Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

Advances in mathematics

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

Theory of Functions of a Complex Variable

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Cohomology Of Number Fields

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some Hopf algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer Hopf algebra of permutations. Among the results surveyed are the Littlewood-Richardson rule and other symmetric function identities, Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, the theory of Lyndon words, the Gessel-Reutenauer bijection, and Hazewinkel's polynomial freeness of QSym. The notes are written with a graduate student reader in mind, being mostly self-contained but requiring a good familiarity with multilinear algebra and -- for the representation-theory applications -- basic group representation theory.

Hopf Algebras in Combinatorics

Let $p$ be an odd prime number and $F$ a field containing a primitive 
$p$th root of unity. We prove a new restriction on the group-theoretic 
structure of the absolute Galois group $G_F$ of $F$. Namely, the third 
subgroup $G_F^{(3)}$ in the descending $p$-central sequence of $G_F$ is 
the intersection of all open normal subgroups $N$ such that $G_F/N$ is 
$1$, ${\Bbb Z}/p^2$, or the extra-special group $M_{p^3}$ of order $p^3$ 
and exponent $p^2$.

On the descending central sequence of absolute Galois groups

Witt rings and galois groups

For a prime power q = pd and a field F containing a root of unity of order q we show that the Galois cohomology ring ${H^*(G_F,\mathbb{Z}/q)}$ is determined by a quotient ${G_F^{[3]}}$ of the absolute Galois group GF related to its descending q-central sequence. Conversely, we show that ${G_F^{[3]}}$ is determined by the lower cohomology of GF. This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.

https://www.math.uni-bielefeld.de/lag/man/358.pdf

Quotients of absolute Galois groups which determine the entire Galois cohomology

We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any $n$-Massey product for some prime $p$ has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro-$p$-groups, where we do not require that all cup products vanish. We prove that all $n$-Massey products, $n\\geq 3$, vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch-Kato conjecture.

Triple Massey products and Galois theory

For fieldF of characteristic notp containing a primitivepth root of unity, we determine the Galois module structure of the group ofpth-power classes ofK for all cyclic extensionsK/F of degreep.

Jan Minac

Papers

On the descending central sequence of absolute Galois groups

Witt rings and galois groups

Quotients of absolute Galois groups which determine the entire Galois cohomology

Triple Massey products and Galois theory

Galois module structure ofpth-power classes of extensions of degreep