PMA Level 3 Modules

PMA3008 Computer Algebra (1st and 2nd semester)
Prerequisite: There are no specific prerequisites 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.
Lecturers: Prof M Paternostro, Dr D Barnes 
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 (1^{st} semester)
Prerequisite: MTH2001 Linear Algebra and Complex Variables
Introduction
Lecturer: Dr YF LinThe 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 subdisciplines such as group theory, representation theory, Lie theory, algebraic geometry, homological algebra, to name but a few.
Content
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)
PreRequisites:
Lecturer: Dr F Pausinger, Dr S Shkarin
Introduction
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 (1^{st} semester)
Prerequisite: PMA2007 Linear Algebra. This module is a compulsory component of the MSci course in Pure Mathematics.
Lecturer: Dr M MathieuIntroduction
Set Theory is a rich and beautiful subject of Pure Mathematics whose fundamental concepts permeate every branch of mathematics. No undergraduate mathematics education is complete without a thorough study of this discipline. This module will teach fluency in the language of elementary axiomatic 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’).
Content
Naïve vs. Axiomatic Set Theory, ZermeloFraenkel axioms, Axiom of Choice, WellOrdering Principle, Zorn’s Lemma, Transfinite Induction, Construction of the number systems, Ordinal and cardinal numbers and their arithmetic.
Assessment70% of the module mark is provided by a written 3hour exam in May 2021.
30% of the module mark are made up by two marked homework assignments due in week 4 and in week 9.Textbooks
D. W. Cunningham, Set theory: A first course, Cambridge Math. Textbooks, Cambridge Univ. Press, New York, 2017.
I. Kaplansky, Set theory and metric spaces, Chelsea Publ. Company, New York, 1977.

PMA3017 Metric and Normed Spaces (2^{nd} semester)
Prerequisite: PMA2002 Analysis
Lecturers: Dr D Barnes, Dr YF LinIntroduction
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 fixedpoint 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 (2^{nd} semester)
Prerequisites: PMA2007 Linear Algebra, PMA2008 Group Theory and PMA3012 Ring Theory.
Lecturer: Dr M MathieuIntroduction
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.
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 straightedge 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% Project 10%