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 and should also be of interest to those in mathematical physics. In what follows, and in the course itself, I will write `smooth manifolds' for brevity.
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.
If time permits the course will also cover some basic ideas of symplectic geometry and Poisson manifolds, or alternatively connection theory in principal bundles.
The course will concentrate on how to work with smooth manifolds. Most proofs will be omitted, but references 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.
The main content of the course will be presented in a series of short videos, which can be watched and rewatched when you choose. Click on http://kchmackenzie.staff.shef.ac.uk/063/ There will be ten `live sessions' in which you can raise questions. The beamer notes from spring 2018 will also be available, but note that these contain several topics that will not be covered this year.
If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course.
Autumn 2020 (Monday, October 5 to Friday, December 11)
Advanced 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). Some knowledge of rigorous analysis is helpful, but not essential.
The differential geometry of curves and surfaces in three dimensional Euclidean space will be used for illustration. A knowledge of general topology is not required.
Outline syllabus (details will be added gradually):
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No assessment information is available yet.
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