This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.

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Differentiable dynamical systems

Group representations in probability and statistics

How to get masses from extra dimensions

Group theory for unified model building

Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.

Robert Hermann

Papers

Lie Groups, Lie Algebras, and Some of Their Applications

The Theory of Spinors.

Differential geometry and the calculus of variations

Lie groups for physicists

Geometry, physics, and systems