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6CCM320a Topics in Mathematics

This module will consist of four ‘mini-modules’ of 10-12 hours duration, and will thus enable you to gain a satisfactory understanding of the key concepts and applications of a selection of important topics in both pure and applicable mathematics. 

Module Coordinator: Professor Simon Salamon

You need only attend any selection of THREE of these four mini-modules

Semester 1: Markov Chains (w/c 26 September - 24 October 2016)

Semester 1: Time Series  (w/c 7 November - 12 December 2016)

Semester 2: Game Theory (w/c 16 January – 13 February 2017)

Semester 2: Prime Numbers (w/c 27 February - 27 March 2017) 

Semester: 1 and 2

Credit Level: 6       Credit Value: 15


Mathematics BSc/MSci  Third Year

Mathematics with a Year Abroad BSc    Third Year

Mathematics with Management and Finance BSc  Third Year

Mathematics and Computer Science BSc  Third Year

Mathematics and Physics BSc/MSci   Third Year

Graduate Diploma

Teaching arrangements   :

Two hours of teaching each week.

Suggested reading: (link to My Reading Lists)

Summative Assessment:

This is solely by means of a single two-hour examination at the end of the academic year, consisting of one section for each of the mini-modules. Each of the four selections will have equal weight and you may answer questions from any of the sections

 Type Weight Marking Model
 2hr written examination 100%  Model 2
  Distribution of Prime Numbers

Lecturer:  Dr Yanki Lekili


4CCM111a Calculus I, 4CCM121a/5CCM121b Introduction to Abstract Algebra


The aim of this module is to discuss several important results about the distribution of prime numbers, and to give an understanding of some of the techniques used to prove these results.


Divisibility theory of the integers, basic distribution issues, the prime number theorem, the Riemann zeta function, arithmetic functions and Dirichlet series, primes and arithmetic progressions.

Formative assessment

Problem sheets will be handed out each week.



Markov Chains

Lecturer:  Dr Gechun Liang


4CCM113a (CM113A) Linear methods, 4CCM111a (CM111A) Calculus I and 4CCM112a (CM112A) Calculus II, 4CCM141a (CM141A) Probability and Statistics

Advice/Help for this topic is also available from: Professor Reimer Kuehn

Aims and objectives

The module aims to introduce the basic concepts of finite-state, discrete-time Markov chains and to illustrate these with applications to a range of problems.


1. Review of Probability theory: (a) Finite, Countable and Uncountable Sets (b) Probability Space (c) Discrete Random Variables (Vectors)

2. Introduction to Discrete Stochastics Processes and Markov Chains: (a) Discrete Stochastic Processes (b) Independence and Conditional Probability (c) Discrete-time Markov Chains (d) Transition Matrix

3. Representations of Markov Chains: (a) Computation of n-step Transition Matrix (b) Transition Graph (c) Markov Recursive Equations

4. Absorption Probability and Strong Markov Property: (a) First-step Analysis of Absorption Probability (b) Strong Markov Property

5. Recurrence, Transcience and Stationary Distributions: (a) Recurrence and Transcience (b) Stationary Distribution

Formative assessment 

Problems will be set each week. Solutions will be made available later.



Game Theory

Lecturer:  Professor Simon Salamon


4CCM113a Linear Methods

Advice/Help for this topic is also available from: Dr John Armstrong


To understand what is meant by best strategy in both the pure (discrete) and mixed (probabilistic) sense, to prove some basic theorems in the subject, and to show how to "win" at matrix games. The course also touches on questions concerning the interpretation of probability and the nature of human behaviour.


Two-person games described by matrices and trees, information sets, pure and mixed strategies, backward induction.  Solving matrix games, dominated strategies, Minimax theorems, the concept of Nash equilibrium, the Shapley-Snow algorithm, extreme solutions.

Formative assessment

Exercise sheet handed out each week. Solutions will be provided


Time Series

Lecturer:  Professor Michael Pitt

Advice/Help for this topic is also available from: tbc


Calculus II (4CCM112a), Linear Methods (4CCM113a), Probability and Statistics I (4CCM141a/5CCM141b), Probability and Statistics II (5CCM241a/6CCM214b)


The module aims at giving you a basic understanding of the methods and mathematical theory of time series.


Stationary processes, auto-correlation and autocovariance functions; Moving Average (MA) processes, Auto-Regressive (AR) processes and Auto-Regressive/Moving Average (ARMA) processes; correlogram; spectral analysis, periodogram; elements of estimation and forecasting and applications to empirical data.

Formative assessment

Problem Sheets will be made available via the module webpage.

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11 May 2018
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