# Modules

Students choose eight taught modules which account for two-thirds of their overall grade. Part-time students choose four modules in their first-year and four modules in their second-year.

**Note: The modules listed here are for the current academic year. The programme is substantially the same from year to year, but modules are subject to change depending on your year of entry.**

The Taught Course Centre (TCC) is a collaboration between the Mathematics Departments at the Universities of Bath, Bristol, Imperial, Oxford and Warwick. The lectures are open to all postgraduate students and are taking place in the room 6M42 (Huxley Building). The link to TCC can be found here.

## Applied analysis

### MATH97104 Introduction to Partial Differential Equations (M5M3)

**Dr M. Coti Zelati**

**Term 1**

1. Basic concepts: PDEs, linearity, superposition principle. Boundary and Initial value problems.

2. Gauss Theorem: gradient, divergence and rotational. Main actors: continuity, heat or diffusion, Poisson-Laplace, and the wave equations.

3. Linear and Qasilinear first order PDEs in two independent variables. Well-posedness for the Cauchy problem. The linear transport equation. Upwinding scheme for the discretization of the advection equation.

4. A brief introduction to conservation laws: The traffic equation and the Burgers equation. Singularities.

5. Derivation of the heat equation. The boundary value problem: separation of variables. Fourier Series. Explicit Euler scheme for the 1d heat equation: stability.

6. The Cauchy problem for the heat equation: Poisson’s Formula. Uniqueness by maximum principle.

7. The ID wave equation. D’Alembert Formula. The boundary value problem by Fourier Series. Explicit finite difference scheme for the 1d wave equation: stability.

8. 2D and 3D waves. Casuality and Energy conservation: Huygens principle.

9. Green’s functions: Newtonian potentials. Dirichlet and Neumann problems.

10. Harmonic functions. Uniqueness: mean property and maximum principles.

### MATH97102 Function Spaces and Applications (M5M11)

**Dr A. Giunti**

**Term 2**

The purpose of this course is to introduce the basic function spaces and to train the student into the basic methodologies needed to undertake the analysis of Partial Differential Equations and to prepare them for the course ‘Advanced topics in Partial Differential Equations’’ where this framework will be applied. The course is designed as a stand-alone course. No background in topology or measure theory is needed as these concepts will be reviewed at the beginning of the course.

The course will span the basic aspects of modern functional spaces: integration theory, Banach spaces, spaces of differentiable functions and of integrable functions, convolution and regularization, compactness and Hilbert spaces. The concepts of Distributions, Fourier transforms and Sobolev spaces will be taught in the follow-up course ‘’Advanced topics in Partial Differential Equations’’ as they are tightly connected to the resolution of elliptic PDE’s and the material taught in the present course is already significant.

In addition to the material below, this Masters version of the module will have additional extension material for self-study. This will require a deeper understanding of the subject than the corresponding M3 module.

The syllabus of the course is as follows:

1) Elements of metric topology

2) Elements of Lebesgue’s integration theory.

3) Normed vector spaces. Banach spaces. Continuous linear maps. Dual of a Banach space. Examples of function spaces: continuously differentiable function spaces and Lebesgue spaces. Hölder and Minkowski’s inequalities. Support of a function; Convolution. Young’s inequality for the convolution. Mollifiers. Approximation of continuous or Lebesgue integrable functions by infinitely differentiable functions with compact support.

4) Compactness. Non-compactness of the unit ball in infinite-dimensional normed vector spaces. Criteria for compactness in space of continuous functions: the Ascoli theorem. Compact operators. Additional reading about the weak topology.

5) Hilbert spaces. The projection theorem. The Riesz representation theorem. The Lax-Milgram theorem. Hilbert bases and Parseval’s identity. Application to Fourier series. Compact operators in Hilbert spaces. The Fredholm alternative. Spectral decomposition of compact self-adjoint operators in Hilbert spaces.

### MATH97098 Stochastic Differential Equations (M5A51)

**Professor G. Pavliotis**

**Term 1**

This is a basic introductory course on the theory and applications of stochastic differential equations. The goal will be to present the basic theory of SDEs and, time permitting, to also present some specific applications such as stochastic optimal control, applications of SDEs to Partial differential equations etc.

The contents of the module are:

1) Modelling with SDEs, overview of applications.

2) Background on probability theory and the theory of stochastic processes.

3) Ito’s theory of stochastic integration and Ito’s formula.

4) Stochatic differential equations, basic theory including existence and uniqueness of solutions.

5) Ergodic properties of SDEs.

6) Connection between SDEs and the forward and backward Kolmogorov equations.

7) Applications including stochastic optimal control, Feynman-Kac formulas, etc.

### MATH97288 Markov Processes (M5P70)

**Prof X-M.Li**

**Term 2**

Markov processes are widely used to model random evolutions with the Markov property `given the present, the future is independent of the past’. The theory connects with many other subjects in mathematics and has vast applications. This course is an introduction to Markov processes. We aim to build intuitions and foundations for further studies in stochastic analysis and in stochastic modelling.

The module is largely self-contained, but it would be useful for students to also take Measure and Integration (M345P19). A good knowledge of real analysis would be helpful (M2PM1).

It is related to:

Applied probability (M345S4), Random Dynamical Systems and Ergodic Theory (M4PA40), Probability theory (M345P6), Stochastic Calculus with Applications to non-Linear Filtering (M45P67), Stochastic Differential Equations (M45A51),

Stochastic simulation (M4S9*), Ergodic Theory (M4PA36), Computational Stochastic Processes (M4A44), and many Mathematical Finance modules. **Contents: **

- Discrete time and finite state Markov chains : Chapman-Kolmogorov equations, irreducible, Perron-Froebenius theorem for stochastic matrices, recurrent and transient.
- Discrete time Markov processes on general state space. Conditional expectations, Chapman-Kolmogorov equation, Feller property, strong Feller property, Kolmogorov's theorem, stopping times, strong Markov, stationary process, weak convergence and Prohorov's theorem, Existence of invariant measures : Krylov-Bogolubov method, Lyapunov method. Ergodicity by contraction method and Doeblin's criterion. Structures of invariant measures, ergodic theorems.

## Numerical analysis and computation

### MATH97139 Numerical Solution of Ordinary Differential Equations (M5N7)

**Dr I. ShevchenkoTerm 1**

** **An analysis of methods for solving ordinary differential equations. Totally examined by project.

Runge-Kutta, extrapolation and linear multistep methods. Analysis of stability and convergence. Error estimation and automatic step control. Introduction to stiffness. Boundary and eigenvalue problems. Solution by shooting and finite difference methods. Introduction to deferred and defect correction.

### MATH97140 Computational Linear Algebra (M5N9)

**Professor C. Cotter**

Term 1

Examined solely by project. Competence in Python is a prerequisite.

Whether it be statistics, mathematical finance, or applied mathematics, the numerical implementation of many of the theories arising in these fields relies on solving a system of linear equations, and often doing so as quickly as possible to obtain a useful result in a reasonable time. This course explores the different methods used to solve linear systems (as well as perform other linear algebra computations) and has equal emphasis on mathematical analysis and practical applications.

Topics include:

1. Direct methods: Triangular and banded matrices, Gauss elimination, LU-decomposition, conditioning and finite-precision arithmetic, pivoting, Cholesky factorisation, QR factorisation.

2. Symmetric eigenvalue problem: power method and variants, Jacobi's method, Householder reduction to tridiagonal form, eigenvalues of tridiagonal matrices, the QR method

3. Iterative methods:

(a) Classic iterative methods: Richardson, Jacobi, Gauss - Seidel, SOR

(b) Krylov subspace methods: Lanczos method and Arnoldi iteration, conjugate gradient method, GMRES, preconditioning.

### MATH97095 Finite Elements: Numerical Analysis and Implementation (M5A47)

**Professor C. Cotter and Dr D. HamTerm 2**

Finite element methods form a flexible class of techniques for numerical solution of PDEs that are both accurate and efficient. The finite element method is a core mathematical technique underpinning much of the development of simulation science. Applications are as diverse as the structural mechanics of buildings, the weather forecast, and pricing financial instruments. Finite element methods have a powerful mathematical abstraction based on the language of function spaces, inner products, norms and operators.

This module aims to develop a deep understanding of the finite element method by spanning both its analysis and implementation. in the analysis part of the module you will employ the mathematical abstractions of the finite element method to analyse the existence, stability, and accuracy of numerical solutions to PDEs. At the same time, in the implementation part of the module you will combine these abstractions with modern software engineering tools to create and understand a computer implementation of the finite element method.

Syllabus:

• Basic concepts: Weak formulation of boundary value problems, Ritz-Galerkin approximation, error estimates, piecewise polynomial spaces, local estimates.

• Efficient construction of finite element spaces in one dimension, 1D quadrature, global assembly of mass matrix and Laplace matrix.

• Construction of a finite element space: Ciarlet’s finite element, various element types, finite element interpolants.

• Construction of local bases for finite elements, efficient local assembly.

• Sobolev Spaces: generalised derivatives, Sobolev norms and spaces, Sobolev’s inequality.

• Numerical quadrature on simplices. Employing the pullback to integrate on a reference element.

• Variational formulation of elliptic boundary value problems: Riesz representation theorem, symmetric and nonsymmetric variational problems, Lax-Milgram theorem, finite element approximation estimates.

• Computational meshes: meshes as graphs of topological entities. Discrete function spaces on meshes, local and global numbering.

• Global assembly for Poisson equation, implementation of boundary conditions. General approach for nonlinear elliptic PDEs.

• Variational problems: Poisson’s equation, variational approximation of Poisson’s equation, elliptic regularity estimates, general second-order elliptic operators and their variational approximation.

• Residual form, the Gâteaux derivative and techniques for nonlinear problems.

The course is assessed 50% by examination and 50% by coursework (implementation exercise in Python).

### MATH97138 Computational Partial Differential Equations (M5N10)

**Dr S. MughalTerm 2**

The module will introduce a variety of computational approaches for solving partial differential equations, focusing mostly on finite difference methods, but also touching on finite volume and spectral methods. Students will gain experience implementing the methods and writing/modifying short programs in Matlab or other programming language of their choice. Applications will be drawn from problems arising in Mathematical Biology, Fluid Dynamics, etc. At the end of the module, students should be able to solve research-level problems by combining various techniques.

Topics (as time permits).

- Finite difference methods for linear problems: order of accuracy, consistency, stability and convergence, CFL condition, von Neumann stability analysis, stability regions; multi-step formula and multi-stage techniques.

- Solvers for elliptic problems: direct and iterative solvers, Jacobi and Gauss-Seidel method and convergence analysis; geometric multigrid method.

- Methods for the heat equation: explicit versus implicit schemes; stiffness.

- Techniques for the wave equation: finite-difference solution, characteristic formulation, non-reflecting boundary conditions, one-way wave equations, perfectly matched layers. Lax-Friedrichs, Lax Wendroff, upwind and semi-Lagrangian advection schemes.

- Domain decomposition for elliptic equations: overlapping alternating Schwarz method and convergence analysis, non-overlapping methods.

### MATH97186 Scientific Computation (M5SC)

**Dr P. RayTerm 2**

Scientific computing is an important skill for any mathematician. It requires both knowledge of algorithms and proficiency in a scientific programming language. The aim of this module is to expose students from a varied mathematical background to efficient algorithms to solve mathematical problems using computation.

The objectives are that by the end of the module all students should have a good familiarity with the essential elements of the Python programming language, and be able to undertake programming tasks in a range of common areas (see below).

There will be four sub-modules: 1. A PDE-module covering elementary methods for the solution of time dependent problems. 2. An optimization-module covering discrete and derivative-free algorithms. 3. A pattern recognition-module covering searching and matching methods. 4. A statistics-module covering, e.g., Monte-Carlo techniques.

Each module will consist of a brief introduction to the underlying algorithm, its implementation in the python programming language, and an application to real-life situations.

## Analytical methods

### MATH97106 Asymptotic Analysis (M5M7)

**Dr O. SchnitzerTerm 1**

** **Asymptotic series and expansions. Asymptotic expansion of integrals, Laplace method, Watson's lemma, stationary phase and steepest descent. Singular perturbations, matched asymptotic expansions: inner/outer expansions and the matching principle, boundary layers and interior layers. Multiple-scale method, Poincaré-Lindstedt method, method of strained co-ordinate, averaging method, non-linear oscillations. Differential equations with a large parameter - the WKBJ method, turning point problems, caustics.

### MATH97105 Applied complex analysis (M5M6)

**Dr M. Fasondini**

**Term 2**

** **

The aim of this module is to learn tools and techniques from complex analysis and orthogonal polynomials that are used in mathematical physics. The course will focus on mathematical techniques, though will discuss relevant physical applications, such as electrostatic potential theory. The course also incorporates computational techniques in the lectures.

*Prerequisites: Complex analysis.*

Topics:

- Revision of complex analysis: Complex integration, Cauchy’s theorem and residue calculus [Revision]
- Singular integrals: Cauchy, Hilbert, and log kernel transforms
- Potential theory: Laplace’s equation, electrostatic potentials, distribution of charges in a well
- Riemann–Hilbert problems: Plemelj formulae, additive and multiplicative Riemann–Hilbert problems
- Orthogonal polynomials: recurrence relationships, solving differential equations, calculating singular integrals
- Integral equations: integral equations on the whole and half line, Fourier transforms, Laplace transforms
- Wiener–Hopf method: direct solution, solution via Riemann–Hilbert methods
- Singularities of differential equations: analyticity of solutions, regular singular points, Hypergeometric functions
- Riemann–Hilbert problems: Plemelj formulae, additive and multiplicative Riemann–Hilbert problems
- Orthogonal polynomials: recurrence relationships, solving differential equations, calculating singular integrals
- Integral equations: integral equations on the whole and half line, Fourier transforms, Laplace transforms
- Wiener–Hopf method: direct solution, solution via Riemann–Hilbert methods
- Singularities of differential equations: analyticity of solutions, regular singular points, Hypergeometric functions

## Mathematical biology, Data science and Machine learning

### MATH97096 Mathematical Biology

**Dr E. Keaveny**

**Term 1**

The aim of the module is to describe the application of mathematical models to biological phenomena. A variety of contexts in human biology and diseases are considered, as well as problems typical of particular organisms and environments.

**The syllabus includes topics from:**1. Population dynamics. Growth and spatial spread of organisms. Fisher's equation.

2. Epidemiology - the spread of plagues.

3. Reaction-Diffusion models: Turing mechanism for pattern formation. How the leopard got his spots (and sometimes stripes).

4. Enzyme Kinetics and chemical reactions: Michaelis-Menten theory. Hormone cycles, neuron-firing.

5. Mass transport; Taylor dispersion.

6. Biomechanics: Blood circulation, animal locomotion: swimming, flight. Effects of scale and size.

7. Other particular problems from biology.

### MATH97097 Methods For Data Science

**Prof M. Barahona**

**Term 2**

This course is in two halves: machine learning and complex networks. We will begin with an introduction to the R language and to visualisation and exploratory data analysis. We will describe the mathematical challenges and ideas in learning from data. We will introduce unsupervised and supervised learning through theory and through application of commonly used methods (such as principle components analysis, k-nearest neighbours, support vector machines and others). Moving to complex networks, we will introduce key concepts of graph theory and discuss model graphs used to describe social and biological phenomena (including Erdos-Renyi graphs, small-world and scale-free networks). We will define basic metrics to characterise data-derived networks, and illustrate how networks can be a useful way to interpret data.

## Dynamical systems

### MATH97181 Dynamics of Games

**Dr S. van Strien**

**Term 1**

Recently there has been quite a lot of interest in modeling learning through studying the dynamics of games. The settings to which these models may be applied is wide-ranging, from ecology and sociology to business, as actively pursued by companies like Google. Examples include

(i) optimization of strategies of populations in ecology and biology

(ii) strategies of people in a competitive environment, like online auctions or (financial) markets.

(iii) learning models used by technology companies

This module is aimed at discussing a number of dynamical models in which learning evolves over time, and which have a game theoretic background. The module will take a dynamical systems perspective. Topics will include replicator dynamics and best response dynamics.

### MATH97176 Dynamical Systems

**Professor J. Lamb****Term 1**

The theory of Dynamical Systems is an important area of mathematics which aims at describing objects whose state changes over time. For instance, the solar system comprising the sun and all planets is a dynamical system, and dynamical systems can be found in many other areas such as finance, physics, biology and social sciences. This course provides a rigorous treatment of the foundations of discrete-time dynamical systems, which includes the following subjects:

- Periodic orbits

- Topological and symbolic dynamics

- Chaos theory

- Invariant manifolds

- Statistical properties of dynamical systems

### MATH97177 Bifurcation Theory

**Professor D. TuraevTerm 2**

This module serves as an introduction to bifurcation theory, concerning the study of how the behaviour of dynamical systems (ODEs, maps) changes when parameters are varied.

The following topics will be covered:

1) Bifurcations on a line and on a plane.

2) Centre manifold theorem; local bifurcations of equilibrium states.

3) Local bifurcations of periodic orbits – folds and cusps.

4) Homoclinic loops: cases with simple dynamics, Shilnikov chaos, Lorenz attractor.

5) Saddle-node bifurcations: destruction of a torus, intermittency, blue-sky catastrophe.

6) Routes to chaos and homoclinic tangency.

### MATH97246 Random Dynamical Systems and Ergodic Theory: Seminar Course

**Professor J. Lamb**

**Term 2**

This is an introductory course on the theory and applications of random dynamical systems and ergodic theory. Random dynamical systems are (deterministic) dynamical systems driven by a random input. The goal will be to present a solid introduction and, time permitting, touch upon several more advanced developments in this field.

The contents of the module are:

- Random dynamical systems; definition in terms of skew products and elementary examples (including iterated function systems, discrete time dynamical systems with bounded noise and stochastic differential equations).
- Introduction to random dynamical systems theory in iterated function systems context.
- Background on measure theory and probability theory.
- Introduction to Ergodic Theory: Birkhoff Ergodic Theorem and Oseledets Ergodic Theorem.
- Dynamics of random circle maps: synchronisation.
- Chaos in random dynamical systems.

** **

### MATH97178 Dynamics, Symmetry and Integrability

**Prof D. Holm** **Term 2**

The following topics will be covered:

- Introduction to smooth manifolds as configuration spaces for dynamics.
- Transformations of smooth manifolds as flows of smooth vector fields.
- Introduction to differential forms, wedge products and Lie derivatives.
- Adjoint and coadjoint actions of matrix Lie groups and matrix Lie algebras.
- Action principles on matrix Lie algebras, their corresponding Euler-Poincaré ordinary differential equations and the Lie-Poisson Hamiltonian formulations of these equations.
- EPDiff: the Euler-Poincaré partial differential equation for smooth vector fields acting on smooth manifolds.
- The Hamiltonian formulation of EPDiff: Its momentum maps and soliton solutions.
- Integrability of EPDiff: Its bi-Hamiltonian structure, Lax pair and isospectral problem, as well as the relationships of these features to the corresponding properties of KdV.

** **

### MATH97238 Classical dynamics

##### Dr Chris Ford

##### Term 1

Learning objectives:

To understand how to reformulate Newton's laws through variational principles.

Ability to construct Lagrangians or Hamiltonians for dynamics problems in any coordinate system.

To be able to solve the equations of motion for a wide variety of problems in dynamics (numerous examples will be presented in the teaching sessions and non-assessed problem sheets).

To understand how to identify and exploit constant of the motion in solving dynamics problems.

On completing this module students should be in a position to apply Lagrangian and Hamiltonian methods to a variety of fields (e.g. Statistical Mechanics, Quantum Mechanics and Geometric Mechanics).

Syllabus:

Calculus of Variations: The Euler-Lagrange equation as a stationarity condition, Beltrami identity.

Lagrangian Mechanics: Review of Newtonian Mechanics, Hamilton's Principle, Lagrangians for conservative and non-conservative systems, generalised coordinates and momenta, cyclic coordinates, Noether's theorem (conservation of angular momentum as an example).

Hamiltonian Mechanics: Phase Space, Hamilton's equations, Poisson brackets, canonical transformations, generating functions, Hamilton-Jacobi theory, action-angle variables, integrability, application of Hamiltonian mechanics to rigid bodies.

## Mathematical Finance

### MATH97101 Mathematical Finance

**Dr P. Siorpaes****Term 2**

*Prerequisites: Differential Equations , Multivariable Calculus, Real Analysis and Probability and Statistics 2 .*

The mathematical modeling of derivatives securities, initiated by Bachelier in 1900 and developed by Black, Scholes and Merton in the 1970s, focuses on the pricing and hedging of options, futures and other derivatives, using a probabilistic representation of market uncertainty. This module is a mathematical introduction to this theory, in a discrete-time setting. We will mostly focus on the no-arbitrage theory in market models described by trees; eventually we will take the continuous-time limit of a binomial tree to obtain the celebrated Black-Scholes model and pricing formula.

We will cover and apply mathematical concepts -such as conditional expectation, filtrations, Markov processes, martingales and martingale transforms, the separation theorem, and change of measure- and financial concepts such as self-financing portfolios, replication and delta hedging, risk-neutral probability, complete markets, non-anticipative strategies, and the fundamental theorem of asset pricing.

## Fluid dynamics

### MATH97088 Fluid Dynamics 1

**Professor X. WuTerm 1**

This module is an introduction to the Fluid Dynamics. It will be followed by Fluid Dynamics 2 in Term 2.

Fluid Dynamics deals with the motion of liquids and gases. Being a subdivision of Continuum Mechanics the fluid dynamics does not deal with individual molecules. Instead an ‘averaged’ motion of the medium is of interest. Fluid dynamics is aimed at predicting the velocity, pressure and temperature fields in flows past rigid bodies. A theoretician achieves this goal by solving the governing Navier-Stokes equations. In this module a derivation of the Navier-Stokes equations will be presented, followed by description of various techniques to simplify and solve the equation with the purpose of describing the motion of fluids at different conditions.

**Aims of this module:**To introduce students to fundamental concepts and notions used in fluid dynamics. To demonstrate how the governing equations of fluid motion are deduced, paying attention to the restriction on their applicability to real flows. Then a class of exact solutions to the Navier-Stokes equations will be presented. This will follow by a discussion of possible simplifications of the Navier-Stokes equations. The main attention will be a wide class of flows that may be treated as inviscid. To this category belong, for example, aerodynamic flows. Students will be introduced to theoretical methods to calculate inviscid flows past aerofoils and other aerodynamic bodies. They will be shown how the lift force produced by an aircraft wing may be calculated.

**Content:**Introduction: The continuum hypothesis. Knudsen number. The notion of fluid particle. Kinematics of the flow field. Lagrangian and Eulerian variables. Streamlines and pathlines. Vorticity and circulation. The continuity equation. Streamfunction and calculation of the mass flux in 2D flows. Governing Equations: First Helmholtz’s theorem. Constitutive equation. The Navier-Stokes equations. Exact Solutions of the Navier-Stokes Equations: Couette and Poiseuille flows. The flow between two coaxial cylinders. The flow above an impulsively started plate. Diffusion of a potential vortex. Inviscid Flow Theory: Integrals of motion. Kelvin’s circulation theorem. Potential flows. Bernoulli’s equation. Cauchy-Bernoulli integral for unsteady flows. Two-dimensional flows. Complex potential. Vortex, source, dipole and the flow past a circular cylinder. Adjoint mass. Conformal mapping. Joukovskii transformation. Flows past aerofoils. Lift force. The theory of separated flows. Kirchhoff and Chaplygin models.

### MATH97087 Fluid Dynamics 2

**Professor J. MestelTerm 2**

*Prerequisites: Fluid Dynamics 2 is a continuation of the module Fluid Dynamics 1 given in Term 1.*

In Fluid Dynamics 1 the main attention was with exact solutions of the Navier-Stokes equations governing viscous fluid motion. The exact solutions are only possible in a limited number of situations when the shape of the body is rather simple. A traditional way of dealing with more realistic shapes (like aircraft wings) is to seek possible simplifications in the Navier-Stokes formulation. We shall start with the case when the internal viscosity of the fluid is very large, and the Navier-Stokes equations may be substituted by the Stokes equations. The latter are linear and allow for simple solutions in various situations. Then we shall consider the opposite limit of very small viscosity, which is characteristic, for example, of aerodynamic flows. In this cast the analysis of the flow past a rigid body (say, an aircraft wing) requires Prandtl’s boundary-layer equations to be solved. These equations are parabolic, and in many situations may be reduced to ordinary differential equations. Solving the Prandtl equations allows us to calculate the viscous drag experienced by the bodies. The final part of the module will be devoted to the theory of separation of the boundary layer, known as Triple-Deck theory.

** ****Aims of the module:**To introduce the students to various aspects of Viscous Fluid Dynamics, and to demonstrate the power (and beauty) of modern mathematical methods employed when analysing fluid flows. This includes the Method of Matched Asymptotic Expansions, which was put forward by Prandtl for the purpose of mathematical description of flows with small viscosity. Now this method is used in all branches of applied mathematics.

**Content:**Dynamic and Geometric Similarity of fluid flows. Reynolds Number and Strouhal Number. Fluid Flows at Low Values of The Reynolds Number: Stokes equations. Stokes flow past a sphere. Stokes flow past a circular cylinder. Skokes paradox. Large Reynolds Number Flows: the notion of singular perturbations. Method of matched asymptotic expansions. Prandtl’s boundary-layer equations. Prandtl’s hierarchical concept. Displacement thickness of the boundary layer and its influence on the flow outside the boundary layer. Self-Similar Solutions of the Boundary-Layer Equations: Blasius solution for the boundary layer on a flat plate surface. Falkner-Skan solutions for the flow past a wedge. Schlichting’s jet solution. Tollmien’s far field solution. Viscous drag of a body. Shear layers. Prandtl transposition theorem. Triple-Deck Theory: The notion of boundary-layer separation. Formulation of the triple-deck equations for a flow past a corner. Solution of the linearised problem (small corner angle case).

### MATH97092 Vortex Dynamics

**Professor D. CrowdyTerm 2**

*Prerequisites: A knowledge of basic applied mathematical methods is the only prerequisite. A basic knowledge of inviscid fluid dynamics is desirable but not required.*

The module will focus on the mathematical study of the dynamics of vorticity in an ideal fluid in two and three dimensions. The module will be pitched in such a way that it will be of interest both to fluid dynamicists and as an application of various techniques in dynamical systems theory.

Fundamental properties of vorticity.

Helmholtz Laws and Kelvin's circulation theorem. Singular distributions of vorticity; Biot-Savart law.

Dynamics of line vortices in 2d and other geometries; dynamics of 2d vortex patches, contour dynamics.

Axisymmetric vortex rings. Dynamics of vortex filaments.

Stability problems.

Miscellaneous topics (effects of viscosity, applications to turbulence, applications in aerodynamics).

### MATH97091 Hydrodynamic Stability

**Prof X. Wu** **Term 2**

*Prerequisites: a knowledge of basic applied mathematical methods in years 1 and 2 and Fluids I (M3/4A2). Wave Theory (M2AM) and Asymptotic Analysis (M3/4/M7) are beneficial but neither is essential.*

Fluid flows in nature and engineering applications may exist in two distinct forms: the simple laminar state which exhibits a high degree of order, and the turbulent state characterised by its complex chaotic behaviours in both time and space. The transition from a laminar state to turbulence is due to hydrodynamic instability, which refers to the phenomenon that small disturbances to a simple state amplify significantly thereby destroying the latter. Hydrodynamic instability and resulting transition are of profound scientific and technological importance because they are critically related to mixing and transports in the atmosphere and oceans, drag and aerodynamic heating experienced by air/spacecrafts, jet noise, combustion in engines and even the operation of the proposed nuclear fusion devices.

This course is an introduction to the basic concepts and techniques of modern hydrodynamic stability theory.

**Aims of this module:**

To present several fundamental hydrodynamic instability mechanisms and the associated mathematical formulations. The theoretical and computational techniques for analysing the stability will be introduced and discussed. In addition to the established textbook material, ongoing research topics will be introduced. The present course will lead students to appreciate the fundamental importance of hydrodynamic stability in modern science and technology, and prepare them for research.

**Content:**

Topics covered will be a selection from the following list.

- Basic concepts of stability; linear and nonlinear stability, initial-value and eigenvalue problems, normal modes, dispersion relations, temporal/spatial instability.
- Buoyancy driven instability: Rayleigh-Benard instability, formulation of the linearised stability problem, Rayleigh number, Rayleigh-Benard convection cells, discussion of the neutral stability properties.
- Centrifugal instability: Taylor-Couette flow, formulation of the linear stability problem, Taylor number, Taylor vortices; inviscid approximation, Rayleigh's criterion; viscous theory and solutions, characterization of stability properties; boundary layers over concave walls, Görtler number, Görtler instability, Görtler vortices.
- Inviscid/viscous shear instabilities of parallel flows: Inviscid/Rayleigh instability, Rayleigh equation, Rayleigh's inflection point theorem, Fjortoft's theorem, Howard's semi-circle theorem, solutions for special profiles, Kelvin-Helmholtz instability, general characteristics of instability, critical layer, singularity; Viscous/Tollmien-Schlichting instability, Orr-Sommerfeld (O-S) equation, Squire's theorem, numerical methods for solving the linear stability problem, discussion of instability properties.
- Inviscid/viscous shear instabilities of (weakly) non-parallel flows: local-parallel-flow approximation and application to free shear layers and boundary layers; non-parallel-flow effects, rational explanation of viscous instability mechanism, high-Reynolds-number asymptotic theory, multi-scale approach, parabolised stability equations; transition process and prediction (correlation); receptivity.
- Nonlinear instability: limitations of linear theories, bifurcation and nonlinear evolution; weakly nonlinear theory, derivation of Stuart-Landau and Ginzburg-Landau equations; nonlinear critical-layer theory.

**Recommended References:**

P. G. Drazin & W. H. Reid, Hydrodynamic Stability, Cambridge University Press

W. O. Criminale, T.L. Jackson & R. D. Joslin Theory and computation in hydrodynamic stability. Cambridge University Press.

H. Schlichting & K. Gersten, Boundary layer theory, Springer.

P. Huerre & M. Rossi, Hydrodynamic Instabilities in Open Flows, Cambridge University Press.

## Mathematical physics

### MATH97093 Quantum Mechanics 1

**Dr E-M Graefe**

**Term 1**

** **Quantum mechanics is one of the most successful theories in modern physics and has an exceptionally beautiful underlying mathematical structure. It provides the basis for many areas of contemporary physics, including atomic and molecular, condensed matter, high-energy particle physics, quantum information theory, and quantum cosmology, and has led to countless technological applications.

This module aims to provide an introduction to quantum phenomena and their mathematical description. Quantum theory combines tools and concepts from various areas of mathematics and physics, such as classical mechanics, linear algebra, probability theory, numerical methods, analysis and even geometry. However, most of the concepts are basic, and little background knowledge is required before we can put them to practical use.

Core topics: the mathematics and foundations of quantum mechanics; Schrodinger equation and wave functions; quantum dynamics; one-dimensional systems; harmonic oscillator; angular momentum; spin-1/2 systems; multiparticle systems; entanglement. Additional topics may include the hydrogen atom and approximation methods.

Additional optional topics may include: Approximation techniques; explicitly time-dependent systems; geometric phases; numerical techniques; many-particle systems; cold atoms; entanglement and quantum information.

### MATH97100 Special Relativity and Electromagnetism

**Dr G. Pruessner**

**Term 1**

This module presents a beautiful mathematical description of a physical theory of great historical, theoretical and technological importance. It demonstrates how advances in modern theoretical physics are being made and gives a glimpse of how other theories (say quantum chromodynamics) proceed.

At the beginning of special relativity stands an experimental observation and thus the insight that all physical theories ought to be invariant under Lorentz transformations. Casting this in the language of Lagrangian mechanics induces a new description of the world around us. After some mathematical work, but also by interpreting the newly derived objects, Maxwell’s equations follow, which are truly fundamental to all our every-day interaction with the world. In particular, Maxwell’s equations can be used to characterise the behaviour of charges in electromagnetic fields, which is rich and beautiful.

This module does not follow the classical presentation of special relativity by following its historical development, but takes the field theoretic route of postulating an action and determining the consequences. The lectures follow closely the famous textbook on the classical theory of fields by Landau and Lifshitz.

Special relativity: Einstein’s postulates, Lorentz transformation and its consequences, four vectors, dynamics of a particle, mass-energy equivalence, collisions, conserved quantities.

Electromagnetism: Magnetic and electric fields, their transformations and invariants, Maxwell’s equations, conserved quantities, wave equation.

### MATH97099 Quantum Mechanics 2

**Dr R. Barnett**

**Term 2**

Quantum mechanics is one of the most successful theories in modern physics and has an exceptionally beautiful underlying mathematical structure. It provides the basis for many areas of contemporary physics, including atomic and molecular, condensed matter, high-energy particle physics, quantum information theory, and quantum cosmology, and has led to countless technological applications. Quantum theory combines tools and concepts from various areas of mathematics and physics, such as classical mechanics, linear algebra, probability theory, numerical methods, analysis and even geometry. However, most of the concepts are basic, and little background knowledge is required before we can put them to practical use.

This module is intended to be a second course in quantum mechanics and will build on topics covered in Quantum Mechanics I.

In addition to the material below, this level 7 (Masters) version of the module will have additional extension material for self-study. This will require a deeper understanding of the subject than the corresponding level 6 (Bachelors) module.

Core topics: Quantum mechanics in three spatial dimensions and the hydrogen atom, the Heisenberg picture, perturbation theory, addition of spin, adiabatic processes and the geometric phase, Floquet-Bloch theory, second quantization and introduction to many-particle systems, Fermi and Bose statistics, quantum magnetism. Additional topics may include WKB theory and the Feynman path integral.

### MATH97005 Tensor Calculus and General Relativity

##### Dr C. Ford

**Term 2 **

The mathematical description of a theory, which is fundamental to gravitation and to behaviour of systems at large scales.

Tensor calculus including Riemannian geometry; principle of equivalence for gravitational fields; Einstein’s field equations and the Newtonian approximation; Schwarzschild’s solution for static spherically symmetric systems; the observational tests; significance of the Schwarzschild radius; black holes; cosmological models and ‘big bang’ origin of the universe.

Variational principles.