Regular Solids and Isolated Singularities
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Advanced Ledures in Mathematics Edited by Gerd Fischer
Jochen Werner Optimization. Theory and Applications
Manfred Denker Asymptotic Distribution Theory in Nonparametric Statistics
Klaus Lamotke Regular Solids and Isolated Singularities
Klaus Lamotke
Regular Solids and Isolated Singularities
Friedr. Vieweg & Sohn
Braunschweig IWiesbaden
CIP·Kurztitelaufnahme der Deutschen Bibliothek Lamotke, Klaus: Regular solids and isolated singularities / Klaus Lamotke. - Braunschweig; Wiesbaden: Vieweg, 1986. (Advanced lectures in mathematics)
ISBN 978-3-528-08958-0 ISBN 978-3-322-91767-6 (eBook) DOI10.1007/978-3-322-91767-6
AMS Subject Classification: 14J 17, 20G 20, 32830, 32C45, 51 F25, 51 M 20,57 MX.
1986 All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, 8raunschweig 1986 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright holder. Produced by Lengericher Handelsdruckerei, Lengerich
ISBN 978-3-528-08958-0
- v Preface The last book XIII of Euclid's Elements deals with the regular solids which therefore are sometimes considered as crown of classical geometry. More than two thousand years later around 1850 Schl~fli extended the classification of regular solids to four and more dimensions. A few decades later, thanks to the invention of group and invariant theory the old threedimensional regular solid were involved in the development of new mathematical ideas: F. Klein (Lectures on the Icosahedron and the Resolution of Equations of Degree Five, 1884) emphasized the relation of the regular solids to the finite rotation groups. He introduced complex coordinates and by means of invariant theory associated polynomial equations with these groups. These equations in turn describe isolated singularities of complex surfaces. The structure of the singularities is investigated by methods of commutative algebra, algebraic and complex analytic geometry, differential and algebraic topology. A paper by DuVal from 1934 (see the References), in which resolutions play an important rele, marked an early stage of these investigations. Around 1970 Klein's polynomials were again related to new mathematical ideas: V.I. Arnold established a hierarchy of critical points of functions in several variables according to growing complexity. In this hierarchy Kleinls polynomials describe the "simple" critical points. The present book grew out of a two semester course at the University of Cologne for students with some basic knowledge of algebra, complex analysis, and topology, who wanted or needed another special course, mostly as their last encounter with Pure Mathematics. The book still reflects this situation: It presents basic material from commutative algebra to differential topology with some emphasis on several complex variables as weIl as the special topics from regular solids and isolated singularities mentioned above. As each chapter has its ow
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