Lecturer: Dmitriy Rumynin
Term(s): Term 2
Status for Mathematics students: List C
Commitment: 30 Lectures
Assessment: 100% 3 hour exam
Formal registration prerequisites: None
- MA260 Norms, Metrics and Topologies or MA222 Metric Spaces : topological spaces
- MA259 Multivariable Calculus : calculus of several variables including the Implicit Function and Inverse Function Theorems
- MA3H5 Manifolds : knowledge of manifolds, tangent spaces and vector fields will help, although all necessary results from Manifolds will be reviewed in this course
Useful background: A knowledge of calculus of several variables including the Implicit Function and Inverse Function Theorems, as well as the existence theorem for ODEs. A basic knowledge of manifolds, tangent spaces and vector fields will help. Results needed from the theory of manifolds and vector fields will be stated but not proved in the course.
- MA254 Theory of ODEs : the existence theorem for ODEs
- MA3F1 Introduction to Topology : homotopy groups will play a role in the later parts of this course
Synergies: Lie groups have both algebraic and geometric sides. These sides are studied deeply in the following two modules:
Content: The concept of continuous symmetry suggested by Sophus Lie had an enormous influence on many branches of mathematics and physics in the twentieth century. Created first as a tool in a small number of areas (e.g. PDEs) it developed into a separate theory which influences many areas of modern mathematics such as geometry, algebra, analysis, mechanics and the theory of elementary particles, to name a few.
In this module we shall introduce the classical examples of Lie groups and basic properties of the associated Lie algebra and exponential map.
The lectures will not follow any particular book and there are many in the Library to choose from. See section QA387. Some examples:
C. Chevalley, Theory of Lie Groups, Vol I, Princeton.
J.J. Duistermaat, J.A.C. Kölk, Lie Groups, Springer, 2000.
F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, (Graduate Texts in Mathematics), Springer, 1983.