MAGIC063: Differentiable Manifolds

Course details

A core MAGIC course

Semester

Autumn 2023
Monday, October 2nd to Friday, December 8th

Hours

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

Timetable

Mondays
12:05 - 12:55 (UK)
Wednesdays
10:05 - 10:55 (UK)

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. 

Smooth manifolds underlie a great deal of modern mathematics: differential geometry, global analysis, the theory of Lie groups, dynamical systems, 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, tensors and connexions on vector bundles. These are irreducible requirements for work with smooth manifolds. It will conclude with a brief introduction to Riemannian geometry.

The course will concentrate on how to work with smooth manifolds, with plenty of explicit computations and concrete examples . Some proofs will be only sketched, but references for complete arguments 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. 

There will be twenty live lecture sessions. PDF lecture notes will be released in advance of these.

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.

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).

Syllabus

Outline syllabus: 

  • Preliminaries: A brief review of point set topology and calculus in R^n
  • Smooth manifolds: atlases, differentiable structures, orientation, calculus on manifolds
  • The tangent bundle: vector fields, the Lie derivative
  • Vector bundles: basic operations and constructions, tensors, connexions, curvature, holonomy
  • Riemannian metrics: the canonical connexion, geodesics, curvature versus topology

Lecturer

  • Professor Martin Speight

    Professor Martin Speight

    University
    University of Leeds

Bibliography

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 8th January 2024 at 00:00 and is due in before Friday 19th January 2024 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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