Courses

Course Information

	Differential Geometry I (Math 380)
	Quantum Mechanics (Mast 684)
	Numerical Analysis (Mast 680B Mast 683)
	Computational Applied Mathematics (Mast 680C)
	Spectral Geometry (Mast 855)
	Reading List

Last update 15 November 1999

Numerical Analysis in C++ (Mast 680B)

Numerical Analysis (Mast 683)

References Students will need to consult books on Numerical Analysis and Scientific Computing. NR = Numerical Recipes by Press et al (Cambridge, 1986) and Introducing C++ for Scientists, Engineers, and Mathematicians by Capper (Springer, 1994) are recommended. NR is well described by its title: for mathematical proofs and detailed results, one must look at the references, and other mathematical literature.
Prerequisites Undergraduate analysis, differential equations, and linear algebra. The C++ language itself will be introduced and studied as part of this course.
Evaluation There will be a sequence of assigments gradually increasing in difficulty. In the final assignment, which will have the flavour of a small project, students will choose one problem from a list of about 10.
Aims This is an elementary course in numerical analysis and computing. Various fundamental topics in numerical analysis will be included. There will be a bias towards analytical problems involving roots, integration, differential equations, optimization, and Fourier transforms. The use of `functional programming' and graphical techniques will be strongly encouraged. Students will be shown how to extend an initial C++ graphics class so that they can take advantage of graphical methods for exploratory purposes. By the end of this course students should have made a good start on the construction of a personal library of tools for exploring and solving mathematical problems numerically.
Computing Students registered in the course will be able to receive a computer account from H925. At the University, the supported dialects are Turbo C++ and Visual C++. These are available in PC LAB C (H923 and H925-1). Students are strongly encouraged to use more private computing environments to which they may have convenient access. For graphics the best dialects for the course are those such as Turbo and Visual that are based on 'Windows'. NB Students who are new to computing should expect that this course will occupy much more time than an ordinary course of mathematics.

Examples and Assignments
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Computational Applied Mathematics (Mast 680C)

References Students will need to consult books on many different topics in mathematics and computing. An annotated reading list is accessible from the home page.
Prerequisites Undergraduate analysis, differential equations, and linear algebra, and the C++ computer language. Students who take this course are expected already to have a basic working knowledge of C++, such as that provided by the courses Mast 683 or Mast 680B.
Evaluation There will be two projects, chosen in consulatation with the professor. The first one will be similar to Assignment 4 of Mast 683 (680B); the second will be more ambitious. They will each require detailed design and planning, both mathematically and computationally.
Aims In the lectures mathematical and computational aspects of problems in the following areas will be considered: approximation theory, ordinary differential equations, calculus of variations, control theory, dynamical systems, partial differential equations, fast Fourier transforms, integral equations, Sturm-Liouville problems, and the discrete spectra of Schrödinger operators. By arrangement a student may choose to work on a project in another area. The lectures will cover the underlying mathematics and computational design problems. It is a principal aim of the course that the students learn how to generate well-documented C++ programs that solve mathematical problems. Although such goals could equally well be achieved in other contemporary computing environments, this course will focus specifically on good design in C++.
Computing Students registered in the course will be able to receive a computer account from H925. At the University, the supported dialects are Turbo C++ and Visual C++. These are available in PC LAB C (H923 and H925-1). Students are strongly encouraged to use more private computing environments to which they may have convenient access. For graphics the best dialects for the course are those such as Turbo and Visual that are based on 'Windows'. NB Students should expect this course to occupy much more of their time than an ordinary course of mathematics.

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Quantum Mechanics (Mast 684)

Books An annotated reading list on mathematics and quantum mechanics may be obtained from the RH web page. Students are advised to acquire a copy of the book An introduction to quantum theory by K. Hannabuss (Oxford, 1997). It may also be necessary to consult a good introductory QM text such as that by Greiner (Springer, 1989), books on classical mechanics, such as Goldstein (Addison-Wesley, 1980), and on operator theory, such as Schechter (North Holland, 1981) or Lieb and Loss (AMS, 1997).
Prerequisites Analysis and a first course in functional analysis
Evaluation There will be a number of problem sets.
Aim To introduce mathematics students to quantum mechanics. The course will start with classical mechanics and then go on to the quantum mechanics of non-relativistic bound systems. Students will be expected to solve problems in mechanics, applied functional analysis, and quantum mechanics. It is hoped that students with a background in mathematics will acquire a basic understanding of quantum mechanics without getting lost in the mysteries of physics or the difficulties of operator theory.

Differential Geometry I (Math 380)

Text Book Elementary Differential Geometry by B. O'Neill (Academic Press, 1997)
Evaluation There will be weekly assigments (10%), one class test (30%), and a final examination (60%). A missed assignment or test cannot be made up. If it benefits the student, the weight of the final exam will be increased to 100%.
The following table gives an indication of the scope and approximate pace of the course, in terms of sections of the text book. Students are expected to read Chapter 3 on Euclidean Geometry the subject matter of which will be examined.

Topics	Sections	weeks
Introduction: a first look at some important concepts	1.1-1.7	1
Curves and Frame fields: the Serret-Frenet apparatus and some extensions	2.1-2.8	4
Test on Chapters I and II	-	1
Euclidean geometry: essentially to be read by the student	3.1-3.5	1
Calculus on a manifold: differentiation, integration, and Stokes's theorem	4.1-4.6	6

Spectral Geometry (Mast 855)

References A list of suitable books and papers on mathematics and quantum mechanics will be given out.
Prerequisites This course has three aspects: mathematics, quantum mechanics, and approximation theory. The techniques are mostly from real analysis. Previous experience with quantum mechanics and operator theory would be helpful, so would some knowledge of computing.
Evaluation There will be assigments (50%) and a small project (50%).
Description Geometrical considerations of the type we shall study in this course enter a quantum-mechanical spectral problem when the Hamiltonian operator H = -Delta + V(x,v), which acts on a suitable domain in L²(Rⁿ), depends smoothly on certain real parameters v. Spectral objects such as eigenvalues depend in turn on v and in so doing they generate curves or surfaces in a Euclidean space. It is the goal of spectral geometry to study these manifolds. Reference to established results from functional analysis is needed to set up the basic problem after which standard techniques of real analysis become the principal tools. For a full appreciation of the significance of these efforts some knowledge of quantum mechanics would be very useful. Since exact results are not always possible, the course will explore what might be called spectral approximation theory. Students will be encouraged to use numerical and graphical methods to investigate specific problems. Access to a micro-computer laboratory will be available, including a set of elementary graphics procedures in C++ for those who wish to use them. Solutions obtained with the aid of Mathematica or Maple are also welcome. Some computer work will be unavoidable.