Content: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.
However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a `pure' subject like group theory can be part of the daily armoury of the `applied' mathematician , so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.
In this module we will cover relatively simple examples, first order equations
,
linear second order equations
and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, i.e. its `qualitative' properties. This approach is invaluable for equations for which we cannot find an explicit solution.
We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.
The course looks at solutions to differential equations in the cases where we are concerned with one- and two-dimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in three-dimensions we see new phenomena, and have a first glimpse of chaotic solutions.
Aims: To introduce simple differential and difference equations and methods for their solution, to illustrate the importance of a qualitative understanding of these solutions and to understand the techniques of phase-plane analysis.
Objectives: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour; and to do the same for simple difference equations.
Books:
The primary text will be:
J. C. Robinson An Introduction to Ordinary Differential Equations, Cambridge University Press 2003.
Additional references are:
W. Boyce and R. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley 1997.
C. H. Edwards and D. E. Penney, Differential Equations and Boundary Value Problems, Prentice Hall 2000.
K. R. Nagle, E. Saff, and D. A. Snider, Fundamentals of Differential Equations and Boundary Value Problems, Addison Wesley 1999.
I have been asked about content of the exam. Here is the basic information:
The only non-examinable section is the last chapter devoted to computational complexity (Chapter 12).
Otherwise, only simple proofs will feature in the compulsory question 1.
More involved proofs might be part of the remaining optional questions.
Also, check the last year exam with solutions.
Lecturer: Christian Boehning
Term(s): Term 1
Status for Mathematics students: List A for Mathematics
Commitment: 30 lectures plus weekly worksheets
Assessment: The weekly worksheets carry 15% assessed credit; the remaining 85% credit by 2-hour examination.
Prerequisites: None, but an understanding of MA125 Introduction to Geometry will be helpful.
Leads To: Third and fourth year courses in Algebra and Geometry, including: MA3D9 Geometry of Curves and Surfaces, MA3E1 Groups and Representations, MA4A5 Algebraic Geometry, MA4E0 Lie Groups, MA473 Reflection Groups, MA4H4 Geometric Group Theory, MA448 Hyperbolic Geometry and others
Content: Geometry is the attempt to understand and describe the world around us and all that is in it; it is the central activity in many branches of mathematics and physics, and offers a whole range of views on the nature and meaning of the universe.
Klein's Erlangen program describes geometry as the study of properties invariant under a group of transformations. Affine and projective geometries consider properties such as collinearity of points, and the typical group is the full matrix group. Metric geometries, such as Euclidean geometry and hyperbolic geometry (the non-Euclidean geometry of Gauss, Lobachevsky and Bolyai) include the property of distance between two points, and the typical group is the group of rigid motions (isometries or congruences) of 3-space. The study of the group of motions throws light on the chosen model of the world.
Aims: To introduce students to various interesting geometries via explicit examples; to emphasize the importance of the algebraic concept of group in the geometric framework; to illustrate the historical development of a mathematical subject by the discussion of parallelism.
Objectives: Students at the end of the module should be able to give a full analysis of Euclidean geometry; discuss the geometry of the sphere and the hyperbolic plane; compare the different geometries in terms of their metric properties, trigonometry and parallels; concentrate on the abstract properties of lines and their incidence relation, leading to the idea of affine and projective geometry.
Books:
M Reid and B Szendröi, Geometry and Topology, CUP, 2005 (some Chapters will be available from the General office).
E G Rees, Notes on Geometry, Springer
HSM Coxeter, Introduction to Geometry, John Wiley & Sons
Aims: To develop further and to continue the study of linear algebra, which was begun in Year 1.
To point out and briefly discuss applications of the techniques developed to other branches of mathematics, physics, etc.
Objectives: By the end of the module students should be familiar with: the theory and computation of the the Jordan canonical form of matrices and linear maps; bilinear forms, quadratic forms, and choosing canonical bases for these; the theory and computation of the Smith normal form for matrices over the integers, and its application to finitely generated abelian groups.
The focus of combinatorial optimisation is on finding the "optimal" object (i.e. an object that maximises or minimises a particular function) from a finite set of mathematical objects. Problems of this type arise frequently in real world settings and throughout pure and applied mathematics, operations research and theoretical computer science. Typically, it is impractical to apply an exhaustive search as the number of possible solutions grows rapidly with the "size" of the input to the problem. The aim of combinatorial optimisation is to find more clever methods (i.e. algorithms) for exploring the solution space.
This module provides an introduction to combinatorial optimisation. Our main focus is on several fundamental problems arising in graph theory and linear programming and algorithms developed to solve them. These include problems related to shortest paths, minimum weight spanning trees, linear programming, matchings, network flows, cliques, colourings, dynamic programming, multicommodity flows and matroids. We will also discuss "intractible" (e.g. NP-hard) problems.