This course is designed for PhD students in pure or in applied
mathematics and should also be of interest to those in mathematical physics.
In what follows, and in the course itself,
I will write `smooth manifolds' since `smooth' is a shorter
word (and because there has never been agreement as to whether
the other word should be `differentiable' or `differential').
Smooth manifolds underlie a great deal of modern mathematics:
differential geometry (of course), global work in differential
equations, the theory of Lie groups, geometric mechanics and
much else, 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 and differential
forms, the tangent and cotangent bundles and the general notion
of vector bundle. These are irreducible requirements for work
with smooth manifolds.
After that the course will cover one or both of (i) connections
in vector bundles, and (ii) Poisson manifolds and their
The connection theory of vector bundles is part of differential
geometry and is a good way to get a feel for curvature and for
its relationship with tensor structures.
Poisson geometry is a relatively recent field. It provides an
easy route into symplectic manifolds, and involves multivector
fields and the Schouten (or Gerstenhaber) bracket, tools which
are of wide use in many parts of mathematics and physics today.
The course will include a few detailed proofs, but the main focus
will be on giving a feel for the various topics and methods; 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.
The course is presented using a beamer file with support from the
visualizer. There are also background notes with detailed references.
The course will be broadly similar to that in spring 2018. The treatment of partitions of unity will be done this year as a `user guide' and most of the detailed proofs in this topic will ne omitted.
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 (integration, implicit function theorem). Linear algebra (axioms of a vector space, linear operators, bases).
The differential geometry of curves and surfaces in 3-dimensional Euclidean space is neither a prerequisite nor part of the syllabus, but if you know something about this you will see how it fits in as a special case.