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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Introduction to Differential Topology (MATH11122)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 4 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummarySmooth manifolds are a universally occurring type of metric space, looking locally like familiar n-dimensional euclidean spaces, but differing from them `in the large'. For n=1, the first example beyond the real line is the circle, and for n=2 (surfaces) we have a family of topologically different possibilities, such as the (surface of the) sphere, the torus, pretzels and so on.

Manifolds provide the natural setting for a general study of differentiable (smooth) functions and mappings and differential topology looks at their basic properties. It provides some basic tools with which to study manifolds and, as is the case with the best mathematics, gives proofs of results about familiar objects that are not easy to obtain by elementary means. For example, we shall give a proof of the fundamental theorem of algebra from this point of view, as well as the famous Brouwer fixed-point theorem, which asserts that any continuous map from the closed ball to itself must have a fixed point. We shall also define and study the Euler characteristic for compact orientable manifolds; this distinguishes the different types of surface mentioned above.

This course is part of our geometry/topology sequence. However, its methods also underlie part of the basic theory of nonlinear partial differential equations, which appears, roughly speaking, as an infinite-dimensional extension of these ideas. It is therefore also of
relevance to students with an interest in these matters as well.

Aims :
1. To define smooth manifolds as certain subspaces of euclidean spaces
2. To define smooth maps between manifolds and give the implicit function theorem
3. To study regular and singular values of smooth maps
4. To define the degree of a smooth map and give standard topological applications
5. To define and study the Euler characteristic (or number) of a compact orientable manifold, including the classification of compact oriented surfaces.
Course description Smooth manifolds and maps, implicit function theorem, manifolds with boundary, regular values and critical values, Sard's theorem, degree of maps, Euler number, vector fields and the Hopf index theorem.

Essentially, chapters 1-6 of Milnor's book, see Reading List below.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: ( Foundations of Calculus (MATH08005) AND Several Variable Calculus (MATH08006) AND Linear Algebra (MATH08007) AND Methods of Applied Mathematics (MATH08035))
Co-requisites It is RECOMMENDED that students also take Topology (MATH10027)
Prohibited Combinations Other requirements It is recommended that students should have taken the first part of MATH10027 Topology. Differential Geometry also provides useful background, but is not essentially.
Information for Visiting Students
Course Delivery Information
Not being delivered
Learning Outcomes
1. Gain experience of smooth manifolds, smooth maps and their basic properties
2. Gain experience of topological methods and results in the context of smooth manifolds
Reading List
We shall follow Milnor's book closely, referring perhaps to the others for further applications.

1. Milnor, Topology from the differentiable viewpoint, QA613.6 Mil.
2. Hirsch, Differential Topology, QA613.6 Hir
3. Guillemin and Pollack, Differential topology, QA613.6 Gui
Additional Information
Graduate Attributes and Skills Not entered
Course organiserDr Tom Mackay
Tel: (0131 6)50 5058
Course secretaryMrs Alison Fairgrieve
Tel: (0131 6)50 5045
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