Differentiable Manifolds (MAGIC063)
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This course is part of the MAGIC core.
This course is designed for PhD students in pure or in applied mathematics. It should also be of interest to mathematical physicists. 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 symplectic leaves.
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 some 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 2017 but there will be more use of coordinate formulations.
If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course.
Spring 2018 (Monday, January 22 to Friday, March 16; Monday, April 23 to Friday, May 4)
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.
Outline syllabus (details will be added gradually):
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