This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

You are then told of a rule that if a card has a vowel on one side, it has an even number on the other. Which cards do you need to turn over to make sure they all satisfy this rule? The blurb speaks of 'thorough revisions that improve accessibility'; that accessibility has been improved cannot be disputed, but the thoroughness can. For example, reference is made incorrectly to a theorem 4.1; an exercise requires an understanding of idempotence although the concept is not discussed in the text; and most curiously, a section on how to solve algebraic inequalities is sandwiched in the chapter on set theory, between a description of the notation for real intervals and an explanation of indexed sets. Again, when so much has been done to improve accessibility in other sections, it is disappointing to note that the description of injections and surjections is inferior to that in many other texts; some schematic diagrams would have helped enormously.

Set theory: the third millennium edition , by Thomas Jech. Pp. 769. £77. 2003. ISBN 3 540 44085 2 (Springer).

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension $V\P$ and all subsequent extensions V\P*\Qdot holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\possible\necessaryφ)\implies\necessaryφ$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with \ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $Vδ\elesub V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that $\ORD$ is Mahlo. The strongest principle along these lines is $\necessary\MPtilde$, which asserts that $\MPtilde$ holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

A simple maximality principle

Consequences of Martin's axiom: When m K > ω1

Review: Robert M. Solovay, A Model of Set-Theory in which Every Set of Reals is Lebesgue Measurable

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the finitely proper posets of small size is compatible with the continuum being larger than the second uncountable cardinal . In particular, this answers a question of Justin Moore by showing that ? does not follow from this arithmetical assumption.

/pdf/forcing-consequences-of-pfa-together-with-the-continuum-4g51hj0nl4.pdf

Forcing consequences of PFA together with the continuum large

We show that Martin’s Maximum++ implies Woodin’s Pmax axiom (∗). This answers a question from the 1990’s and amalgamates two prominent axioms of set theory which were both known to imply that there are א2 many real numbers.

Martin’s Maximum${}^{++}$ implies Woodin’s axiom $(*)$

Large cardinals and locally defined well-orders of the universe

We define the ℵ1.5-chain condition. The corresponding forcing axiom is a generalization of Martin’s Axiom; in fact, ${\text{MA}}_{ < \kappa }^{1.5}$ implies ${\text{M}}{{\text{A}}_{ < \kappa }}$. Also, ${\text{MA}}_{ < \kappa }^{1.5}$ implies certain uniform failures of club-guessing on ω1 that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal κ ≥ ω2 such that ${\mu ^{{\aleph _0}}} < \kappa $ for all µ < κ and such that ⋄({α < κ: cf(α) ≥ω1}) holds, that there is a proper ℵ2-c.c. partial order of size κ forcing ${2^{{\aleph _0}}} = \kappa $ together with MA<κ1.5.

/pdf/a-generalization-of-martin-s-axiom-1tb2a89dcm.pdf

A generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.

David Asperó

Papers

Forcing consequences of PFA together with the continuum large

Martin’s Maximum${}^{++}$ implies Woodin’s axiom $(*)$

Large cardinals and locally defined well-orders of the universe

A generalization of Martin's Axiom

A Generalization of Martin's Axiom