Symmetries play a decisive role in the natural sciences and throughout mathematics. Infinite-dimensional Lie theory deals with symmetries depending on infinitely many parameters. Infinite-dimensional Lie Groups provides a comprehensive introduction to this important subject by developing a global infinite-dimensional Lie theory on the basis that a Lie group is simply a manifold modeled on a locally convex space, equipped with a group structure with smooth group operations. The focus is on the local and global level, as well as on the translation mechanisms allowing or preventing passage between Lie groups and Lie algebras. Starting from scratch, the reader is led from the basics of the theory through to the frontiers of current research.

This second volume subtitled, Geometry and Topology, builds on its companion volume, General Theory and Main Examples, by consistently placing emphasis on geometric considerations involving differential forms of various types. This framework is applied to homotopy groups, extensions of Lie groups, integrability of infnite-dimensional Lie algebras, and relations to symplectic geometry. The aim is to lay the foundation for the development of a substantive body of literature addressing the global geometric perspective to compliment the abundance of existing Lie-algebraic results.

Together, these essentially self-contained texts provide all necessary background as regards generally locally convex spaces, finite-dimensional Lie theory and differential geometry, with its modest prerequisites limited to a basic knowledge of abstract algebra, point set topology, differentiable manifolds, and functional analysis in Banach spaces. The clear exposition includes careful explanations, illustrative examples, numerous exercises, and detailed cross-references to simplify a non-linear reading of the material.