The Geometry of Momentum

Abstract

Although the idea of the momentum map associated with a symplectic action of a group is already contained in work of Lie, the geometry of momentum maps was not studied extensively until the 1960's. Centering around the relation between symmetries and conserved quantities, the study of momentum maps was very much alive at the end of the 20th century and continues to this day, with the creation of new notions of symmetry. A uniform framework for all these momentum maps is still to be found; groupoids should play an important role in such a framework.

Expanded version of talk at conference on "Geometry in the 20th Century: 1930-2000" (Paris, September 2001); 15 pages