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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UCLA and the University of California on immigration and refugee ban

A message from the UCLA Office of the Chancellor to our campus community:

This past week, as most of you are well aware, President Trump signed an executive order that suspends entry into the United States for various categories of travelers. The order includes refugees, immigrants, non-immigrant visa holders, and possibly lawful U.S. permanent residents from seven majority-Muslim countries: Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.

The executive order directly challenges the core values and mission of universities to encourage the free exchange of scholars, knowledge and ideas. It may affect the ability to travel for thousands of students and scholars now in the US diligently pursuing their scholarly careers as well as countless others who wish to take advantage of our open universities to pursue knowledge and truth. Although the breadth of the Order is not yet clear, it also could adversely affect the ability to travel for many faculty, students, and staff in our own community.

Already, universities across the US as well as scholarly societies such as the APLU and the AAU have issued powerful statements decrying this action. UCLA joins this rising chorus in expressing opposition to the executive order. As your Chancellor and Executive Vice Chancellor, we want to reassure the campus community as a whole and especially those directly affected by this order that the University of California and our campus leadership stand by our core values.

We are actively engaged with the UC Office of the President to understand the full implications of the order and to find ways of protecting members of our community. The integrity of our mission as a research university and the well-being of our campus community are paramount.

The UC Office of the President has advised UC community members from these seven countries who hold a visa to enter the United States or who are lawful permanent residents do not travel outside of the United States. In the meantime, if you are a student, scholar or faculty who have visa issues or questions that deserve our attention, please contact the UCLA Dashew Center for International Students and Scholars at (310) 825-1681.

Please also see below a message that was sent today from UC President Janet Napolitano and signed by leadership from throughout the University of California.

Gene D. Block, Chancellor
Scott L. Waugh, Executive Vice Chancellor and Provost

President Janet Napolitano and the Chancellors of the University of California today (Jan. 29) issued the following statement:

We are deeply concerned by the recent executive order that restricts the ability of our students, faculty, staff, and other members of the UC community from certain countries from being able to enter or return to the United States. While maintaining the security of the nation’s visa system is critical, this executive order is contrary to the values we hold dear as leaders of the University of California. The UC community, like universities across the country, has long been deeply enriched by students, faculty, and scholars from around the world, including the affected countries, coming to study, teach, and research. It is critical that the United States continues to welcome the best students, scholars, scientists, and engineers of all backgrounds and nationalities.

We are committed to supporting all members of the UC community who are impacted by this executive action.

President Janet Napolitano, University of California
Chancellor Nicholas B. Dirks, University of California, Berkeley
Interim Chancellor Ralph Hexter, University of California, Davis
Chancellor Howard Gillman, University of California, Irvine
Chancellor Gene Block, University of California, Los Angeles
Chancellor Dorothy Leland, University of California, Merced
Chancellor Kim A. Wilcox, University of California, Riverside
Chancellor Pradeep Khosla, University of California, San Diego
Chancellor Sam Hawgood, University of California, San Francisco
Chancellor Henry T. Yang, University of California, Santa Barbara
Chancellor George R. Blumenthal, University of California, Santa Cruz

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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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Petition against cuts at the University of Leicester

The large majority of faculty in the mathematics department at the University of Leicester (in England) are being forced to re-apply for their jobs, on roughly one month’s notice. Of the 21 permanent faculty, 6 will not be re-appointed. (More details are available on Tim Gowers’s blog.)

The rationale offered by management for the cuts is a large predicted budget deficit across the University, but mathematics will suffer disproportionately: the plan is for 4.5% funding cuts overall, but more like 20% to the mathematics department. (In any case, analysis by the University and College Union disputes management’s financial claims.)

One central issue here is that universities operate on a long time scale. It takes decades to build a strong university, but only a few months to lose the confidence of students and faculty. Leicester administrators are acting with a speed and recklessness that would hardly be acceptable in the corporate world, let alone in higher education. The university needs to pause and rethink its whole plan, for mathematics as well as for other departments.

Please sign the petition, calling for these measures to be stopped.

Thanks to Artie Prendergast-Smith for alerting me to this story.

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My book at 25% discount ($63.75) as of 6 Sept 2016

totaro-090616My book is currently available at Amazon for $63.75.

UPDATE (1 Sept 2016): The sale appears to be over.

My book is currently available at Amazon for $29.11.


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WAGS @ Colorado State, 15–16 October

Cat_TailThe Fall 2016 edition of the Western Algebraic Geometry Symposium (WAGS) will be held at Colorado State University on the weekend of 15–16 October. In addition to an excellent program of talks, there will be a lively poster session.

Speakers are:
Enrico Arbarello, Sapienza Universita di Roma/Stony Brook
Emily Clader, San Francisco State University
Luis Garcia, University of Toronto
Diane Maclagan, University of Warwick
Sandra Di Rocco, KTH
Brooke Ullery, University of Utah

For more information and to register, see the WAGS Fall 2016 site.

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Our friend the Tate elliptic curve

Cat discovers genusRigid analytic spaces are all the rage these days, thanks to the work of Peter Scholze and his collaborators on perfectoid spaces. In this post, I want to briefly describe the example that inspired the whole subject of rigid analytic spaces: the Tate elliptic curve. Tate’s original 1959 notes were not published until 1995. (My thanks to Martin Gallauer for his explanations of the theory.)

Let {\bf C}_p be the completion of the algebraic closure of the p-adic numbers {\bf Q}_p. The difficulty in defining analytic spaces over {\bf C}_p, by analogy with complex analytic spaces, is that {\bf C}_p is totally disconnected, and so there are too many locally analytic (or even locally constant) functions. Tate became convinced that it should be possible to get around this problem by his discovery of the Tate elliptic curve. Namely, by explicit power series, he argued that some elliptic curves X over {\bf Q}_p could be viewed as a quotient of the affine line minus the origin as an analytic space: {\bf Q}_p^*/\langle q^{\bf Z}\rangle \cong X({\bf Q}_p).

Trying to make sense of the formulas led Tate to his definition of rigid analytic spaces. In short, one has to view a rigid analytic space not just as a topological space, but as a space with a Grothendieck topology — that is, a space with a specified class of admissible coverings. So, for example, the closed unit disc \{ z: |z| \leq 1\} acts as though it is connected, because its covering by the two disjoint open subsets \{ z: |z| < 1\} and \{ z: |z| = 1\} is not an admissible covering. (“Affinoids,” playing the role of compact open sets, include closed balls such as |z|\leq a for any real number a, but not the open ball |z|<1. An admissible covering of an affinoid such as \{ z: |z| \leq 1\} is required to have a refinement by finitely many affinoids.)

Tate’s formulas for the p-adic analytic map G_m \rightarrow X, modeled on similar formulas for the Weierstrass p-function, are as follows.

Theorem. Let K be a complete field with respect to a non-archimedean absolute value, and let q \in K^* have 0<|q|<1. Then the following power series define a isomorphism of abelian groups K^*/q^{\bf Z}\cong X(K), for the elliptic curve X below:

x(w)=\sum_{m\in {\bf Z}}\frac{q^m w}{(1-q^mw)^2} -2s_1

y(w)=\sum_{m\in {\bf Z}}\frac{q^{2m} w}{(1-q^mw)^2} +s_1,

where s_l=\sum_{m\geq 1}\frac{m^lq^m}{1-q^m} for positive integers l. The corresponding elliptic curve X in {\bf P}^2 is defined in affine coordinates by y^2+xy=x^3+Bx+C, where B=-5s_3 and C=(5s_3+7s_5)/12. Its j-invariant is j(q)=1/q+744+196884q+\cdots. For every element j\in K with |j|>1 (corresponding to an elliptic curve over K that does not have potentially good reduction), there is a unique q\in K with j(q)=j.

It is worth contemplating why the formulas for x(w) and y(w) make sense, for w\in K^*. The series both have poles when w is an integer power of {q}, just because these points map to the origin of the elliptic curve, which is at infinity in affine coordinates. More important, these formulas make it formally clear that x(qw)=x(w) and y(qw)=y(w), but the series do not obviously converge; the terms are small for m \rightarrow \infty, but they are large for m\rightarrow -\infty.

To make sense of the formulas, one has to use the identity of rational functions \frac{w}{(1-w)^2} = \frac{w^{-1}}{(1-w^{-1})^2}. As a result, the series for x(w) (for example) can be written as

x(w)=\frac{w}{(1-w)^2}+\sum_{m\geq 1}\big(\frac{q^mw}{(1-q^mw)^2}+\frac{q^mw^{-1}}{(1-q^mw^{-1})^2} -2\frac{q^m}{(1-q^m)^2}\big),

which manifestly converges. One checks from this description that the series x(w) satisfies x(qw)=x(w), as we want.


S. Bosch, U. Güntzer, R. Remmert. Non-Archimedean Analysis. Springer (1984).

B. Conrad. Several approaches to non-Archimedean geometry. P-adic Geometry, 9–63, Amer. Math. Soc. (2008).

W. Lütkebohmert. From Tate’s elliptic curve to abeloid varieties. Pure and Applied Mathematics Quarterly 5 (2009), 1385–1427.

J. Tate. A review of non-Archimedean elliptic functions. Elliptic Curves, Modular Forms, & Fermat’s Last Theorem (Hong Kong, 1993), 162–184. Int. Press (1995).


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