MA4TPA: Topics in Pure and Applied Mathematics

Module code: MA4TPA

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 4 (Undergraduate Masters)

When you'll be taught: Semester 1

Module convenor: Professor Simon Chandler-Wilde, email: s.n.chandler-wilde@reading.ac.uk

Module co-convenor: Dr Jochen Broecker, email: j.broecker@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST TAKE MA2DE AND TAKE MA2RA2 (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded: IN TAKING THIS MODULE YOU CANNOT TAKE MA3TPA (Compulsory)

Placement information: NA

Academic year: 2024/5

Available to visiting students: No

Talis reading list: Yes

Last updated: 21 May 2024

Overview

Module aims and purpose

This module tackles two topics in pure and applied mathematics, namely i) integral equations; ii) stochastic processes. The integral equations part of the module provides a first introduction to the theory of integral equations, numerical methods for their solution, and applications, including in meteorology. The stochastic processes part of the module provides a rigorous introduction to the theory of Brownian motion, stochastic analysis (including stochastic integrals and stochastic differential equations) and martingales.

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Formulate integral equations as problems in a Banach space, apply a range of approximation and numerical solution techniques, and draw conclusions about their accuracy
2. Formulate wave-scattering problems as integral equations
3. Carry out analyses of Brownian Motion and Martingales
4. Carry out analyses using Ito calculus, in particular on stochastic differential equations

Module content

This module will tackle two topics in pure and applied mathematics, namely: i) integral equations, their theory and applications; ii) stochastic processes.

In the integral equations section of the module we will provide a general introduction to the key aspects of the theory, introducing functional analysis ideas as needed, following this up by a discussion of widely used approximation and numerical techniques, and a detailed examination of the application of integral equations to model real-world wave-scattering problems, arising in our research in mathematics, and in applications in meteorology. The main elements of the integral equations section are:

• Classification of integral equations
• Exact solution of degenerate kernel Fredholm integral equations
• Questions of uniqueness and existence of solution (tackled by functional analysis methods): the Fredholm alternative and Neumann series
• Numerical methods for Fredholm and Volterra integral equations, namely degenerate kernel approximations and Trapezium rule time-stepping
• Applications of integral equation methods to wave scattering: the Lippmann Schwinger integral equation and application in atmospheric particle scattering
• A complete numerical analysis of the trapezium rule method for Volterra integral equations via Gronwall inequalities, their discrete counterparts, and regularity results

In the stochastic processes section of the module we will construct Brownian motion, a stochastic process which can be interpreted as the integral of white noise. Then we discuss stochastic integrals, that is, integrals against Brownian motion, leading to the theory of stochastic differential equations. Key aspects of the theory will be discussed, including applications of Stochastic Differential Equations as well as other, related processes called Martingales. The main elements are:

• Construction of Brownian motion
• Properties of Brownian motion: Scaling, Law of large numbers, law of the iterated logarithm, Hölder continuity of paths
• Construction of the Ito integral
• Continuity of the Ito integral as a function of the upper limit
• Martingales
• Ito formula
• Stochastic differential equations

Structure

Teaching and learning methods

Lectures supported by problem sheets and tutorials as well as a small peer-to-peer reading group on measure and integration basics.

Study hours

At least 30 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 50% to pass this module

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Peer-to-peer reading group on measure and integration basics: students will produce a short report which will be discussed during two feedback sessions.

Reassessment

Additional costs