# AMA Level 3 Modules

**AMA3002 Quantum Theory (1**^{st}semester)

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr G Gribakin

**Introduction**

Quantum mechanics brought about the most fundamental change in our understanding of how the world works. It describes the behaviour of microscopic particles (electrons, photons, atomic nuclei, etc.) and their interactions in a way that is very different from our everyday experience. Key points of the theory is wave-particle duality (e.g., the ability of particles to display typical wave-like properties, such as interference), quantisation (i.e., restriction of possible values of some physical observables, such as energy or angular momentum, to a discrete set of values), statistical nature of its predictions, and the role played by the observer in a measurement process.

In this module we develop the mathematical methods that enable one to describe how nature behaves at small scales, e.g., that of individual atoms. The mathematical setting of Quantum Theory is that of Hilbert spaces and linear operators, and its practical aspects involve dealing with matrices, eigenvalue problems and differential equations.

Students completing this module will understand the basic principles of quantum mechanics and its mathematical tools, learn how to solve a number of simple, fundamental problems, and practice a number of approximate methods that greatly widen the range of problems that can be solved.

**Contents**

- Overview of classical physics and the need for new theory.
- Basic principles: states and the superposition principle, amplitude and probability, linear operators, observables, commutators, uncertainty principle, time evolution (Schrödinger equation), wavefunctions and coordinate representation.
- Elementary applications: harmonic oscillator, angular momentum, spin.
- Motion in one dimension: free particle, square well, square barrier.
- Approximate methods: semiclassical approximation (Bohr-Sommerfeld quantisation), variational method, time-independent perturbation theory, perturbation theory for degenerate states (example: spin-spin interaction, singlet and triplet states).
- Motion in three dimensions: Schrödinger equation, orbital angular momentum, spherical harmonics, motion in a central field, hydrogen atom.
- Atoms: hydrogen-like systems, Pauli principle, structure of many-electron atoms and the Periodic Table.

**Assessment**

Exam 70% Assignments 1 and 2 15% each

**AMA3003 Tensor Field Theory (2**^{nd}semester)

*Pre-requisite:* While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or a BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed. Useful background includes vector algebra and dynamics, linear algebra, vector calculus, fluid mechanics, calculus of variations, classical mechanics, electromagnetism, quantum mechanics.** **

*Lecturer*: Dr D Green, Dr A Swann

**Introduction**

Physical theories are associated with a particular mathematical machinery, often catalyzing each other’s development. For example, electromagnetism goes with vector calculus, quantum mechanics - with linear algebra, etc. Tensor analysis is the machinery of general relativity, but has applications in many other areas. Tensors are a generalization of vectors and matrices. Their defining property is how they behave under coordinate transformations. Their usefulness is that they represent intrinsic, coordinate-independent properties and relations. This module covers the algebra and calculus of tensors, with applications in special and general relativity. The mathematician will enjoy the tensors and differential geometry, and the theoretical physicist - the relativity; this module appeals to a range of backgrounds and interests.

**Contents**

**Coordinate transformations; scalar; gradient; contravariant and covariant vectors; tensor field; outer and inner product; contraction; quotient rule; metric tensor; symmetry; scalar product, length, orthogonality.**

*Tensor algebra:*** Special relativity and electromagnetism:** Lorentz transformation; velocity and acceleration transformation; time dilation; Lorentz contraction; proper time; Minkowski space-time; 4-velocity; 4-acceleration; 4-momentum; 4-force; mass-energy-momentum relations; scattering; Maxwell’s equations; gauge transformation; 4-current; 4-potential; electromagnetic tensor; Maxwell’s equations in tensor form; transformation of fields; electromagnetic energy tensor.

** Tensor calculus: **Curvature; geodesics; Christoffel symbols; covariant derivative; divergence and Laplacian; geodesic coordinates; curvature tensor; Bianchi identity; Ricci tensor; Einstein tensor.

** General relativity: **Newtonian gravity; energy-momentum tensor; Einstein field equations; weak-field approximation; Schwarzschild solution; tests of general relativity; black holes.

**Assessment**

Exam 80% Continuous Assesssment 20%

**AMA3006 Partial Differential Equations (1**^{st}semester)

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr G Kiss

**Introduction**

This module develops further some ideas and methods initially introduced in MTH2002. It concerns those problems in Applied Mathematics which can be formulated as differential equations involving functions of more than one variable (e.g., position and time, or several coordinates). These are partial differential equations (PDE). Besides MTH2002, students deal with examples of such equations in AMA2005, AMA3002, AMA3007, AMA3014 and SOR3012. This course studies properties of these equations, and develops methods for solving them.

**Contents**

*Introduction:*Terminology: order, linear and nonlinear equations, initial and boundary conditions. Derivation of the wave equation (in one dimension, 1D) and heat equations. Further examples: Laplace's equation, Poisson's equation, Schrödinger equation. Method of Separation of Variables: for the wave equation and heat equation in 1D, and for the circular membrane. Bessel functions. Eigenvibrations (modes).

*Fourier Method:*Fourier expansion of a function. Piecewise continuous functions. Convergence of Fourier series (FS). FS for even and odd functions. Half-range FS. FS near discontinuities - the Gibbs phenomenon. Application of FS to solving PDE. Laplace's equation for a disk: Poisson's integral.

*Integral Transform Methods:*Fourier transform (FT) as a limit of the complex Fourier series for the infinite interval. Notion of the Dirac delta function. FT of even and odd functions, sine and cosine FT, and FT of partial derivatives. Application to PDE (heat, wave, Laplace). Laplace transform (LT). LT of some common functions, convolution and shift theorems. Applications of LT to ordinary and partial differential equations.

*Orthogonal expansions:*Sturm-Liouville Theory. Inner product and norm, orthogonal systems of functions. Gram-Schmidt process. Expansion of functions in orthogonal systems. Convergence in the mean and completeness. Self-adjoint differential operators and Green's formula. Singular, periodic and homogeneous boundary conditions. Sturm-Liouville theory: properties of the eigenfunctions and eigenvalues; degeneracy. Generalised Fourier series. Fourier-Bessel expansion.

*Green's Functions:*Dirac delta function. Green's function of the Sturm-Liouville equation. Green's functions in several dimensions (Dirichlet problem for the Laplace equation; heat and wave equations with source terms).

*Normal Forms of 2nd-order PDE in Two Variables*: Linear and quasi-linear PDE. Hyperbolic, parabolic and elliptic types. Reduction to the normal form, and use of this method for solving PDE.

**Assessment**

Exam 80% Project 20%

**AMA3007 Financial Mathematics (2**^{nd}semester)

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr A Ferraro

**Introduction**

Mathematical skills are highly sought after in the financial services industries, and this employment sector remains a favoured destination for graduates. Around 40% of Mathematics graduates entering employment across the UK (see www.prospects.ac.uk for recent data) go into financial services, which includes, accountancy, retail and investment banking, mergers and acquisitions, insurance and actuarial work, capital market trading, and hedge fund employment, and so on.

At the low end of this sector, retail banking for example, a degree in mathematics is certainly not essential. This work is mainly concerned with simple arithmetic operations. However, at the high end of financial services, in a hedge fund for example, employers expect to see PhD-level qualifications in mathematics from their applicants along with excellent software skills. These mathematicians are involved in the business of derivative pricing and trading and earn salaries well over 100k. Derivatives are financial products (instruments as they are called in the trade) derived from assets that have an unpredictable price. The total outstanding notional value of derivatives contracts today has grown beyond a quadrillion dollars (that's 10^{15} to you and me). It is a perilous and lucrative business!

Derivatives were originally devised to avoid risk by providing an insurance on a risky asset. Nowadays, they are an essential part of risk taking in capital markets. Indeed the speculation in buying and selling these instruments, specifically credit derivatives, precipitated the current credit crunch. Of course, this trade relies upon knowing the fair price of a derivative. Pioneering work by Black, Merton and Scholes, showed that, under certain assumptions for the unpredictability of the asset, the price of the derivative obeys a partial-differential equation. The construction of such equations and their solution is where mathematicians come in!

The objective of the course is to provide an introduction to the mathematical techniques which can be applied to pricing problems for financial derivatives. Specifically, our focus is on stochastic calculus and the theory and practice of pricing simple derivatives such as contracts and options.

We are grateful to First Derivatives plc for their support of this course and the provision of prizes for the best examination performance.

**Contents**

Introduction to financial derivatives: forwards, futures and options. Future markets and prices. Option markets. Binomial models and the risk-free portfolio. Stochastic calculus and random walks. Black-Scholes equation. Pricing European options. Various option pricing models. Interest-rate derivatives. Credit derivatives. Swaps.

**Assessment**

Exam 70% Report 20% Presentation 10%

**AMA3011 Applied Mathematics Project (1st and 2nd semester)**

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of BSc Mathematics, Applied Mathematics and Physics or Theoretical Physics programmes, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Module Co-ordinators*: Prof J Kohanoff (1st semester), Dr C Ramsbottom (2nd semester)

**Introduction**

This module (or alternatively, PMA3013) is compulsory for all students on the BSc pathway and constitutes a self-study project on an advanced mathematical topic under the supervision of a member of staff. Students will be offered a choice of topic subjects which can span the entire range of applied mathematics and statistics, including some theoretical physics.

**Contents**

Self-study of an advanced mathematical topic under the supervision of a member of staff. Students will be offered a choice of subjects, which can span the entire range of applied mathematics, including theoretical physics. The study concludes with a written report and a poster presentation.

**Assessment**

Report 80% Presentation 20%

**AMA3013 Calculus of Variations and Hamiltonian Mechanics (2**^{nd}semester)

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr P Siegl, Dr D Dundas

**Introduction**

The aim of this course is to develop the essential ideas and mathematics of the Calculus of Variations, with applications to various problems.

In its simplest form the Calculus of Variations is concerned with finding a function which makes a given integral a maximum or minimum, or, more generally, ‘stationary’. A famous historical example (solved by the Bernoullis, Newton and others) is the following. A particle, which is initially at rest, slides under gravity along a smooth wire connecting two points: find the shape of the wire, which minimizes the time taken for the particle to travel between the two points. This corresponds to finding the function which minimizes the time integral. More generally, ‘variational’ problems may involve integrals containing more than one function or higher derivatives of the functions than the first, or involve constraints on the functions, which may be given in integral form or point-wise. In addition, the variational integral may be multi-dimensional.

In the formulation of physical laws variational methods have a much deeper significance. This was first suggested by the elegant work of Lagrange and the Irish mathematician Hamilton who looked at the basic mathematical structure of classical mechanics. The formalism that they developed is not only essential for a full understanding of quantum mechanics and statistical mechanics but turns out to have a much wider application in that it can be extended to systems that are not normally considered in dynamics, e.g., the Electromagnetic Field. In this sense variational methods provide a ‘unifying’ principle of physics.

The first part of this course concentrates on the basic mathematics of the Calculus of Variations. The second part deals with Lagrangian and Hamiltonian mechanics and their variational basis. While the course is more or less self-contained, it does requires a knowledge of basic first-year calculus, including functions of several variables and a smattering of ordinary differential equations. Newton's laws of motion in vector form and the elementary ideas of kinetic and potential energy are also needed.

**Contents**

*Part I: The Calculus of Variations*(approximately 15 lectures): Motivation: the Brachistochrone and Isoperimetric problems. Functional; extremum, stationary point. Function classes. Weak and strong variations. The simplest variational problem: necessary condition for an extremum; Euler-Lagrange lemma; Euler's equation. Several unknown functions. Fermat's principle. Geodesics. Functionals depending upon higher order derivatives. Variational problems with subsidiary conditions: Lagrange multipliers; finite subsidiary conditions. Variable end-point theorem: broken extremals; Weierstrass-Erdmann corner condition. Second variation of a functional: Legendre's necessary condition for a minimum. Direct methods: the Ritz method; the method of finite differences; the Sturm-Liouville problem.

*Part II: Analytical Mechanics*(approximately 15 lectures): Constraints and generalized coordinates; holonomic constraints. Virtual displacement; D'Alembert's principle; Lagrange's equations. Action integral. Hamilton's principle. Generalized momentum; cyclic coordinates; conservation laws. The Hamiltonian; Hamilton's equations; derivation of Hamilton's equations from a variational principle. Principle of least action; Jacobi's form of the principle of least action. Canonical transformations: generating function; symplectic matrices and canonical transformations; Hamilton's equations in symplectic form. Poisson brackets: Jacobi's identity; canonical transformations of Poisson brackets; Hamilton's equations in Poisson bracket form; Poisson's theorem. The Hamilton-Jacobi equation; Hamilton's characteristic function. Action-angle variables. Phase-space diagram.

**Assessment**

Exam 70% Assignments 1 & 2 (15% each)

*Pre-requisite*: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or BSc Mathematics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturer*: Dr A Zhigun

**Introduction**

Mathematics is a powerful instrument for revealing mechanisms behind biological processes ranging from formation of spots on animal coats to metastasis. The module is concentrated on modelling growth, interaction, and migration by means of ordinary and partial differential equations. Where the resulting problems cannot be solved by hand, we will apply various analytical tools, as well as perform numerical simulations. Thus we will gain both quantitative and qualitative information about the solutions.

In this module no previous knowledge of biology is assumed. With each topic a brief description of the biological background will be provided sufficient to understand the models being studied. The emphasis throughout the course is on the practical application of mathematical models in helping unravel the underlying mechanisms involved in the biological processes. The mathematics in the module is dictated by the biology and not vice-versa.

You will gain a good appreciation of the art of modelling which relies on:

- a sound understanding and appreciation of the biological problem,
- a realistic mathematical representation of the important biological phenomena,
- finding useful solutions and studying their properties,
- a biological interpretation of the mathematical results in terms of predictions and insights.

**Contents**

- Population models for single species
- Models for interacting populations
- Reaction kinetics
- Biological waves for single species models
- Pattern formation
- Biological motion including reaction diffusion models and chemotaxis
- Tumour migration

**Assessment**

Exam 70% Project 30%

*Pre-requisite*: MTH1001 and MTH1002

*Lecturer*: Dr F Pausinger

**Introduction**

This module gives an introduction to basic techniques in discrete mathematics. The main objects of study in discrete mathematics are integers, graphs and logical statements. Discrete objects can often be enumerated by integers and, thus, discrete mathematics can be characterised as the branch of mathematics dealing with countable sets. Discrete mathematics is particulary important in our modern world since digital computers typically operate in discrete steps and store data in discrete bits. Thus, the main applications of the methods presented in this module can be found in such diverse areas as computer algorithms, cryptography, automated theorem proving or operations research.

**Contents**

- Logic: formal statements, truth values, repetition of proof principles
- Elementary Number Theory: divisibility and primes, modular arithmetic
- Combinatorics: basic counting, pigeonhole principle, inclusion-exclusion, recurrence relations, generating functions
- Introduction to graph theory: trees, Eulerian graphs, Hamiltonian graphs, bipartite graphs, planar graphs, vertex colouring, connectivity, matchings, Ramsey numbers
- Algorithms: Big Oh notation, asymptotic analysis of algorithms, basic algorithmic techniques and their analysis.

**Assessment**

Exam 70%, Class test 15%, Project 15%

**AMA3020 Investigations (2**^{nd}semester)

*Pre-Requisites*: While there are no specific pre-requisites, this course is intended for MSci students in Mathematics, Mathematics and Statistics & Operational Research, Applied Mathematics and Physics, Mathematics and Computer Science, and Theoretical Physics, and a mathematical knowledge and ability commensurate with this stage is assumed.

*Lecturers*: Prof M Paternostro, Dr C Ballance

**Introduction**:

Problem-solving is a key skill in many domains, from finance to academia, from software development to data analysis. Being able to address accurately a previously unseen problem, finding creative manners to solve it, and presenting complex concepts in a simple and effective way are invaluable assets. This module will facilitate the development of problems-solving skills that are expendable in a broad domain of fields and areas’

**Content**:

This module provides an introduction to project development and management in topics of Applied Mathematics. Students are trained in research methods by working on a range of projects.

In the first part of the module the students conduct a short practice investigation, followed by two short investigations (one in small groups [typically in pairs] and one individually) in a range of problems in Applied Mathematics and Theoretical Physics. The results are presented in the form of typed reports.

This is followed by a long investigation, which is a literature study each a topic in Mathematical or Theoretical Physics not covered in the offered (or chosen) modules. The results of the investigation are presented in the form of a set of notes and a presentation that takes the form of a short lecture and is delivered by the students individually.

The two short investigations are typed up in reports and submitted for assessment. The set of notes on which the presentation for the long investigation is based, is typed up and submitted for assessment.

**Assessment**: Reports x2 80% Group Presentation 20%

**AMA3022 Team Project: Mathematics with Finance (1st semester)**

*Pre-requisite*: Only available to students on the BSc Mathematics with Finance programme.

*Lecturers*: Dr D Dundas, Dr S Moutari

**Introduction**

As a result of taking this module, students will learn to respond to a briefing on a problem by a client. They will be able to work successfully as part of a team to address the problem. They will also be able to make a final presentation on the outcome of the work.

**Contents**

Business skills workshop. Presentation skills. Negotiation skills. Customer relationship. Project management/team building. Teams required to negotiate, plan, develop and deliver a completed task working as a group, commissioned by the `client' company. The project will require software development skills.

**Assessment**

Business Proposals 30% Report & Presentation 40% Business Plan 10% Peer Evaluation 20%