Lie Algebras of Diffeomorphism Groups of Non-compact Manifolds - Part 1

Explore a mathematical lecture examining the complex relationship between diffeomorphism groups and non-compact manifolds in infinite-dimensional geometry. Delve into the canonical structure of diffeomorphism groups of compact manifolds as infinite-dimensional manifolds, and investigate why non-compact manifolds present unique challenges. Learn about Omori's no-go theorem and its implications for exponential mapping from vector fields to diffeomorphisms. Discover a novel solution demonstrating that all diffeomorphism groups are elastic as diffeological spaces, existing within a subcategory carrying tangent structure. Understand how the Lie algebra of elastic groups consists of invariant vector fields, and examine why the Lie algebra of a diffeomorphism group encompasses all vector fields, even in non-compact cases.