Topological Matrix Groups (MAE826)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	MAE826
Semester	8
Course Title	Topological Matrix Groups
Independent Teaching Activities	Interactive, Presentations (Weekly Teaching Hours: 3, Credits: 6)
Course Type	Special Background, skills development
Prerequisite Courses	-
Language of Instruction and Examinations	Greek, English
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The aim of the course is to provide an introduction to Lie theory through matrix groups. The main subject of study is the closed subgroups of the general linear group. Our study is extended from real to complex and quaternion numbers. The corresponding linear groups are in fact topological groups and an introduction of basic properties of topological group is also provided. The Lie algebra of a matrix group is defined. The special orthogonal, unitary and symplectic groups provide important example of Lie algebras. Lie algebras are studied using the exponential map. Finally Lie groups are defined.

General Competences

Study particular characteristics of group theory in topology and geometry Independent and team work Working in an interdisciplinary.

Syllabus

• General linear groups
• Real and Complex algebras, Quaternions. Matrix algebras
• Inner product, orthogonal, unitary and symplectic groups
• Homomorphisms
• Differential curves, tangent vectors. Dimension of a matrix group
• Differential homomorphisms
• Expontential and logarithmic funcions. Lie algebras
• Special orthogonal and symplectic groups
• Topological groups, manifolds
• Maximal tori
• Differential manifolds, Lie groups.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face, Distance learning
Use of Information and Communications Technology	Communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working hours in class 8 Project 30 Assignments 33 Final exam 41 Course total 150
Student Performance Evaluation	Written Examination, Oral Presentation, written assignments.

Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• J. F. Adams, Lectures on Lie groups, University of Chicago Press, 1969.
• M. L. Curtis, Matrix Groups, Springer-Verlag, 1979.
• R. Howe. Very basic Lie theory, American math. monthly,90, 1983.