank at most 1.
The Hodge conjecture belongs to several other families of conjectures. There is Bloch’s conjecture that the Hodge theory of an algebraic surface should determine whether the Chow group of zero cycle is finite-dimensional. There is the Beilinson–Lichtenbaum conjecture, recently proved by Voevodsky and Rost, which asserts that certain motivic cohomology groups with finite coefficients map isomorphically to etale cohomology.
This web of conjectures mutually support each other. Mathematicians continually make progress on one or the other of them. Trying to prove them has led to a vast amount of progress in number theory, algebra, and algebraic geometry. For me, this is the best reason to believe the Hodge conjecture.