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Publisher:

Springer Verlag

Publication Date:

2004

Number of Pages:

784

Format:

Hardcover

Price:

79.95

ISBN:

3-540-43826-2

Category:

Anthology

[Reviewed by , on ]

Fernando Q. Gouvêa

01/1/2005

What better way to commemorate the bicentennial of Niels Henrik Abel's birth than to bring together a bunch of mathematicians to talk about his work and where it has led? The conference was held in Oslo in June of 2002, and coincided with the launch of the Abel Prize in Mathematics. There is no doubt that it was a grand event.

The book on *The Legacy of Niels Henrik Abel* is a result of the conference, but it is not your typical proceedings volume. Contributions to the book were solicited before the conference, and there is only partial overlap between the book's content and the talks given at the conference.

Several of the articles in the book are historical and/or biographical. Some, such as Arild Stubhaug's short account of the life of Abel, are mostly on the historical end of the spectrum. Others, such as Steven Kleiman's "What is Abel's Theorem Anyway?" are a blend of mathematics and history, inspecting four theorems that are often called "Abel's Theorem", looking at what Abel actually did, and looking at what came after. Several papers are straight mathematics connected, closely or distantly, to Abel's work. The result is a massive book with a lot of interesting material in it. About half of this material should be accessible to a wide audience; the other half will be of interest mostly to specialists.

With the book comes a CD-ROM containing still more material related to Abel and to the conference. In particular, one can find information on the Centennial Conference held in 1902 (including talks by Sylow and Picard), many pictures, reproductions of stamps, statues, and coins featuring Abel. There is information about the Abel Prize and about honorary doctorates granted at the celebration. There is even an ad for a special edition of Abel's collected works.

It's a delightful package, particularly for those of us who are interested in the history of mathematics and/or in number theory, algebraic geometry, and complex function theory. Just the images on the CD would make me want to have a copy. Given the size of this book and the extras on the CD, this isn't even very expensive; no library should be without a copy.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College; he is the editor of MAA Reviews.

The Legacy of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

King Harald V

Opening address

The Abel Bicentennial Conference University of Oslo, June 3, 2002 . . . . . 5

Arild Stubhaug

The Life of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Christian Houzel

The Work of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Phillip Griffiths

The Legacy of Abel in Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 179

Daniel Lazard

Solving Quintics by Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Birgit Petri and Norbert Schappacher

From Abel to Kronecker: Episodes from 19th Century Algebra . . . . . . . . . 227

GÃ¹unther Frei

On the History of the Artin Reciprocity Law in Abelian Extensions

of Algebraic Number Fields: How Artin was Led to his Reciprocity Law . 267

Aldo Brigaglia, Ciro Ciliberto, Claudio Pedrini

The Italian School of Algebraic Geometry and Abel's Legacy . . . . . . . . . . 295

Fabrizio Catanese

From Abel's Heritage: Transcendental Objects

in Algebraic Geometry and Their Algebraization . . . . . . . . . . . . . . . . . . . . 349

Steven L. Kleiman

What is Abel's Theorem Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Torsten Ekedahl

On Abel's Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Mark Green, Phillip Griffiths

Formal Deformation of Chow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Herbert Clemens

An Analogue of Abel's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Tom Graber, Joe Harris, Barry Mazur, Jason Starr

Arithmetic Questions Related to Rationally Connected Varieties . . . . . . . . 531

Yum-Tong Siu

Hyperbolicity in Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

G. M. Henkin

Abel-Radon Transform and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Simon Gindikin

Abel Transform and Integral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

V. P. Palamodov

Abel's Inverse Problem and Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . 597

Jan-Erik BjÃ¹ork

Residues and D-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Mikael Passare, August Tsikh

Algebraic Equations and Hypergeometric Series . . . . . . . . . . . . . . . . . . . . . 653

HÃ¸akan Hedenmalm

Dirichlet Series and Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Yuri I. Manin

Real Multiplication and Noncommutative Geometry . . . . . . . . . . . . . . . . . 685

William Fulton

On the Quantum Cohomology of Homogeneous Varieties . . . . . . . . . . . . . 729

Christian Kassel

Quantum Principal Bundles up to Homotopy Equivalence . . . . . . . . . . . . . 737

Michel van den Bergh

Non-commutative Crepant Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749

Moira Chas, Dennis Sullivan

Closed String Operators in Topology Leading to Lie Bialgebras

and Higher String Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

King Harald V

Opening address

The Abel Bicentennial Conference University of Oslo, June 3, 2002 . . . . . 5

Arild Stubhaug

The Life of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Christian Houzel

The Work of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Phillip Griffiths

The Legacy of Abel in Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 179

Daniel Lazard

Solving Quintics by Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Birgit Petri and Norbert Schappacher

From Abel to Kronecker: Episodes from 19th Century Algebra . . . . . . . . . 227

GÃ¹unther Frei

On the History of the Artin Reciprocity Law in Abelian Extensions

of Algebraic Number Fields: How Artin was Led to his Reciprocity Law . 267

Aldo Brigaglia, Ciro Ciliberto, Claudio Pedrini

The Italian School of Algebraic Geometry and Abel's Legacy . . . . . . . . . . 295

Fabrizio Catanese

From Abel's Heritage: Transcendental Objects

in Algebraic Geometry and Their Algebraization . . . . . . . . . . . . . . . . . . . . 349

Steven L. Kleiman

What is Abel's Theorem Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Torsten Ekedahl

On Abel's Hyperelliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Mark Green, Phillip Griffiths

Formal Deformation of Chow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Herbert Clemens

An Analogue of Abel's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Tom Graber, Joe Harris, Barry Mazur, Jason Starr

Arithmetic Questions Related to Rationally Connected Varieties . . . . . . . . 531

Yum-Tong Siu

Hyperbolicity in Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

G. M. Henkin

Abel-Radon Transform and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Simon Gindikin

Abel Transform and Integral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

V. P. Palamodov

Abel's Inverse Problem and Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . 597

Jan-Erik BjÃ¹ork

Residues and D-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Mikael Passare, August Tsikh

Algebraic Equations and Hypergeometric Series . . . . . . . . . . . . . . . . . . . . . 653

HÃ¸akan Hedenmalm

Dirichlet Series and Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Yuri I. Manin

Real Multiplication and Noncommutative Geometry . . . . . . . . . . . . . . . . . 685

William Fulton

On the Quantum Cohomology of Homogeneous Varieties . . . . . . . . . . . . . 729

Christian Kassel

Quantum Principal Bundles up to Homotopy Equivalence . . . . . . . . . . . . . 737

Michel van den Bergh

Non-commutative Crepant Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749

Moira Chas, Dennis Sullivan

Closed String Operators in Topology Leading to Lie Bialgebras

and Higher String Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

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