MAGIC063: Differentiable Manifolds

This course is designed for PhD students in pure or in applied mathematics and should also be of interest to those in mathematical physics. 
Smooth manifolds underlie a great deal of modern mathematics: differential geometry, global analysis, the theory of Lie groups, dynamical systems, as well as large areas of mathematical physics. 
The main part of this course will cover the basic theory of smooth manifolds and smooth maps, vector fields, tensors and connexions on vector bundles. These are irreducible requirements for work with smooth manifolds. It will conclude with a brief introduction to Riemannian geometry.

The course will concentrate on how to work with smooth manifolds, with plenty of explicit computations and concrete examples . Some proofs will be only sketched, but references for complete arguments will be provided. I hope that at the end of the course you will be able to make use of the literature to learn more of what is particularly important for you in your own work. 
There will be twenty live lecture sessions. PDF lecture notes will be released in advance of these.

If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course. 

Calculus of several variables.

Linear algebra (axioms of a vector space, linear operators in finite dimensions, bases, inner product spaces, dual spaces).

Basic topology of Euclidean spaces (open and closed sets, compactness, open covers).

Outline syllabus: 

• Preliminaries: A brief review of point set topology and calculus in R^n
• Smooth manifolds: atlases, differentiable structures, orientation, calculus on manifolds
• The tangent bundle: vector fields, the Lie derivative
• Vector bundles: basic operations and constructions, tensors, connexions, curvature, holonomy
• Riemannian metrics: the canonical connexion, geodesics, curvature versus topology