# PMA Level 3 Modules

• #### PMA3008    Computer Algebra   (1st or 2nd semester)

Pre-requisite: There are no specific pre-requisites for this module.  You should know what expanding or factorising an algebraic expression means, what the differential and integral calculus are all about, what vectors and matrices are etc. Recent practice at performing calculations in these topics is not needed as the program will do that for you.

Introduction When most people think of computers in the context of mathematics, they think of some numerical calculation being performed thousands of times in order to approximate something. There is much more to the mathematical use of computers than this. Modern computer programs (such as Mathematica, Maple or Sagemath) can, for example, integrate and differentiate symbolically, so when asked for the integral of (1+x+x2)/(1+x+x2 +x3) from 0 to 1, they will return π/8 + 3 log(2)/4. Whilst any calculation that such a program can perform could also be done by a human, they can remember a large number of rules for you and not make silly mistakes. Furthermore these programs can produce beautiful and complicated graphs, evaluate numerical expressions to accuracies of thousands of significant figures and manipulate primes, vectors and matrices.

This module will provide a practical, problem–based introduction to the use of one such program. This module is taught as a mixture of demonstration lectures and computer lab sessions.

Contents:

Using the computer package; simple calculations; manipulating expressions; lists; graph drawing; defining functions; calculus; Boolean expressions and conditional statements; loops; solving equations; vectors and matrices.

The remainder of the module consists of case studies from many areas of pure mathematics.

Assessment:

Exam 100%

During the examination you may bring in any material that you wish, as long as it is on paper. Access to email and the internet will be disabled for the duration of the examination.

• #### PMA3012    Ring Theory   (1st semester)

Pre-requisite: PMA2007 Linear Algebra

The purpose of this module is to give a general introduction to the theory of rings, which is a subject of central importance in algebra. Historically, some of the major discoveries have helped to shape the course of developments of modern abstract algebra. Today, ring theory is a possible meeting ground for many algebraic sub-disciplines such as group theory, representation theory, Lie theory, algebraic geometry, homological algebra, to name but a few.

Main topics:
Rings, subrings, ideals, quotient rings, homomorphisms, canonical factorisation, isomorphism theorems, integral domains, principal ideal rings, fields, simple rings, Noetherian rings, polynomial rings, Hilbert's basis theorem.

Assessment

Exam 80%    Project 20%

Textbooks
J. Beachy, Introductory lectures on rings and modules (London Math. Soc. Student Text No. 47, Cambridge University Press)
T. W. Hungerford, Algebra (Springer GTM, 1971)

• #### PMA3013  Mathematical Investigations   ( 2nd semester)

Pre-Requisites:

Lecturers:     Dr S Shkarin

Course Content

This module is concerned with the investigation processes of mathematics, including the construction of conjectures based on simple examples and the testing of these with further examples, aided by computers where appropriate. A variety of case studies will be used to illustrate these processes. A series of group and individual investigations will be made by students under supervision, an oral presentation will be made on one of these investigations. While some of the investigations require little more than GCSE as a background, students will be required to undertake at least one investigation which needs knowledge of Mathematics at Level 2 or Level 3 standard and/or some background reading.

Assessment

Solo Project 50%   Group Project 40%   Presentation 10%

• #### PMA3014   Set Theory    (1st semester)

Pre-requisite: PMA2007 Linear Algebra. This module is a compulsory component of the MSci course in Pure Mathematics.

Set theory is the language in which most of contemporary Pure Mathematics is most readily expressed. It is also a subject of study in its own right, whose techniques and insights find application across the entire discipline and whose unresolved/unresolvable issues compel us to question our “intuitive expectation of certainty” in many areas. This module will seek to teach fluency in the language of elementary set theory, facility in the use of key techniques such as transfinite induction and maximality principles, and basic arithmetic of cardinal and ordinal numbers (the ‘arithmetic of infinity’). It will also develop an axiomatic description of set theory to allow some discussion of the issues of completeness and consistency.

The chapters and their approximate numbers of lectures are as follows:

• The language of sets and mappings [7 lectures]
• Construction of number systems [5 lectures]
• Cardinal numbers (cardinality, theorems of Schroeder-Bernstein and Cantor, elementary cardinal arithmetic) [5 lectures]
• Axiomatic set theories (an introduction to the Zermelo-Fraenkel approach and the axiom of choice, and to Zorn's lemma and the well ordering principle) [5 lectures]
• Ordinal numbers (including their application to defining cardinality, and some uses of transfinite induction) [7 lectures]

Assessment

Exam 70%   Project x2 30%

Textbooks
No prescribed text will be followed in detail. The following references may be of use:
Lipschutz, Set Theory and Related Topics (Schaum).
Simmons, Topology and Modern Analysis (McGraw-Hill).
Rotman and Kneebone, Theory of Sets and Transfinite Numbers (Oldbourne).
Stewart, Foundations of Mathematics (OUP).
In addition, the first (or zeroeth!) chapter of almost any book on modern algebra, analysis or topology will provide some discussion of set theory.

• #### PMA3017   Metric and Normed Spaces   (2nd semester)

Pre-requisite: PMA2002 Analysis

Introduction

Analysis is the study of convergence and continuity, it is fundamentaly linked to the structure of the real numbers. The aim of this course is to move gradually away from real analysis to the more general setting of metric and normed spaces. The fundamental question is: what aspects of real analysis survive to the setting of metric and normed spaces?

A metric space is a set with a notion of distance, called a metric. Them most familiar example is the real line, with the distance from x to y given by | x - y | . This notion of distance allows us to define convergence and continuity in a much more abstract setting.

Often, we want the set underlying our metric space to have an addition operation. To get interesting structures, we must then add some extra compatibility conditions to our metric. This leads us to the concept of a normed vector space, this is a vector space as we have encountered before, but now it comes with a measure of the size of a vector, called the norm. As every normed vector space is a metric space, we will begin with the study of metric spaces.

Contents

Definition and examples of metric spaces; open sets, closed sets, closure points, sequential convergence, compactness; completeness; continuous mappings between metric spaces; Banach's fixed-point theorem and applications; Baire category theorem and applications. Normed spaces; Banach spaces; finite dimensional normed spaces; subspaces and quotient spaces; linear operators; boundedness; compact operators; dual spaces.

Assessment

Exam 80%   Presentation 10%   Continuous Assessment 10%

• #### PMA3018   Algebraic Equations   (2nd semester)

Pre-requisite: PMA2007 Linear Algebra and PMA2008 Group Theory

The theory of algebraic equations is the study of solutions of polynomial equations. Although the problem originates in explicit manipulations of polynomials, the modern treatment is in terms of field extensions and groups of ‘symmetries’ of fields.

The content includes: review of polynomial rings and characteristic of rings and fields, factoring polynomials, extension fields, construction of some extension fields, algebraic and transcendental elements, constructions with straight-edge and compass, splitting fields, the fundamental theorem of Galois theory, groups of automorphisms of fields, separable, normal and Galois extensions, examples.

Assessment

Exam 80%   Presentation 10%   Assignment 10%