2020/21

https://www.mathematical-modelling.science/index.php/lectures/warwick-2020-2021

This is the third module in the Warwick algebraic topology sequence.  

The abelian group structure on homology is upgraded via dualization to a natural ring structure on cohomology. This additional structure makes cohomology a more powerful invariant than homology. Cohomology is also a more natural invariant in many contexts, and various cohomology theories play a key role in a number of fields, including differential geometry, mathematical physics, algebraic geometry, and number theory, among others.

Poincaré duality is a relation between the homology and cohomology of an oriented manifold. The ring structure of cohomology plays a critical role in its proof.   

The module MA4K0 Introduction to Uncertainty Quantification sits at the meeting point of mathematics, statistics, and many applications.  We draw on techniques from functional analysis, numerical analysis, probability theory, Bayesian statistics, and computational mathematics to answer questions such as

• How do we represent random or uncertain quantities?
• How can we propagate uncertainty forwards through systems of interest in the real world to make predictions?
• How can we propagate uncertainty backwards through systems of interest in the real world to learn about them, e.g. initial conditions or governing parameters?

Overview

There is much active mathematical research into aeroacoustics (the study of sound in aircraft engines).  This field is closely followed, and often contributed to (sometimes helpfully) by engineers in both academia and industry (e.g. Airbus, Boeing, NASA, etc).  The aim of this course is to give an overview of the mathematical techniques needed to understand the current research problems, and read current papers in the area.  This could lead on to several possible PhD projects, including in asymptotics, numerical analysis, and stability theory.

Aims

The application of wave theory to problems involving the generation, propagation and scattering of acoustic and other waves is of considerable relevance in many practical situations. These include, for example, underwater sound propagation, aircraft noise, remote sensing, the effect of noise in built-up areas, and a variety of medical diagnostic applications. This course would aim to provide the basic theory of wave generation, propagation and scattering, and an overview of the mathematical methods and approximations used to tackle these problems, with emphasis on applications to aeroacoustics.  The ultimate aim is for students to understand the underlying mathematical tools of acoustics sufficiently to read current research publications on acoustics, and to be able to apply these techniques to current research questions within mathematics, engineering and industry.

Learning Outcomes

• Reproduce standard models and arguments for sound generation and propagation.
  
• Apply mathematical techniques to model sound generation and propagation in simple systems.
• Understand and apply Wiener-Hopf factorisation in the scalar case.
  

Approximate Syllabus

• Some general acoustic theory.
• Sound generation by turbulence and moving bodies (including the Lighthill and Ffowcs Williams Hawkings acoustic analogies).
• Scattering (including the scalar Wiener-Hopf technique applied to the Sommerfeld problem of scattering by a sharp edge)
• Long-distance sound propagation including nonlinear and viscous effects.
• Wave-guides.
• High frequencies and Ray Tracing.

Reading List

• D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, et al, "Modern Methods in Analyticial Acoustics", Springer 1992.
• M. Howe, "Acoustics & Aerodynamic Sound", Cambridge 2015 (available online through Warwick Library).
  
• S.W. Rienstra & A. Hirschberg, "An Introduction to Acoustics", (available online).

Content: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.

However, this theory is not only of interest to the applied mathematician: indeed, it is an integral part of any rigorous mathematical training, and is developed here in a systematic way. Just as a `pure' subject like group theory can be part of the daily armoury of the `applied' mathematician , so ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology.

In this module we will cover relatively simple examples, first order equations, linear second order equations and coupled first order linear systems with constant coefficients, for most of which we can find an explicit solution. However, even when we can write the solution down it is important to understand what the solution means, i.e. its `qualitative' properties. This approach is invaluable for equations for which we cannot find an explicit solution.

We also show how the techniques we learned for second order differential equations have natural analogues that can be used to solve difference equations.

The course looks at solutions to differential equations in the cases where we are concerned with one- and two-dimensional systems, where the increase in complexity will be followed during the lectures. At the end of the module, in preparation for more advanced modules in this subject, we will discuss why in three-dimensions we see new phenomena, and have a first glimpse of chaotic solutions.

Aims: To introduce simple differential and difference equations and methods for their solution, to illustrate the importance of a qualitative understanding of these solutions and to understand the techniques of phase-plane analysis.

Objectives: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations) and to interpret their qualitative behaviour; and to do the same for simple difference equations.

Books:

The primary text will be:
J. C. Robinson An Introduction to Ordinary Differential Equations, Cambridge University Press 2003.

Additional references are:
W. Boyce and R. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley 1997.
C. H. Edwards and D. E. Penney, Differential Equations and Boundary Value Problems, Prentice Hall 2000.
K. R. Nagle, E. Saff, and D. A. Snider, Fundamentals of Differential Equations and Boundary Value Problems, Addison Wesley 1999.