Modules

Dr N. Boulle

Term 2

Machine learning techniques such as deep learning have recently achieved remarkable results in a very wide variety of applications such as image recognition, self-driving vehicles, partial differential equation solvers, trading strategies. However, how and why the recent (deep learning) models work is often still not fully understood. In this course we will begin with a general introduction into machine learning and continue to deep learning. We will focus on better some observed phenomena in deep learning aiming to gain insight into the impact of the optimization algorithms and network architecture through mathematical tools. This module will be 100% coursework.

Dr A. Menegaki
Term 2

This course develops the analysis of boundary value problems for elliptic and parabolic PDE’s using the variational approach. It is a follow-up of ‘Function spaces and applications’ but is open to other students as well provided they have sufficient command of analysis. An introductory Partial Differential Equation course is not needed either, although certainly useful.

Learning Outcomes

On successful completion of this module you will be able to:

• appreciate the concepts of distribution (differentiation, convergence);
• manipulate the main properties of the Sobolev space H^m for integer m (inbeddings and compactness theorems, Poincaré inequality);
• derive the variational formulation of a specific elliptic boundary value problem and to provide the reasoning leading to the proof of the existence and uniqueness of the solution;
• develop the spectral theory of an elliptic boundary value problem;
• solve a parabolic boundary value problem using the spectral theory of the associated elliptic operator.
• interpret results from advanced textbooks and research papers on the theory of Partial Differential Equations;
• independently appraise and evaluate an advanced topic on Partial Differential Equations, namely the theory of nonlinear elliptic and parabolic equations on the whole space.

Module Content

The course consists of three parts. The first part (divided into two chapters) develops further tools needed for the study of boundary value problems, namely distributions and Sobolev spaces. The following two parts are devoted to elliptic and parabolic equations on bounded domains. They present the variational approach and spectral theory of elliptic operators as well as their use in the existence theory for parabolic problems.

The aim of the course is to expose the students to some important aspects of Partial Differential Equation theory, aspects that will be most useful to those who will further work with Partial Differential Equations be it on the theoretical side or on the numerical one.

The syllabus of the course is as follows:

1. Distributions. The space of test functions. Definition and examples of distributions. Differentiation. Convergence of distributions.
2. Sobolev spaces: The space H1. Density of smooth functions. Extension lemma. Trace theorem. The space H10. Poincare inequality. The Rellich-Kondrachov compactness theorem (without proof). Sobolev imbedding (in the simple case of an interval of R). The space Hm. Compactness and Sobolev imbedding for arbitrary dimension (statement without proof).
3. Linear elliptic boundary value problems: Dirichlet and Neumann boundary value problems via the Lax-Milgram theorem. Spectral theory. The maximum principle. Regularity (stated without proofs). Classical examples: elasticity system, Stokes system.
4. Linear parabolic initial-value problems: Bochner Spaces. Existence and uniqueness of weak solutions by the Galerkin method. Application to the incompressible Navier-Stokes equations.
5. Additional reading for the mastery question: fixed point methods for solving nonlinear elliptic and parabolic problems.

Dr E. Keaveny
Term 1

The module is an introductory course in numerical methods for ordinary differential equations. The purpose of this module is to learn how to use the computer to find numerical solutions to ordinary differential equations as well as to provide you with theoretical knowledge and practical skills to lay the solid groundwork necessary to advance in scientific computing.

Learning Outcomes

On the successful completion of the module you will be able to

• use classical numerical methods for ordinary differential equations;
• analyse different properties of numerical methods (e.g. accuracy and stabilty);
• develop your own methods with prescribed properties;
• compare different methods with respect to accuracy, stability, computational and space complexity;
• create efficient numerical algorithms;
• construct numerical methods to solve boundary value problems for partial differential equations.

Module Content

This module will cover the following topics:

1. Taylor series methods;
2. Linear multi-step methods;
3. Runge-Kutta methods;
4. Adaptive step size control;
5. Boundary value problems for ordinary differential equations;
6. Introduction to the finite difference method and energy Inequalities method;
7. Introduction to boundary value problems for partial differential equations.

Professor C. Cotter
Term 1

Linear systems of equations arise in countless applications and problems in mathematics, science and engineering.  Often these systems are large and require a computer to solve.  This course provides an overview of the algorithms used to solve linear systems and eigenvalue problems, in terms of their development, stability properties, and application.

Learning Outcomes

On successful completion of this module, you will be able to

• describe, select and use algorithms for QR decomposition of matrices;
• solve least-squares problems using QR decomposition;
• apply LU decomposition to solve linear systems;
• analyse and modify algorithms that take advantage of matrix structure;
• find numerical solutions to eigenvalue problems;
• critically analyse various iterative methods for solving linear systems.
• combine the techniques you have mastered in order to assess unseen algorithms;
• adapt the techniques to analyse related topics such as functions of matrices.

Module Content

The module will cover the following topics:

1. Direct methods: Triangular and banded matrices, Gauss elimination, LU-decomposition, conditioning and finite-precision arithmetic, pivoting, Cholesky factorisation, QR factorisation and their numerical implementation.
2. Eigenvalue problems: power method and variants, Jacobi's method, Householder reduction to tridiagonal form, eigenvalues of tridiagonal matrices, the QR method.
3. Iterative methods: Krylov subspace methods: Lanczos method and Arnoldi iteration, conjugate gradient method, GMRES, preconditioning.

Dr S. Mughal
Term 2

This 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 another programming language of their choice. Applications will be drawn from problems arising in areas such as Mathematical Biology and Fluid Dynamics.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate the physical and mathematical differences between different types of PDES;
• design suitable finite difference methods to solve each type of PDE;
• outline a theoretical approach to testing the stability of a given algorithm;
• determine the order of convergence of a given algorithm;
• demonstrate familiarity with the implementation and rationale of multigrid methods;
• develop finite-difference based software for use on research level problems;
• communicate your research findings as a poster, in a form suitable for presentation at a scientific conference.

Module Content

The module will cover the following topics:

1. Introduction to Finite Differences
2. Classificaton of PDEs
3. Explicit and Implicit methods for Parabolic PDEs
4. Iterative Methods for Elliptic PDEs. Jacobi, Gauss-Seidel, Overrelaxation
5. Multigrid Methods
6. Hyperbolic PDEs. Nonlinear Advection/Diffusion systems. Waves and PMLs

as well as various advanced practical topics from Fluid Dynamics, which will depend on the final project.

Dr Z. Zhang
Term 1

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.

Learning Outcomes

On successful completion of this module, you should be able to:

• demonstrate your understanding of the concepts and results associated with the elementary theory of Markov processes, including the proofs of a variety of results
• apply these concepts and results to tackle a range of problems, including previously unseen ones          
• apply your understanding to develop proofs of unfamiliar results
• demonstrate additional competence in the subject through the study of more advanced material
• combine ideas from across the module to solve more advanced problems
• communicate your knowledge of the area in a concise, accurate and coherent manner.

Module Content

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 good 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 have taken the second-year module Lebesgue Measure and Integration. A good knowledge of real analysis will be assumed.

The module is related to a number of  modules in stochastic analysis, probability theory, dynamical systems and mathematical finance.

An indicative list of contents is:

1. Discrete time  and finite state Markov chains :  Chapman-Kolmogorov equations, irreducible, Perron-Frobenius theorem for stochastic matrices, recurrent and transient.
2. 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.

There will also be extra self-study of extension material (in the form of a book chapter, additional notes or a research paper) applying or extending material from the above topics.

Professor C. Cotter 
Term 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.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate the core mathematical principles of the finite element method;
• employ the finite element method to formulate and analyse numerical solutions to linear elliptic PDEs;
• implement the finite element method on a computer;
• compare the application of various software engineering techniques to numerical mathematics;
• generalize the concept of a directional derivative;
• appraise and evaluate techniques for solving nonlinear PDEs using the finite element method.

Module Content

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, students 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 students will combine these abstractions with modern software engineering tools to create and understand a computer implementation of the finite element method.

This module is composed of the following sections:

1. Basic concepts: weak formulation of boundary value problems, Ritz-Galerkin approximation, error estimates, piecewise polynomial spaces, local estimates;
2. Efficient construction of finite element spaces in one dimension: 1D quadrature, global assembly of mass matrix and Laplace matrix;
3. Construction of a finite element space: Ciarlet’s finite element, various element types, finite element interpolants;
4. Construction of local bases for finite elements: efficient local assembly;
5. Sobolev Spaces: generalised derivatives, Sobolev norms and spaces, Sobolev’s inequality;
6.  Numerical quadrature on simplices: employing the pullback to integrate on a reference element;
7. Variational formulation of elliptic boundary value problems: Riesz representation theorem, symmetric and nonsymmetric variational problems, Lax-Milgram theorem, finite element approximation estimates;
8. Computational meshes: meshes as graphs of topological entities, discrete function spaces on meshes, local and global numbering;
9. Global assembly for Poisson equation: implementation of boundary conditions, general approach for nonlinear elliptic PDEs;
10. Variational problems: Poisson’s equation, variational approximation of Poisson’s equation, elliptic regularity estimates, general second-order elliptic operators and their variational approximation;
11. Residual form and the Gâteaux derivative;
12. Newton solvers and convergence criteria.

Professor P. Germain

Term 1

The purpose of this course is to introduce the basic function spaces and to train the student in 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. Most of the topics contained in the module do not require preliminary knowledge. However, knowledge of the material in the Y2 module on “Lebesgue Measure and Integration” (or a suitable equivalent) is recommended.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate the main concepts of metric topology and integration theory (Fatou’s lemma, monotone and dominated convergence theorems);
• manipulate concepts associated with Banach spaces (Cauchy sequence, completeness concept, bounded operators, continuous linear forms, dual space);
• apply the concept of uniform convergence of functions, and those related to spaces of differentiable functions;
• interpret the concept of convergence in Lebesgue spaces;
• manipulate convolutions and sequences of mollifiers to approximate continuous or Lebesgue integrable functions by infinitely differentiable functions with compact support;
• appreciate the notion of compactness and the difference between finite and infinite-dimensional normed vector spaces;
• interpret results from advanced textbooks and research papers;
• independently appraise and evaluate an advanced topic (namely the notions of weak and weak-star compactness in Banach spaces).

Module Content

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, Hilbert spaces. The concepts of Distributions 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 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. The extra material will relate to the concept of compactness in Banach spaces.

The syllabus of the course is as follows:

1. Review of metric topology and Lebesgue’s integration theory
2. Normed vector spaces. Banach spaces. Continuous linear maps. Dual of a Banach space.
3. Examples of function spaces: continuously differentiable function spaces and Lebesgue spaces. Hölder and Minkowski’s inequalities. Convolution and Mollification. 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: weak and weak star topologies and Banach-Alaoglu’s theorem
5. Hilbert spaces. The projection theorem. The Riesz representation theorem. The Lax-Milgram theorem. Hilbert bases and Parseval’s identity. Application to Fourier series.

Dr D. Heydecker
Term 1

This module provides an introduction to stochastic differential equations (SDEs), together with the necessary background material from stochastic analysis and the link between SDEs and partial differential equations. The course covers the following topics: elements of the theory of stochastic processes in continuous time, Brownian motion, construction of the Ito stochastic integral, existence and uniqueness theory for SDEs, methods for solving SDEs, connection between SDEs and Markov processes, the Fokker-Planck equation,  ergodic theory for SDEs.

Learning Outcomes

On successful completion of this module, you will be able to:

• formulate the basics of the theory of stochastic processes in continuous time;
• appreciate the fundamental properties of Brownian motion;
• apply Ito's theory of stochastic integration;
• prove existence and uniqueness of solutions to stochastic differential equations under certain conditions;
• construct the link between stochastic differential equations and Markov processes;
• connect SDEs and the forward and backward (Fokker-Planck) partial differential equations;
• develop techniques for solving the Fokker-Planck equation;
• assemble tools from elementary Hilbert space theory to study the ergodic properties of SDEs.

Module Content

The module is composed of the following sections:

1. Introduction
2. Elements of probability theory and of stochastic processes in continuous time
3. Brownian motion and stochastic calculus
4. Stochastic integrals
5. Stochastic differential equations
6. Applications to partial differential equations
7. Markov processes and invariant measures

Dr B. Bravi
Term 2

This module provides an hands-on introduction to the methods of modern data science. Through interactive lectures, the student will be introduced to data visualisation and analysis as well as the fundamentals of machine learning.

Learning Outcomes

On successful completion of this module, you will be able to:

• Visualise and explore data using computational tools;
• Appreciate the fundamental concepts and challenges of learning from data;
• Analyse some commonly used learning methods;
• Compare learning methods and determine suitability for a given problem;
• Describe the principles and differences between supervised and unsupervised learning;
• Clearly and succinctly communicate the results of a data analysis or learning application;
• Appraise and evaluate new algorithms and computational methods presented in scientific and mathematical journals;
• Design and implement newly-developed algorithms and methods.

Module Content

The module is composed of the following sections: 

1. Introduction to computational tools for data analysis and visualisation;
2. Introduction to exploratory data analysis;
3. Mathematical challenges in learning from data: optimisation;
4. Methods in Machine Learning: supervised and unsupervised; neural networks and deep learning; graph-based data learning;
5. Machine learning in practice: application of commonly used methods to data science problems. Methods include: regressions, k-nearest neighbours, random forests, support vector machines, neural networks, principal component analysis, k-means, spectral clustering, manifold learning, network statistics, community detection;
6. Current research questions in data analysis and machine learning and associated numerical methods.

Dr P. Ray
Term 1

This module introduces students to the analysis and implementation of efficient algorithms used to solve mathematical and computational problems connected to a broad range of scientific topics. Mathematical tools and concepts from linear algebra, calculus, numerical analysis, and statistics will be utilised to develop and analyse computational solutions to mathematical and scientific problems.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 areas.

Learning Outcomes

On successful completion of this module you will be able to:

• analyse the performance of simple sorting and searching algorithms and implement them in Python;
• computationally analyse complex networks and dynamical processes of complex systems;
• effectively utilise important tools for data analysis such as discrete Fourier transforms;
• evaluate and implement numerical methods for mathematical optimisation and the solution of differential equations;                                                               
• assess the correctness and efficiency of simple data structures and algorithms on graphs and implement them in Python;
• independently appraise and evaluate  a range of state-of-the art algorithms and computational methods;
• adapt a range of computational methods and apply them in a coherent manner to an open scientific problem.

Module Content

The module will cover the following topics:

1. Sorting and searching with scientific applications from fields such as bioinformatics;
2. Algorithms on graphs and basic data structures such as queues and hash tables;
3. Methods for data analysis using tools such as discrete Fourier transforms;
4. Analysis and use of common optimisation methods such as Simulated Annealing;
5. Numerical solution of differential equations arising in multiscale problems;                         
6. Computational analysis of complex systems.

Dr M. Sorella
Term 1

In this module, students are exposed to different phenomena which are modelled by partial differential equations. The course emphasizes the mathematical analysis of these models and briefly introduces some numerical methods.

Learning Outcomes

On successful completion of this module you will be able to:

• appreciate how to formally differentiate complicated finite dimensional functionals and simple infinite dimensional functionals;
• describe, select and use a variety of methods for solving partial differential equations;
• outline how various partial differential equations respect conservation laws;
• utilize energy methods to critically analyse the stability of solutions to PDEs;
• develop the general method of characteristics and derive the eikonal equation;
• justify the proper use of the calculus of variations in classical settings.

Module Content

The module is composed of the following sections:

1. Introduction to PDEs 

• Basic Concepts 
• Gauss Theorem

2. Method of Characteristics    

• Linear and Quasilinear first order PDEs in two independent variables.   
• Scalar Conservation Laws   
• Hamilton-Jacobi Equations. General Method of Characteristics.

3. Diffusion    

• Heat equation. Maximum principle    
• Separation of variables. Fourier Series.

4. Waves    

• The 1D wave equation    
• 2D and 3D waves. 

5. Laplace-Poisson equation    

• Dirichlet and Neumann problems.  
• Introduction to calculus of variations. The Dirichlet principle.    
• Finite Element Method.    
• Lagrangians and the minimum action principle.

Dr O. Schnitzer
Term 1

This advanced course presents a systematical introduction to asymptotic methods, which form one of the cornerstones of modern applied mathematics. The foundation of asymptotic approximations is laid down first. The key ideas and techniques for deriving asymptotic representations of integrals, and for constructing appropriate solutions to differential equations will be explained. The techniques introduced find wide applications in engineering and natural sciences.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate the foundation upon which asymptotic approximations are based;
• describe a variety of asymptotic methods and for each method acquire a thorough understanding of the key ideas involved and their mathematical nature;
• demonstrate basic skills in applying each of these methods to solve classical problems;
• combine, modify and extend methods to unfamiliar problems, such as those that emerge from research topics or practical applications;
• outline how asymptotic methods can in principle be applied to a wide variety of problems;
• interpret results from advanced textbooks and research papers on asymptotic methods;
• construct advanced solution techniques by selecting an appropriate combination of different asymptotic methods to solve higher-dimensional problems.

Module Content

1. Asymptotic approximations (fundamentals): Order notation. Diverging series, asymptotic expansions. Parameter expansions, overlap regions, distinguished limits and uniform approximations.Stokes phenomenon.
2. Introduction to perturbation methods: Asymptotic solution of algebraic equations with a small parameter. Regular vs. singular perturbations. Method of dominant balance. Local analysis of ordinary differential equations.
3. Asymptotic analysis of integrals: Method of integration by parts. Integrals of Laplace type: Laplace's method, Watson's Lemma. Integrals of Fourier type: method of stationary phase. Integral in the complex plane: method of steepest descent. Method of splitting the range of integration.
4. Matched asymptotic expansion: Inner and outer expansions, matching principles, notions of 'boundary layer' and interior layer. Composite approximation. Application to relaxation oscillations.
5. Methods of multiple scales: WKB approximations including turning-point problems and eigenvalue quantisation. Secular terms and solvability conditions. Poincare-Lindstedt method for periodic solutions. Multiscale method for quasi-periodic solutions. Application to weakly perturbed oscillators, nonlinear resonance, parametric resonance.
6. A selection of topics from the following: Stokes phenomenon, hyperasymptotics, expansions involving logarithmic terms, homogenisation.

Dr S Brzezicki

Term 1

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

Learning Outcomes

On successful completion of this module, you will be able to:

• apply the technique of contour deformation for calculating integrals;
• appreciate the connection that exists between computational tools such as quadrature and orthogonal polynomials and complex analysis;
• evaluate singular integral equations with Cauchy and logarithmic kernels;
• use the Wiener-Hopf method to solve a class of integral equations;
• compute matrix functions using contour inegration;;
• interpret results from advanced textbooks and research papers;
• independently appraise and evaluate an advanced topic in complex analysis.

Module Content

This module covers the following topics:

1. Revision of complex analysis: complex integration, Cauchy’s theorem and residue calculus;
2. Singular integrals: Cauchy, Hilbert, and log kernel transforms;
3. Potential theory: Laplace’s equation, electrostatic potentials, distribution of charges in a well;
4. Riemann–Hilbert problems: Plemelj formulae, additive and multiplicative Riemann–Hilbert problems;
5. Orthogonal polynomials: recurrence relationships, solving differential equations, calculating singular integrals;
6. Integral equations: integral equations on the whole and half line, Fourier transforms, Laplace transforms;
7. Wiener–Hopf method: direct solution, solution via Riemann–Hilbert methods.

Dr D. Kalise
Term 1

This module is an introduction to the theory and practice of mathematical optimization and its many applications in mathematics, data science, and engineering. The module aims at endowing students with the necessary mathematical background and a thorough methodological toolbox to formulate optimization problems and developing an algorithmic approach to its solution.The module is structured into five parts: (i) formulation and classification of problems; (ii) unconstrained optimization; (iii) stochastic and nature-inspired optimization; (iv) convex optimization; (v) introduction to optimal control and dynamic optimization. The assessed coursework for this module involves a series of computational tasks.

Learning Outcomes

On successful completion of this module you will be able to

• formulate a mathematical optimization problem by identifying a suitable objective and constraints;
• identify the mathematical structure of an optimization problem and, based on this classification, choose an appropriate methodological approach;
• develop a mathematical and computational appreciation of convexity as a fundamental feature in optimization;
• implement different computational optimization algorithms such as gradient descent and related variants;
• analyse the results of a computational optimization method in terms of optimality guarantees, sensitivities, and performance.
• interpret the role played by optimization in its application to computational data science;
• design optimal control approaches relevant to tackling large-scale nonlinear problems.

Module Content

1. Mathematical preliminaries
2. Unconstrained optimization
3. Gradient descent methods
4. Linear and non-linear least squares problems
5. Stochastic gradient descent
6. Nature-inspired optimization
7. Convex sets and functions
8. Convex optimization problems and stationarity
9. KKT conditions
10. Duality
11. Introduction to dynamic optimization and optimal control.

This final topic is linked to the Mastery Material for MSc students which will involve the study of some of the following solution techniques:

• shooting and multiple shooting methods;
• the reduced gradient approach;
• two-point boundary value solvers for optimal control;
• dynamic Programming and the Hamilton-Jacobi PDE;
• the linear-quadratic regulator and the Riccati equation.

This will be examined by way of an extra question on the May examination paper.

Prof D. Holm

Term 2

Brief Description

The course surveys a small section of the road to geometric mechanics (GM), by treating several examples in classical mechanics all in the same geometric framework. This framework is based on linear algebra, transformation theory, differential equations, variational calculus, and Lie group invariant variational principles, and their actions on functions and differential forms defined on manifolds.

Learning Outcomes

On successful completion of this module, you will be able to:

- describe motion on smooth manifolds using the following definitions: tangent bundle (TM), cotangent bundle (T*M, phase space), kinetic energy, Lagrangian, variational derivative, Hamilton's variational principle (LagHvp), Euler-Lagrange equations (EL eqns);

- derive and solve Euler-Lagrange equations for a variety of classical problems, including: oscillators, plane pendulum, charged particle motion, motion on a sphere, spherical pendulum, rotating rigid body, geodesic motion on the Lobachevsky half-plane, equations of motion for Newton's law of gravitation;

- prove and apply Noether's theorem about how Lie group symmetries of Hamilton's principle imply conservation laws in the GM framework for transformations of the matrix Lie groups of rotations, translations and scaling (in the context of the example problems);

- link fibre derivative, Legendre transformation, Hamilton's equation, Poisson brackets, and Noether quantities as momentum maps for Lie group actions;

- apply the Geometric Mechanics framework to rigid body motion and Newton's solution for the Kepler problem of planetary motion;

- demonstrate familiarity with the transformation theory for smooth invertible maps and the calculus of differential forms, including exterior derivative, wedge product, contraction with vector fields, and Cartan's formula for Lie derivative through worked exercises;

- relate Lie-Poisson brackets to coadjoint actions of matrix Lie groups on matrix Lie algebras;

- derive the equations for ideal fluid dynamics from Hamilton's principle with symmetry under smooth invertible maps.

Module Content

This module explains the principles of mechanics as a unified geometric framework, by using

Noether's theorem about Lie-group invariant variational principles to focus on Sophus Lie's momentum map concept as the key for applying Poincaré's geometric approach. The only prerequisites are linear algebra, vector calculus and some familiarity with the Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.

The course presents increasingly advanced ideas as a series of recurring thematic examples and worked exercises for classic problems of oscillation and rotation. For example, it treats the geometry of rigid-body motion from the series of viewpoints of Newton, Lagrange, Hamilton and Poincaré. This unfolding series of treatments of rigid body motion from several viewpoints reveals the framework of geometric mechanics and sets the stage for the introduction of the flows of Hamiltonian vector fields and their Lie-derivative actions on differential forms.

After learning a few useful methods of vector calculus in the language of differential forms, the student recognises the geometric structure of ideal fluid dynamics as something which has become familiar and natural; namely, geodesic motion for the kinetic energy metric on the manifold of smooth invertible maps (L2 norm of the Eulerian fluid velocity). Thus, the course reveals a geometric framework for variational dynamics with symmetry and develops the ideas in the framework from finite to infinite dimensions.

Prof A. Laptev

Term 2

This module introduces some of the partial differential equations (PDE) appearing in physics and geometry, as well as a number of classical techniques to study them analytically. One focus is on building up some intuition for the broad variety of PDEs, the phenomena they describe and for the mathematical techniques that have been developed to study them.

Learning Outcomes

On successful completion of this module you should:

-- have developed an intuition for a variety of partial differential equations (behaviour of their solutions, techniques to study them).

-- understand some of the deep connections of PDE to physics and geometry.

-- be able to state and prove well-posedness theorems for a variety of PDE and understand their relevance.

-- be familiar with elliptic equations and elliptic regularity theory.

-- be familiar with hyperbolic equations (wave equations).

-- be able to attend a research talk in PDE

Module content

--Review of ODE Theory (Picard's Theorem, Gronwall's inequality).

--Theory of first order quasilinear PDE (Methods of Characteristics).

--Cauchy-Kovalevskaya Theorem (with sketch of the proof).

--Holmgren's uniqueness theorem (with proof via Cauchy-Kovalevskaya, examples).

--Laplace's equation (fundamental solution, regularity of harmonic functions, maximum principle, Green's function for a ball),

--General second order elliptic equations (Existence and Regularity Theory, Fredholm Alternative).

--Discussion of Schroedinger and Heat Equation (Schwartz space, Fourier techniques).

--Wave Equation (Energy estimate, domain of dependence, domain of influence, fundamental solution, solution via Fourier techniques, Duhamel's principle.)

It will be helpful if you have taken one or more of the following courses:

Functional Analysis, Measure and Integration (or Lebesgue Measure and Integration), Fourier Analysis and Distributions.

Dr A. Sykulski

Term 2

Data collected in space are common in many applications including climate science, epidemiology, and economics. This module covers theoretical and methodological statistical foundations for spatial data. The module is structured to cover in detail the three fundamental forms in which spatial data are collected: gridded data, network data, and point pattern data. For each data type, stochastic models will be defined and explored including random fields and point processes. Properties including isotropy, stationarity and homogeneity will be formalised and explored in the context of each model and data type. In addition, techniques for spatial interpolation will be studied.

Learning Outcomes

On successful completion of this module students will be able to:

• Recognise the key differences between different types of spatial data (gridded, network and point pattern)
• Apply different classes of spatial covariance models to gridded and network data
• Formulate different intensity functions for point pattern data
• Define the concepts of homogeneity, stationarity and isotropy in the context of spatial models
• Select, derive and apply appropriate methods for spatial interpolation
• Demonstrate an integrated understanding of the concepts of the module by critical, independent study of research articles and books.

Indicative Module Content

1. Introduction to spatial data (gridded, network and point pattern data)
2. Random Fields and Covariance Functions
3. Spatial Interpolation and Kriging
4. Network Data and Markov Random Fields
5. Spatial Point Processes

Dr D. Cheraghi

Term 2

In this module we look at the subject of complex analysis from a more geometric point of view. We shall look at geometric notions associated with domains in the plane and their boundaries, and how they are transformed under holomorphic mappings. In turn, the behaviour of conformal maps is highly dependent on the shape of their domain of definition.

Learning Outcomes
On successful completion of this module, you will be able to:

• identify features of, and develop arguments about, certain holomorphic maps;
• state, apply, and explain aspects of the Riemann Mapping Theorem for arbitrary simply connected plane domains;
• explain the automorphisms of the disk and the upper half plane;
• Explain hyperbolic geometry, and basic notions of length, geodesics, isometries;
• apply area theorem, and derive distortion estimates for arbitrary conformal mappings;
• acquire deeper understanding of holomorphic mappings through generalisation to quasi-conformal mappings;
• appreciate significance of universal bounds in geometric function theory;
• appreciate significance of universal bounds in geometric function theory;

Indicative Module Content
Complex analysis is the study of the functions of complex numbers. It is employed in a wide range of topics, including dynamical systems, algebraic geometry, number theory, and quantum field theory, to name a few. On the other hand, as the separate real and imaginary parts of any analytic function satisfy the Laplace equation, complex analysis is widely employed in the study of two-dimensional problems in physics such as hydrodynamics, thermodynamics, Ferromagnetism, and percolations.
In this module we look at the subject of complex analysis from a more geometric point of view. We shall look at geometric notions associated with domains in the plane and their boundaries, and how they are transformed under holomorphic mappings. In turn, the behaviour of conformal maps is highly dependent on the shape of their domain of definition.
An indicative list of topics is:
-Schwarz lemma and automorphisms of the disk,
-Riemann sphere and rational maps,
-Conformal geometry on the disk, Poincare metric, Isometries, Hyperbolic contractions,
-Conformal Mappings, Conformal mappings of special domains, Normal families, Montel’s theorem, General form of Cauchy integral formula, Riemann mapping theorem,
-Growth and Distortion estimates, Area theorem,
-Quasi-conformal maps and Beltrami equation, Linear distortion, Dilatation quotient, Absolute continuity on lines, Quasi-conformal mappings, Beltrami equation, application of MRMT,
There will also be extra self-study of extension material (in the form of a book chapter, additional notes or a research paper) applying or extending material from the above topics.
Prerequisites: It will be helpful if you have taken (or are taking) one or more of the following courses:
Functional Analysis, Measure and Integration (or Lebesgue Measure and Integration), Fourier Analysis and Distributions.

Dr S. Brzezicki

Term 1

This module will give students an insight into the wide variety of mathematics and its many applications within the area of game theory. The module aims to promote an active learning style, involving many classroom games as well as games to be played as homework.
The module will cover the classical theory of games involving concepts of dominance, best response and equilibria, where we will prove Nash’s Theorem on the existence of equilibria in games. We will see the concept of when a game is termed zero-sum and prove the related Von Neumann’s Minimax Theorem. We will briefly discuss cooperation in games and investigate the interesting Nash bargaining solution which arises beautifully from reasonable bargaining axioms.
Broadening our scope, we will look at the area of combinatorial game theory, building up our intuition through investigating the classical game of Nim in detail. We will also see the concept of a congestion game, often applied to situations involving traffic flow, where we will see the counter intuitive Braess paradox emerge and prove Nash’s theorem in another context.
The module will finish with a small tour through some other areas and applications of game theory.
Learning Outcomes
On successful completion of this module you will be able to:
- define the concepts of dominance, best-response and equilibria in a variety of competitive scenarios (games);
- solve (determine all equilibria or find optimal strategies) small games via a variety of techniques: iterated deletion of dominated strategies, finding equaliser strategies, use of subgames;
- determine when a game may be termed zero-sum, and be able to recognise, find and apply minimax and maximin strategies in these games;
- apply game theory to traffic flow or flow of information through networks, appreciating the differences and importance of optimal societal routing as compared with selfish individual routing;
- calculate bargaining solutions in simple co-operative games;
- determine whether communication is beneficial or not in different strategic situations;
- demonstrate an integrated understanding of the concepts of the module by critical, independent study of research articles and books.
Indicative Module Content
1. Recap of some basic notions in probability, calculus and analysis, some recap of induction in a game theoretic context.
2. Motivational/illustrative classroom games.
3. Dominance, best-response and equilibria.
4. Nash's theorem on equilibria in games.
5. Zero-sum games and Von Neumann's minimax theorem.
6. Subgame solutions as extensions to full game solutions.
7. Coopertaive games, the Nash arbitration procedure and bargaining solutions.
8. Congestion games; Braess paradox, selfish routing vs optimal societal routing, existence of equilibria.
9. Combinatorial games; Nim, Nim sums and Nim values, sums of games.

Dr P. Bressloff
Term 1

Mathematical biology entails the use of mathematics to model biological phenomena in order to understand these systems, as well as predict their behaviour. It is an incredibly diverse field utilising the complete mathematical toolbox to ascertain insight into many areas of biology and medicine including population dynamics, physiology, epidemiology, cell biology, biochemical reactions, and neurology. This module aims to provide a foundational course in the subject area relying primarily on tools from applied dynamical systems, applied PDEs, asymptotic analysis and stochastic processes.

Learning Outcomes

On successful completion of this module you will be able to:

• translate biological phenomena into the language of mathematics;
• appreciate canonical problems in epidemiology, ecology, biochemistry and physiology;
• critically analyse sets of ordinary differential equations especially in the non-linear setting;
• critically analyse sets of partial differential equations especially when either travelling-wave solutions or pattern forming phenomena might emerge;
• utilise the concept of stochastic population processes for exact and approximate solutions;
• use the techniques of order-of-magnitude reasoning and dimensional analysis;
• interpret results from the research literature on Mathematical Biology and analyze how the syllabus content relates to this wider body of work;
• appraise and evaluate an advanced topic in Mathematical Biology from a selection of case studies.

Module Content

Examples and topics include:

1. One-dimensional systems: existence and uniqueness; fixed points and their stability; bifurcations; logistic growth; SIS epidemic model; spruce budworm model; law of mass action; Michaelis-Menten enzyme dynamics.
2. Multidimensional systems: existence, uniqueness, fixed point stability; two-dimensional systems; SIS model for two populations; genetic control systems; population competition models; predator-prey dynamics and the Lotka-Volterra model.
3. Oscillations and bifurcations: Poincaré-Bendixson Theorem; oscillations in predator-prey models; relaxation oscillators; Fitzhugh-Nagumo model; fixed point bifurcations; Hopf bifurcations and limit cycles.
4. Spatial dynamics: reaction-diffusion equations; Fisher-Kolmogorov equation; travelling waves in predator-prey systems; spatial SIS model; spread of rabies in a fox population; Turing instabilities; pattern formation in one and two dimensions.
5. Stochastic processes: continuous-time Markov chains; simple birth and death processes; stationary probability distributions; logistic growth process; branching processes and drug resistance; multivariate processes; stochastic enzyme dynamics; stochastic predator-prey dynamics.

Jupyter notebooks containing codes written in Python will be utilised throughout the course and a working knowledge of, or a willingness to learn and use Python, is expected.

Dr O.Karin

Term 2

This module will provide an introduction to the interdisciplinary field of mathematical systems biology. Drawing on analogies between biological and engineered systems, we will learn about mathematical approaches to model functional aspects of biological systems. We will discuss a wide
range of topics, including control, memory, and computation in biological systems. Each topic will be discussed in the context of specific experimental systems.

Learning Outcomes
On successful completion of this module you will:
- be able to describe the major concepts and principles of systems biology, including design principles and emergent properties in biological systems
- be able to develop mathematical theories on functional aspects of biological systems, including biochemical and cellular circuits
- appreciate the role of feedback regulation in biological systems, and acquire tools to analyze feedback systems
- critically evaluate the relation between theory and experiment in systems biology
- acquire an understanding of a range of mathematical and computational motifs that play an important functional role in a wide range of biological systems
- develop an appreciation for the complexity and diversity of biological systems, and an understanding of the role of interdisciplinary approaches in advancing our understanding of these systems
- Demonstrate an integrated understanding of the concepts of the module by critical, independent study of research articles and books.

Indicative Module Content
1. Introduction to biological circuits
2. Negative and positive feedback (responses, oscillations, memory, differentiation)
3. Integral feedback (adaptation, scale invariance)
4. Gradients: sampling and optimization
5. Bifurcations and feedback tuning to bifurcations
6. Hopfield networks
7. Optimal control and learning

Dr S. van Strien
Term 1

Recently there has been considerable interest in modelling learning. The settings to which these models are applied is wide-ranging. Examples include optimization of strategies of populations in ecology and biology, iterated strategies of people in a competitive environment and learning models used by technology companies such as Google.

This module is aimed at discussing a number of such models in which learning evolves over time and which have a game theoretic background. The module will use tools from the theory of dynamical systems and will aim to be rigorous. Topics will include replicator systems, best response dynamics and fictitious games, reinforcement learning and no-regret learning.

Learning Outcomes

On successful completion of this module, you will be able to:

• analyse 2D replicators systems for one and two player games;
• work comfortably with the notions of Nash, Correlated Equilibrium, Course Correlated Equilibrium and Evolutionary Stable Strategies;
• explain the notion of reciprocity in relation to Iterated Prisoner Dilemma games;
• appreciate the connection between Reinforcement Learning and replicator systems;
• outline the idea behind no regret learning models and the Blackwell approachability theorem;
• derive the proofs behind the methods that are used in the final project;
• appraise and interpret results from advanced textbooks and research papers.

Module Content

The module will cover the following topics:

1. Replicator systems;
2. Rock-paper-scissor games;
3. Iterated prisoner dilemma games;
4. Best response dynamics;
5. Two player games;
6. Fictitious games as a learning model;
7. Reinforcement learning;
8. No regret learning.

TBC
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.

Learning Outcomes

On successful completion of this module, you will be able to:

• demonstrate a familiarity with the basic concepts of topological dynamics;
• provide an outline of the ergodic theory of dynamical systems;
• appreciate the concept of symbolic dynamics through which topological and probabilistic dynamical properties can be understood;
• demonstrate an understanding of precise mathematical characterisations of chaotic dynamics;
• apply the above context in a number of one-dimensional settings, in particular in the context of piecewise affine expanding maps;
• independently appraise and evaluate advanced topological and probabilistic dynamical properties, beyond the foundations;
• independently develop and interpret examples in two and higher dimensions.

Module Content

The module covers the following topics:

1. Introduction: orbits, periodic orbits and their local stability;
2. Topological dynamics: invariant sets and limit sets, coding and sequence spaces, topological conjugacy, transitivity and mixing;
3. Chaotic dynamics: sensitive dependence, topological entropy, topological Markov chains;
4. Ergodic theory: sigma-algebras and measures, invariant measures, Poincaré recurrence, ergodicity and Birkhoff’s Ergodic Theorem, Markov measures and metric entropy;
5. Additional reading material in line with M4 objectives.

Dr D. Li
Term 2

This module serves as an introduction to bifurcation theory, concerning the study of how the behaviour of dynamical systems such as ODEs and maps changes when parameters are varied.The goal is to acquaint the students with the foundations of the theory, its main discoveries and the universal methods behind this theory that extend beyond its remit.

Learning Outcomes

On successful completion of this module, you will be able to:

• exploit basic dimension reduction methods (invariant manifold and invariant foliations);
• apply the method of normal forms;
• demonstrate a sound knowledge of the basics of stability theory;
• appreciate the role of control parameters and to construct bifurcation diagrams;
• describe the mathematical framework associated with classical local and global bifurcations;
• interpret results from advanced textbooks and research papers on bifurcation theory;
• independently appraise and evaluate the transition from periodic to quasiperiodic regimes and to chaos via destruction of quasiperiodicity.

Module Content

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.

Professor J. Lamb
Term 2

This is a 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 to the subject and then to touch upon several more advanced developments in this field.

Learning Outcomes

On successful completion of this module, you will be able to:

• describe the fundamental concepts of random dynamical systems;
• summarize the ergodic theory of random dynamical systems;
• select and critically appraise relevant research papers and chapters of research monographs;
• combine the ideas contained in such papers to provide a written overview of the current state of affairs concerning a particular aspect of random dynamical systems theory;
• thoughtfully engage orally in discussions related to random dynamical systems.

Module Content

Introductory lectures include foundational material on:

1. Invariant measures and ergodic theory
2. Random (pullback) attractors
3. Lyapunov exponents
4. Random circle homeomorphisms

Further material is at a more advanced level, touching upon current frontline research. Students select material from research level articles or book chapters.

Dr O. Street

Term 1

Classical dynamics is developed through variational principles rather than Newtonian force laws. Lagrangian and Hamiltonian formulations are considered. The methods are applied to a variety of problems including pendulums, the Kepler problem, rigid bodies and motion of a charged particle in a magnetic field.  The role of conserved quantities is emphasised. Advanced ideas including Hamilton-Jacobi theory, action-angle variables, adiabatic invariance and Hamiltonian Chaos are developed.

Learning Outcomes

On successful completion of this module, you will be able to:

• reformulate Newton's laws through variational principles;
• construct Lagrangians or Hamiltonians for dynamics problems in any coordinate system;
• solve the equations of motion for a wide variety of problems in dynamics;
• identify and exploit constants of the motion in solving dynamics problems;
• apply Lagrangian and Hamiltonian methods to problems in a variety of fields (e.g. Statistical Mechanics, Quantum Mechanics and Geometric Mechanics).
• independently appraise and evaluate two advanced topics from the following list: classical field theory, canonical perturbation theory, chaos.
• interpret results from advanced textbooks and research papers on two of the topics mentioned above.

Module Content

This module will cover the following topics:

1. Review of the Calculus of Variations.
2. Newtonian Mechanics: momentum, angular momentum, conservative forces.                                            
3. Lagrangian 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).
4. Hamiltonian Mechanics: Phase Space, Hamilton's equations, Poisson brackets, canonical transformations, generating functions,  Hamilton-Jacobi theory, action-angle variables, adiabatic invariance, integrability, application of Hamiltonian mechanics to rigid bodies.                                                                                    
5. Introduction to Hamiltonian Chaos.

Dr P. Siorpaes
Term 1

The mathematical modelling 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.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate the fundamental principles involved in pricing derivatives;
• describe and critically analyse simple market models and explore their qualitative properties;
• confidently perform calculations involving pricing and hedging in discrete market models;
• demonstrate a familiarity with some key concepts in modern probability theory and apply them to perform computations;
• outline a mathematical formulation describing the behaviour of a number of financial derivatives;
• construct dynamic programming techniques to solve problems where inter-temporal relations are important;
• appraise and critically evaluate one or more of the advanced topics listed below.

Module Content

The module will cover the following topics:

financial derivatives, arbitrage, no-arbitrage pricing, self-financing portfolios, non-anticipative trading strategies, hedging of derivatives, domination property, complete markets, ‘risk-neutral’ probabilities, the fundamental theorems of asset pricing, conditional probability and expectation, filtrations, Markov processes, martingales, change of measure.

Extra mastery component will include the following advanced topics: utility, optimal investment.

Prof D. Brigo

Term 2

To deal with valuation, hedging and risk management of financial options, we briefly introduce stochastic differential equations using a Riemann-Stiltjes approach to stochastic integration. We introduce no-arbitrage theory in continuous time based on replicating portfolios, self-financing conditions and Ito's formula. We derive prices as risk neutral expectations. We derive the Black Scholes model and introduce volatility smile models. We illustrate valuation of different options and introduce risk measures like Value at Risk and Expected Shortfall, motivating them with the financial crises.

Learning Outcomes

On successful completion of this module you will be able to

• work comfortably with stochastic differential equations commonly encountered in finance
• explain what is meant by no-arbitrage markets and why no-arbitrage is important operationally;
• connect no-arbitrage by replication to the existence of a risk neutral measure;
• price and hedge several types of financial options with several SDE models;
• calculate risk measures such as Value at Risk and Expected Shortfall;
• write code to price options according to SDE models covered in the module.
• independently appraise and evaluate SDE models for financial products.
• adapt a range of numerical methods and apply them in a coherent manner to unfamiliar and open problems in finance.

Module Content

1.Recap of key tools from probability theory

2.Brownian motion

3.Ito and Stratonovich stochastic integrals

4.Ito and Stratonovich stochastic differential equations (SDEs)

5.No-arbitrage through replication

6.No arbitrage though risk neutral measure

7.Derivation of the Black Scholes formula

8.Introduction of a few volatility smile models

9.Pricing of several types of options

10.Introduction to crises and risk measures

11.The Barings collapse and the introduction of value at risk (VaR)

12.Problems of VaR and an alternative: expected shortfall (ES)

13.Numerical examples and problems with risk measures, including software code.

Dr C. Salvi

Term 2

The theory of rough paths provides a mathematical language to describe the effects a stream can generate when interacting with non-linear dynamical systems. It has had a significant impact on several areas of stochastic analysis, notably on the development of Hairer’s Fields medal winning work on regularity structures for singular stochastic PDEs. In recent years, rough path theory has played a key role in the design of state-of-the-art machine learning algorithms for processing irregularly sampled time series data. This first half of this module will focus on the mathematics of signatures and signature kernels and applications to machine learning. The second half will focus on constructing rough integration, establish solutions to rough differential equations and their consistency with stochastic differential equations, and finally on the interplay between rough paths and modern deep learning models dubbed neural differential equations, which incorporate neural networks as vector fields of classical differential equations.

Learning Outcomes
On completion of this module students will be able to display mastery of a complex and specialised area of knowledge and skills in rough paths and applications to machine learning. In particular students who attended the course will be able to:
• Use the basic properties of the signatures and signature kernels and present their utility in machine learning.
• Understand the basics of rough integration, rough differential equations and their consistency with classical stochastic integration.
• Use neural differential equations models (Neural ODEs, CDEs, SDEs, RDES).
• Implement in Python some of the above models for synthetic and real-world examples
• Engage independently with research literature, critically evaluating the performance of neural differential equation models in practice,
• Apply the techniques developed in the module to solve unfamiliar, non-standard problems.

Indicative Module Content
1. Analytic and algebraic properties of the signature
2. Functional analysis, topology and probability on unparameterised paths
3. Signature kernels and associated reproducing kernel Hilbert spaces
4. Universality and characteristicness of signature kernels
5. Computing signature kernels as solutions to PDEs
6. Applications to machine learning
7. Geometric and controlled rough paths
8. From Young integration to rough integration
9. From stochastic to rough differential equations
10. Neural ODEs, SDEs, RDEs and the log-ODE method
Prerequisites in stochastic calculus, the theory and numerical analysis of ODEs and SDEs are assumed.

Dr M. Autenrieth

Term 2

This module gives a broad mathematical introduction to both microeconomics and macroeconomics, with a particular emphasis on the former. We consider the motivations and optimal behaviours of firms and consumers in the marketplace (profit and utility maximisation, respectively), and show how this leads to the widely observed laws of supply and demand. We look at the interaction of firms and consumers in markets of varying levels of competition. In the final section, we discuss the interplay of firms, households and the government on a macroeconomic scale.

Learning Outcomes
On successful completion of this module, you will be able to:
- Give a mathematical description of the problems of profit maximisation and utility maximisation.
- Apply the concepts of preference relations and utility functions as well as their interaction.
- Determine optimal behaviour by solving the profit or utility maximisation problem for stylised examples.
- Describe the change in demand with respect to a change in price or income.
- Characterise the equilibrium price or quantity as the maximiser of the community surplus, and discuss how deviations lead to a deadweight loss.
- Describe the interplay of firms, households, government, the financial sector and the oversees sector in a stylised way.
- State and apply the definition of Gross Domestic Product (GDP) as a measure of the economic activity.
- Discuss shortcomings of GDP as a measure of social welfare and how to overcome these shortcomings.
- Demonstrate an integrated understanding of the concepts of the module by critical, independent study of research articles and books.

Module Content
An indicative list of sections and topics is:
- Theory of the firm: Profit maximisation for a competitive firm. Cost minimisation. Geometry of costs. Profit maximisation for a non-competitive firm.
- Theory of the consumer: Consumer preferences and utility maximisation. The Slutsky equation.
- Levels of competition in a market: Consumers’ and Producers’ surplus. Deadweight loss.
- Macroeconomic theory: Circular flow of income. Gross Domestic Product. Social welfare and allocation of income.

Professor X. Wu
Term 1

Fluid dynamics investigates motions of both liquids and gases. Being a major branch of continuum mechanics, it does not deal with individual molecules, but with an ‘averaged' motion of the medium (i.e. collections of molecules). The aim is to predict the velocity, pressure and temperature fields in flows arising in nature and engineering applications. In this module, the equations governing fluid flows are derived by applying fundamental physical laws to the continuum. This is followed by descriptions of various techniques to simplify and solve the equations with the purpose of describing the motion of fluids under different conditions.

Learning Outcomes

On successful completion of this module you will be able to

• state the underlying assumptions of the continuum hypothesis;
• compare and contrast the different frameworks that can be used to describe fluid motion and to identify the connections between them;
• derive exact solutions of the Navier-Stokes equations and justify the physical and mathematical assumptions made in obtaining them;
• perform simplifications arising under the assumption of inviscid flow which permit the integration of the Euler equations, leading to results such as Bernoulli's equation and Kelvin's circulation theorem;
• demonstrate a sound understanding of the method of conformal mappings and be able to use this method to analyse various two-dimensional inviscid flows;
• choose the appropriate conformal mapping to solve inviscid flow problems in complicated geometries;
• predict the shape of the flow streamlines for such problems.

Module Content

The module is composed of the following sections:

1. Introduction: The continuum hypothesis. Knudsen number. The notion of fluid particle. Kinematics of the flow field. Lagrangian and Eulerian frameworks. Streamlines and pathlines. Strain rate tensor. Vorticity and circulation. Helmholtz’s first theorem. Streamfunction.
2. Governing Equations: Continuity equation. Stress tensor and symmetry, Constitutive relation. The Navier-Stokes equations.
3. Exact Solutions of the Navier-Stokes Equations: Couette and Poiseuille flows. The flow between two coaxial cylinders. The flow over an impulsively started plate. Diffusion of a potential vortex.
4. 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.

Professor J. Mestel
Term 2

In this module, we deal with a wide class of realistic problems by seeking asymptotic solutions of the governing Navier-Stokes equations in various limits. We shall start with the “slow, small or sticky” case, when the Reynolds number is low and we obtain the linear Stokes equations. Then we consider the lubrication limit, and show how a thin layer of fluid is able to exert enormous pressures and prevent moving solid bodies from touching. Next we shall consider the “fast and vast” limit of high Reynolds number, which is characteristic of most flows we encounter in everyday life. In the final part of the module we consider a mixture of advanced topics, including flight, bio-fluid-dynamics and an introduction to flow stability.

Learning Outcomes

On successful completion of this module you will be able to:

• simplify and solve the governing Navier-Stokes equations in situations where there is a short lengthscale in one of the coordinate directions;
• apply the general properties of low Reynolds number flows to predict the drag on slow-moving bodies, like a solid sphere or spherical bubble, and appreciate the causes of the ‘Stokes paradox’;
• analyse lubrication-like flows in thin layers;
• derive the boundary-layer equations and identify self-similar solutions for flows at large Reynolds number;
• determine stability criteria for various fundamental flows;
• model animal locomotion at low and high Reynolds numbers;
• interpret results from advanced fluid mechanics textbooks and research papers;
• independently appraise and evaluate some advanced topics in viscous fluid mechanics.

Module Content

The module is composed of the following sections:

1. Low-Reynolds-number flows: Dynamic Similarity. Properties of the Stokes equations. Uniqueness and minimal dissipation theorems. The analysis of the flow past a solid sphere and spherical bubble. Stokes paradox.
2. Lubrication Theory: Derivation of Reynolds’ lubrication equation and examples. Hele-Shaw and thin film flows.
3. High-Reynolds-number flows; Boundary-layer theory: The notion of singular perturbations. Derivation of boundary-layer equations. Blasius flow, Falkner-Skan solutions and applications. Von Mises variables and their application to periodic boundary layers. Prandtl-Batchelor Theorem for flows with closed streamlines.
4. Introduction to hydrodynamic stability: Importance of stability. Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Circular flow stability criterion.
5. Swimming and Flight; Animal locomotion: Scallop theorem. Resistive Force Theory. Introduction to 3D-aerofoil theory. Flight strategies.
6. Advanced Topics: Current research in areas such as convection and magnetohydrodynamics.

Professor D. Crowdy
Term 2

This is an advanced module in applied mathematical methods applied to the subfield of fluid dynamics called vortex dynamics. The module will focus on the mathematical study of the dynamics of vorticity in an ideal fluid in two and three dimensions. The material will be pitched in such a way that it will be of interest to those specializing in fluid dynamics but can also be viewed as an application of various techniques in dynamical systems theory.

Learning Outcomes

On successful completion of this module, you will be able to:

• interpret the role of vorticity within a range of problems in fluid mechanics;
• derive and compare a range of vortex models, from the point vortex models to distributed models, including vortex patches;
• combine your knowledge of different branches of mathematics (e.g. vector calculus, complex analysis and the theories of Hamiltonian dynamical systems and partial differential equations) in order to describe the dynamics of vorticity;
• choose from an array of applied mathematical techniques to explicitly solve for vorticity distributions;
• appraise the role that vortex structures play in modelling physical systems.

Module Content

The module will cover the following topics:

1. Eulerian description of fluid flows;
2. Incompressible flows and streamfunctions;
3. Vorticity, vortex lines and vortex tubes;
4. Biot-Savart law;
5. Euler's equations and the vorticity equation;
6. Kelvin's circulation theorem;
7. Bernoulli theorems;
8. Point vortex model, complex potentials;
9. Point vortex equilibria;
10. Dynamics of point vortices;
11. Vortex dynamics on a spherical surface;
12. Vortex patch models;
13. Vortex patch equilibria;
14. Vortex patch dynamics and contour dynamics;
15. Other distributed vortex models.

Prof X. Wu
Term 2

Fluid flows 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. This is of profound scientific and technological importance because of its relevance to mixing and transport in the atmosphere and oceans, drag and aerodynamic heating experienced by air/spacecrafts, jet noise, combustion in engines and even the operation of proposed nuclear fusion devices.

Learning Outcomes

On successful completion of this module you will be able to

• construct the basic concepts underpinning hydrodynamic stability theory;
• predict linear stability properties based on eigenvalue analysis;
• compare and contrast various different instability mechanisms;
• derive various theorems that help us decide whether a flow is stable or unstable;
• model the effects of nonlinearity within an asymptotic framework.

Module Content

Topics covered will be a selection from the following list.

1. Basic concepts of stability; linear and nonlinear stability, initial-value and eigenvalue problems, normal modes, dispersion relations, temporal/spatial instability.
2. Buoyancy driven instability: Rayleigh-Benard instability, formulation of the linearised stability problem, Rayleigh number, Rayleigh-Benard convection cells, discussion of the neutral stability properties.
3. 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.
4. 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, KelvinHelmholtz 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.
5. 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 stabilityeq uations; transition process and prediction (correlation); receptivity.
6. 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.

Dr P. Berloff 
Term 2

This is an advanced-level fluid-dynamics course with geophysical flavours. The lectures target upper-level undergraduate and graduate students interested in the mathematics of planet Earth, and in the variety of motions and phenomena occurring in planetary atmospheres and oceans. The lectures provide a mix of theory and applications.

Learning Outcomes

On successful completion of this module you will be able to:

• demonstrate a deep understanding of the foundations of geophysical fluid dynamics;
• model a broad range of natural phenomena associated with the atmosphere and ocean;
• appreciate the main concepts and terminology used in the field;
• derive the boundary layer equations for flow in a rotating frame and justify the relative importance of various terms in the equations of motion;
• describe, select appropriately and apply a range of methods and techniques for solving practical problems;
• independently appraise an advanced topic in geophysical fluid dynamics;
• evaluate results from research papers in the field of geophysical fluid dynamics.

Module Content

The module is composed of the following sections:

1. Introduction and basics;
2. Governing equations (continuity of mass, material tracer, momentum equations, equation of state, thermodynamic equation, spherical coordinates, basic approximations);
3. Geostrophic dynamics (shallow-water model, potential vorticity conservation law, Rossby number expansion, geostrophic and hydrostatic balances, ageostrophic continuity, vorticity equation);
4. Quasigeostrophic theory (two-layer model, potential vorticity conservation, continuous stratification, planetary geostrophy);
5. Ekman layers (boundary-layer analysis, Ekman pumping);
6. Rossby waves (general properties of waves, physical mechanism, energetics, reflections, mean-flow effect, two-layer and continuously stratified models);
7. Hydrodynamic instabilities (barotropic and baroclinic instabilities, necessary conditions, physical mechanisms, energy conversions, Eady and Phillips models);
8. Ageostrophic motions (linearized shallow-water model, Poincare and Kelvin waves, equatorial waves, ENSO “delayed oscillator”, geostrophic adjustment, deep-water and stratified gravity waves);
9. Transport phenomena (Stokes drift, turbulent diffusion);
10. Nonlinear dynamics and wave-mean flow interactions (closure problem and eddy parameterization, triad interactions, Reynolds decomposition, integrals of motion, enstrophy equations, classical 3D turbulence, 2D turbulence, transformed Eulerian mean, Eliassen-Palm flux).

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. We will use tools and concepts from various areas of mathematics and physics, such as classical mechanics, linear algebra, probability theory, numerical methods, analysis and geometry.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate Schrödinger's formulation of quantum mechanics, wave functions and wave equations;
• construct the mathematical framework of quantum mechanics, including the 4 postulates of quantum mechanics and the Dirac notation;
• solve the eigenvalue problem for basic one-dimensional quantum systems;
• exploit the method of stationary states to deduce the time-evolved quantum state from the initial state of a system;
• communicate fluently using the Dirac notation;
• interpret results from advanced quantum mechanics textbooks and research papers;
• independently appraise and evaluate an advanced (more contemporary) topic in quantum mechanics from those listed in the syllabus below.

Module Content

The module will cover the following topics:

1. Hamiltonian dynamics;
2. Schrödinger equation and wave functions;
3. stationary states of one-dimensional systems;
4. mathematical foundations of quantum mechanics;
5. quantum dynamics;
6. angular momentum.

A selection of topics among the following additional optional topics will be covered depending on students interests:

1. approximation techniques;
2. explicitly time-dependent systems;
3. geometric phases;
4. numerical techniques;
5. many-particle systems;
6. cold atoms;
7. entanglement and quantum information.

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. 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.

Learning Outcomes

On successful completion of this module you will be able to

• demonstrate an understanding of the relation between space and time and apply Lorentz transforms;
• appreciate the structure of special relativity as derived from the principle of least action;
• determine relativistic particle trajectories;
• derive Maxwell’s equations from first principles and apply them to variety of interactions of charges and fields;
• critically analyse various solutions of the electromagnetic wave equations;
• describe electrostatic interactions and motion using Coulomb’s law;
• construct an expansion of electrostatic interactions in terms of multipoles.

Module Content

This course follows closely the following book: L.D. Landau and E.M. Lifschitz, Course on Theoretical Physics Volume 2: Classical Theory of Fields.

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.

Dr R. Barnett
Term 2

Quantum mechanics (QM) is one of the most successful theories in modern physics and has an exceptionally beautiful underlying mathematical structure. Assuming some prior exposure to the subject (such as Quantum Mechanics I), this module aims to provide an intermediate/advanced treatment of 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 geometry.

Learning Outcomes

On successful completion of this module, you will be able to:

• outline key aspects of quantum mechanics at the intermediate/advanced level;
• harness the power of symmetry in understanding quantum mechanics;
• describe many-particle quantum mechanical systems, and demonstrate familiarity with the formalism of second quantisation;
• solve complex quantum mechanical problems using the machinery introduced in this module;
• use the knowledge gained here as a solid foundation for a research project in quantum mechanics;
• interpret results from advanced quantum mechanics textbooks and research papers;
• appraise and evaluate a topic in quantum mechanics from the syllabus at an advanced level.

Module Content

This module will cover the following core topics:

1. quantum mechanics in the momentum basis;
2. the Heisenberg picture;
3. the use of symmetry and general transformations in quantum mechanics;
4. Bloch’s theorem;
5. perturbation theory;
6. adiabatic processes;
7. second quantisation;
8. introduction to many-particle systems;
9. Fermi and Bose statistics.

Additional topics include: WKB theory, the Feynman path integral, quantum magnetism.

Dr C. Ford

Term 2        

This module provides an introduction to General Relativity. Starting with the rather simple Mathematics of Special Relativity the goal is to provide you with the mathematical tools to formulate  General Relativity. Some examples, including the Schwarzschild space-time are considered in detail.

Learning Outcomes

On successful completion of this module, you will be able to:

• appreciate the application of tensors in special relativity;
• demonstrate a working knowledge of tensor calculus;
• explain the concepts of parallel transport and curvature;
• formulate and solve the geodesic equation for a given space-time metric;
• derive Einstein's field equations and analyse Schwarzschild's solution;
• interpret results from advanced general relativity textbooks and research papers;
• appraise and critically evaluate two of the extensions and applications listed below.

Module Content

This module will cover the following topics:

1. Special Relativity
2. Tensors in Special Relativity
3. Tensors in General Coordinates Systems
4. Parallel Transport and Curvature
5. General Relativity
6. The Schwarzschild Spacetime
7. Variational Methods
8. Extensions and Applications (selected from gravitational waves, Einstein-Hilbert action, cosmology, Einstein-Cartan theory, differential geometry)