An Introduction to Smooth Manifolds

The goal of this course is to introduce the student to the basics of smooth manifold theory. The course will start with a brief outline of the prerequisites from topology and multi-variable calculus. After the introducion of differentiable manifolds, a large class of examples, including Lie groups, will be presented. The course will culminate with a proof of Stokes' theorem on manifolds. INTENDED AUDIENCE : Masters and PhD students in mathematics, physics, robotics and control theory, information theory and climate sciences. PREREQUISITES : Real analysis, linear algebra and multi-variable calculus, topology. INDUSTRY SUPPORT : Nil

Syllabus

COURSE LAYOUT Week 1 : Review of topology and multi-variable calculus Week 2 : Definition and examples of smooth manifolds Week 3 : Smooth maps between manifolds, submanifolds Week 4 : Tangent spaces and vector fields Week 5 : Lie brackets and Frobenius theorem Week 6 : Lie groups and Lie algebras Week 7 : Tensors and differential forms Week 8 : Exterior derivative Week 9 : Orientation Week 10 : Manifolds with boundary Week 11 : Integration on manifolds Week 12 : Stokes Theorem