Abel's influence on modern mathematics is substantial. This is seen in many ways, but maybe clearest in the number of mathematical terms containing the adjective Abelian. In algebra, algebraic and complex geometry, analysis, the theory of differential and integral equations, and function theory there are terms like Abelian groups, Abelian varieties, Abelian integrals, Abelian functions. A number of theorems are attributed to Abel. The famous Addition Theorem of Abel, proved in his Paris Mémoire, stands out, even today, as a mathematical landmark.

This book, written by some of the foremost specialists in their fields, contains important survey papers on the history of Abel and his work in several fields of mathematics. The purpose of the book is to combine a historical approach to Abel with an overview of his scientific legacy as perceived at the beginning of the 21st century.

Klappentext

A unique series of fascinating research papers on subjects related to the work of Niels Henrik Abel, written by some of the foremost specialists in their fields. Some of the authors have been specifically invited to present papers, discussing the influence of Abel in a mathematical-historical context. Others have submitted papers presented at the Abel Bicentennial Conference, Oslo June 3-8, 2002. The idea behind the book has been to produce a text covering a substantial part of the legacy of Abel, as perceived at the beginning of the 21st century.

Inhalt

0. Olav Arnfinn Laudal, Ragni Piene: The legacy of Niels Henrik Abel 1. H. M. King Harald V: Opening address 2. Arild Stubhaug: The life of Niels Henrik Abel 3. Christian Houzel: The work of Niels Henrik Abel 4. Phillip Griffiths: The legacy of Abel in algebraic geometry 5. Daniel Lazard: Solving quintics by radicals 6. Birgit Petri, Norbert Schappacher: From Abel to Kronecker - Episodes from 19th century algebra 7. Günther Frei: On the history of the Artin Reciprocity Law in Abelian extensions of algebraic number fields: How Artin was led to his Reciprocity Law 8. Aldo Brigaglia, Ciro Ciliberto, Claudio Pedrini: The Italian school of algebraic geometry and Abel's legacy 9. Fabrizio Catanese: From Abel's heritage: transcendental objects in algebraic geometry and their algebraization 10. Steven L. Kleiman: What is Abel's Theorem anyway? 11. Torstein Ekedahl : On Abel's hyperelliptic curves 12. Mark Green, Phillip Griffiths: Formal deformation of Chow groups 13. Herbert Clemens: An analogue of Abel's theorem 14. Tom Graber, Joe Harris, Barry Mazur, and Jason Starr: Arithmetic questions related to rationally connected varieties 15. Yum-Tong Siu: Hyperbolicity in complex geometry 16. G Henkin: Abel-Radon transform and applications 17. Simon Gindikin: Abel transform and integral geometry 18. V P. Palamodov: Abel's inverse problem and inverse scattering 19. Jan-Erik Bjørk: Residues and D-modules 20. Mikael Passare, August Tsikh: Algebraic equations and hypergeometric series 21. Håkan Hedenmalm: Dirichlet series and functional analysis 22. Yuri I. Manin: Real multiplication and noncommutative geometry 23. William Fulton: On the quantum cohomology of homogeneous varieties 24. Christian Kassel: Quantum principal bundles up to homotopy equivalence 25. Michel Van den Bergh: Non-commutative crepant resolutions 26. Moira Chas, Dennis Sullivan: Closed string operators in topology leading to Lie bialgebras and higher string algebra