MAGIC063: Differentiable Manifolds

Course details

A core MAGIC course

Semester

Spring 2019
Monday, January 21st to Friday, March 29th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
80

Timetable

Mondays
10:05 - 10:55
Fridays
09:05 - 09:55

Announcements

Material for the first two or so lectures is now uploaded.
There is also a `guide' to the books listed in the bibliography.

Description

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 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 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.

Prerequisites

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.

Syllabus

Outline syllabus (details will be added gradually):
§1
Examples
§2
Basics of smooth manifolds and smooth maps
§3
Basics of vector bundles
§4
Tangent vectors, vector fields, and the tangent bundle
§5
Differential forms and the cotangent bundle
§6
Integration of differential forms
§7
Connections in vector bundles
§8
Poisson manifolds and their symplectic leaves

Lecturers

  • KM

    Dr Kirill Mackenzie

    University
    University of Sheffield
    Role
    Main contact
  • SH

    Stuart Hall

    University
    University of Newcastle

Bibliography

Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

Assessment

Description

The assessment for MAGIC063 Differentiable Manifolds will be via a single take-home exam in April with two weeks to complete and submit online. There will be four questions and you will need at least 50 marks to pass

Assessment not available

Assessments are only visible to those being assessed for the course.

Files

Only consortium members have access to these files.

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Lectures

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