Klaus Lamotke Libri

Die Theorie Riemannscher FlAchen wird vom Autor als ein Mikrokosmos der Reinen Mathematik dargestellt, in dem Methoden der Topologie und Geometrie, der komplexen und reellen Analysis sowie der Algebra zusammenwirken, um die reichhaltige Struktur dieser FlAchen aufzuklAren und zu erlAutern. Viele Beispiele und Bilder, die in der historischen Entwicklung eine Rolle spielten, ergAnzen die Darstellung. Wegen seiner Methodenvielfalt enthAlt das Buch gleichzeitig EinfA1/4hrungen in die Topologie (Fundamentalgruppe, Aoeberlagerungen, FlAchen), in die algebraische Geometrie (Kurven und ihre SingularitAten) und in die Potentialtheorie (Perron-Prinzip).Das vorliegende Buch beruht auf Vorlesungen und Seminaren fA1/4r Studenten mittlerer und hAherer Semester im AnschluA an eine EinfA1/4hrung in die komplexe Funktionentheorie.

The text is organized into five main sections, each addressing distinct mathematical concepts. The first section focuses on regular solids and finite rotation groups, covering topics such as Platonic solids, convex polytopes, and the enumeration and realization of regular solids. It delves into the rotation groups associated with these solids and the finite subgroups of the rotation group SO(3), including discussions on normal subgroups and their generators. The second section explores finite subgroups of SL(2, G) and invariant polynomials, detailing finite subgroups of SL(2, C), the role of quaternions, and four-dimensional regular solids. It examines orbit spaces and the algebra of invariant polynomials, including generators and relations. The third section presents the local theory of several complex variables, discussing germs of holomorphic functions, analytic sets, and maps. It addresses key concepts such as the embedding dimension, finite maps, and regular sequences. The fourth section investigates quotient singularities and their resolutions, focusing on invariant holomorphic functions, complex orbit spaces, and the resolution of cyclic and non-cyclic quotient singularities. It also covers modifications, line bundles, and intersection numbers. The final section outlines the hierarchy of simple singularities, introducing basic concepts, the Milnor number, transformation groups, and the classification of holomorphi