Education

MRI Spring School 2004

Lie Groups in Analysis, Geometry and Mechanics

Lie groups are groups that at the same time are smooth manifolds. Although a Lie group itself is a nonlinear object, it is almost completely determined by its Lie algebra, which is the tangent space of the group at its identity element, provided with the so-called Lie brackets. Lie groups often arise as groups of transformations, viewed as symmetries of the spaces or systems on which they act. The analysis of these actions often provides a great deal of information about the spaces or systems in question.

In the Spring School, after a quick review of the basic properties of Lie groups and their actions, we will concentrate on applications to both classical and quantum mechanics, (symplectic) differential geometry as well as algebraic geometry. Furthermore, we will study representation theory, the study of actions of Lie groups by means of linear operators on spaces of functions or distributions on a manifold on which the group acts. This situation is quite familiar from quantum mechanics.

Programme

The programme consists of 3 weeks of courses, which consist of plenary lectures complemented by exercise classes, on the following topics:

• Structure theory of Lie groups and Lie algebras (J.A.C. Kolk, Utrecht)
• Group actions (J.A.C. Kolk, Utrecht)
• Symplectic geometry (J.J. Duistermaat, Utrecht)
• Symmetry in mechanics (R.H. Cushman, Utrecht)
• Representation theory and applications in classical quantum mechanics (E.P. van den Ban, Utrecht)
• Analysis on principal fiber bundles (J.J. Duistermaat, Utrecht)
• The momentum map from an algebraic and differential geometric point of view (G.J. Heckman, Nijmegen)

A course usually consists of 8 lectures of 45 minutes and 4 exercise sessions of 45 minutes. The courses will be followed by a three-day conference for which we will invite specialists in the field, who will present lectures at a level comprehensible to the participants.

Dates and location

Details about the contents

Schedule of lectures

Programme of the conference

Lecture notes of the courses

Information on the lecturers

Prerequisites

Background material for the Spring School is already available and more precise information will be distributed some months before the courses start. In principle, participants can use this to independently prepare for the school. Participants receive help with these preparations at a distance, or in collaboration with the home university.

Admission to the Spring School is closed

Information on Mathematical Research Institute

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Further information on the Spring School