Announcements


The first live session has been postponed from Monday 7th to Monday 14th. You can view the short videos already, see below, and you are welcome to email me with any query about the course.
For links to the short videos click here:
http://kchmackenzie.staff.shef.ac.uk/063/

Forum

General


This course is part of the MAGIC core.

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' 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.

Semester

Autumn 2019 (Monday, October 7 to Friday, December 13)

Hours

  • Live lecture hours: 10
  • Recorded lecture hours: 10
  • Total advised study hours: 80

Timetable

  • Mon 11:05 - 11:55

Prerequisites

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.

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
Connections in vector bundles
§7
(If time permits) Further topics.

Other courses that you may be interested in:

Lecturer


Kirill Mackenzie
Email K.Mackenzie@sheffield.ac.uk
Phone 07899754395
Photo of Kirill Mackenzie


Bibliography


Multidimensional Real Analysis I: DifferentiationDuistermaat and Kolk
Multidimensional Real Analysis II: IntegrationDuistermaat and Kolk
Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics ...Abraham and Marsden
A comprehensive introduction to differential geometry, Volume ISpivak
Differentiable Manifolds: A First CourseConlon
Foundations of differentiable manifolds and Lie groupsWarner
Introduction to Smooth ManifoldsLee
Differential Geometry of Curves and Surfacesdo Carmo


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will 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



A take-home examination of four questions. A total mark of 50 or above must be obtained in order to pass.

No assignments have been set for this course.

Files



No files have yet been uploaded for this course.

Recorded Lectures


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