Fachbereich
Mathematik und Statistik
Universität
Konstanz
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Schwerpunkt Reelle Geometrie und Algebra > Prof. Dr. Salma Kuhlmann , Dr. Annalisa Conversano

Proseminar Matrizengruppen
(WS2010/2011)

Prof. Dr. Salma Kuhlmann
Dr. Annalisa Conversano


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:
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