Announcements


A replacement for the lecture on Wednesday January 25th, which I was unable to give, will be arranged later in the semester.

There is a set of background notes, notes-date.pdf, which will be enlarged as the course progresses.

There is also an annotated version of the bibliography.

Forum

General


This course is part of the MAGIC core.

Description

This course is designed for PhD students in pure or in applied mathematics. 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.

If you are enrolled in the course, or considering enrolling, please feel free to email me any questions or comments about the course.

Semester

Spring 2017 (Monday, January 23 to Friday, March 31)

Timetable

  • Mon 10:05 - 10:55
  • Wed 10:05 - 10:55

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

Students


Photo of Thomas Baker
Thomas Baker
(*Ext_Assessed)
Photo of Jinrong Bao
Jinrong Bao
(Loughborough)
Photo of John Blackman
John Blackman
(Durham)
Photo of Bobby Cheng
Bobby Cheng
(Sussex)
Photo of Paul Druce
Paul Druce
(Nottingham)
Photo of Michael Foskett
Michael Foskett
(Surrey)
Photo of Raffaele Grande
Raffaele Grande
(Cardiff)
Photo of Neil Hansford
Neil Hansford
(Sheffield)
Photo of Thomas Morley
Thomas Morley
(Sheffield)
Photo of Erik Paemurru
Erik Paemurru
(Loughborough)
Photo of Norbert Pintye
Norbert Pintye
(Loughborough)
Photo of Matthew Poulter
Matthew Poulter
(Lancaster)
Photo of Roberto Sisca
Roberto Sisca
(Surrey)
Photo of Ville Syrjanen
Ville Syrjanen
(Sussex)
Photo of Andrew Turner
Andrew Turner
(Birmingham)
Photo of Patrick WRIGHT
Patrick WRIGHT
(Leeds)


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
Connections, Curvature and Cohomology: Volume IGreub, Halperin and Vanstone
Connections, Curvature, and Cohomology: Volume IIGreub, Halperin and Vanstone
Introduction to Smooth ManifoldsLee
Treatise on analysis, Volume IIIDieudonné
Differential Geometry of Curves and Surfacesdo Carmo
Poisson StructuresLaurent-Gengoux, Pichereau and Vanhaecke
Lectures on the Geometry of Poisson ManifoldsVaisman
Introduction to Symplectic TopologyMcDuff and Salamon
Applicable Differential GeometryCrampin and Pirani
Differential Manifolds and Theoretical PhysicsW D Curtis and F R Miller
Geometrical methods of mathematical physicsSchutz


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



Assessment for MAGIC 063 Differentialble Manifolds will be via a single take-home paper in April/May with 2 weeks to complete and submit online.
To pass the exam you are required to complete at least 3 out of the 4 questions and to obtain at least 50 marks.
There will be more information on the paper itself.

No assignments have been set for this course.

Recorded Lectures


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