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.
