Seminar "Matrizengruppen" im Wintersemester 2010/2011 Prof. Dr. Salma Kuhlmann Dr. Annalisa Conversano Termin: Donnerstag, 14-16 Uhr (oder n.V.) Beginn: Donnerstag, 21. Oktober 2010 Zweite Vorbesprechung: 14. Oktober 2010 um 14.00 Uhr, F433 In diesem Proseminar soll die Theorie der Matrizengruppen erarbeitet werden, und wichtige Beispiele studieret werden. Grundlage bildet das Buch: "Matrix Groups" von A. Baker, Springer 2002. Die Teilnehmerinnen und Teilnehmer werden bei der Ausarbeitung ihrer Vorträge durch individuelle Vorbesprechungen unterstützt. Voraussetzungen: Das Proseminar richtet sich in erster Linie an Studierende im Grundstudium (3. Semester), ist aber auch für höhere Semester geeignet. Vorausgesetzt werden nur Kenntnisse aus den Analysis und Lineare Algebra Grundvorlesungen. Zielgruppe: LA, BA, D, MA Prerequisites: A basic course in Linear Algebra and a basic course in Analysis. Evaluation: based on the lecture and typed notes in support of the lecture. Structure of the Proseminar: Lectures 1-4 will provide the preliminaries needed for the study of Matrix groups. Lectures 5-10 are devoted to the basics of the theory of Matrix groups. In Lectures 11-15 important examples are studied. Detailed schedule: (If you want more informations before next Vorbesprechung and/or to pick a Lecture and get the material needed to prapare it, please write an email to annalisa.conversano@uni-konstanz.de) Lecture 01 INTRODUCTION TO GROUPS Definition. Subgroups. The center. Homomorphisms. Lecture 02 FINITE GROUPS AS GROUPS OF MATRICES The Symmetric group. Cayley's Theorem. Representation of the Symmetric group as group of matrices. Lecture 03 THE EUCLIDEAN TOPOLOGY Definition. Basis. Closed sets. Continuous maps. Compactness and connectedness. Lecture 04 METRIC SPACES The supnorm. Properties. The metric topology induced by the supnorm. Lecture 05 THE MATRIX EXPONENTIAL AND LOGARITHM Definition of Exp(A) and Log(A). Main properties. Lecture 06 CALCULATING EXPONENTIAL Diagonalisable matrices. Jordan form. Lecture 07 DIFFERENTIAL EQUATIONS IN MATRICES The derivative of a curve. Solutions of differential equations. Lecture 08 ONE-PARAMETER SUBGROUPS Differentiable curves. One-parameter semigroups and groups. The connection with exponential. Lecture 09 LIE ALGEBRAS Definition. Examples. Subalgebras and ideals. The center. Lecture 10 TANGENT SPACES Definition. Dimension. The connection with Lie algebras. Lecture 11 THE GENERAL LINEAR GROUP Definition. The group structure. Topological properties. The Lie algebra. Lecture 12 THE SPECIAL LINEAR GROUP Definition. The group structure. Topological properties. The Lie algebra. Lecture 13 ORTHOGONAL AND UNITARY GROUPS Definition. The group structure. Topological properties. The Lie algebra. Lecture 14 TRIANGULAR GROUPS Definition. The group structure. Topological properties. The Lie algebra. Lecture 15 AFFINE GROUPS Definition. The group structure. Topological properties. The Lie algebra. Letzte Änderung: 14. 10. 2010