Mathematics MMath

Calculus provides the basic, underpinning mathematics for much of modern technology, from the design of chemical reactors and high-speed trains, to models for gene networks and space missions.

The basic ideas that underpin calculus are functions and limits. To study these rigorously you need to learn about the tools of mathematical analysis. In addition to differential equations and the calculus of functions of one or more variables and their differentiation, integration and analysis, you will learn the basics of logic and how to construct rigorous proofs.

You’ll learn how to construct and analyse differential equations that model real-world systems. Applications that you’ll learn about include systems governed by Newton’s laws of motion, such as sets of interacting particles and the orbits of planets, as well as models of population dynamics. You will also be introduced to the mathematical basis of concepts such as work and energy.

Pure mathematics at university is typically very different to the pure mathematics you've learnt at school or college.

In this module, you'll use the language of sets, functions and relations to study abstract mathematical ideas. You will also learn how to construct mathematical proofs. Topics that you will learn about include:

• set theory
• prime numbers
• symmetry and groups
• rings, fields and integer
• polynomial arithmetic

Linear algebra underpins many areas of modern mathematics. The basic objects that you will study in this module are vectors, matrices and linear transformations. Topics covered include:

• vector geometry
• matrix algebra
• vector spaces
• linear systems of equations
• eigenvalues and eigenvectors
• inner product spaces.

The mathematical tools that you study in this module are fundamental to many mathematical, statistical, and computational models of the real world.

Probability theory allows us to assess risk when calculating insurance premiums. It can help when making investment decisions. It can be used to estimate the impact that government policy will have on climate change or the spread of disease. 

You will study the theory and practice of discrete and continuous probability, including topics such as:

• Bayes’ theorem
• multivariate random variables
• probability distributions
• the central limit theorem

There is no area of modern mathematics that does not use computational methods to make progress on problems with which the human brain is unable to cope due to the volume of calculations required.

Scientific computation underpins many technological developments in all sectors of the economy. You'll learn how to write code for mathematical applications using Python.

Python is a freely available, widely-used computer language. No previous computing knowledge will be assumed. It will be used throughout your degree programme.

Statistics is concerned with methods for collecting, organising, summarising, presenting and analysing data. It enables us to draw valid conclusions and make reasonable decisions based on statistical analysis. It can be used to answer a diverse range of questions in areas such as the pharmaceuticals industry, economic planning and finance. 

The module covers statistical inference, you'll learn how to analyse, interpret and report data. Topics that you’ll learn about include:

• point estimators and confidence intervals
• hypothesis testing
• linear regression
• goodness-of-fit tests

This module introduces various analytical methods for the solution of ordinary and partial differential equations.

You will begin by studying asymptotic techniques, which can be used when the equations involve a small parameter, which is often the case. We will also study some aspects of dynamical systems theory, which has wide applicability to models of real world problems.

This module will further develop your understanding of the tools of real and complex analysis. This provides you with a solid foundation for subsequent modules in metric and topological spaces, relativity, and numerical analysis.

You’ll study topics such as:

• the Bolzano-Weierstrass Theorem
• norms, sequences and series of functions
• differentiability
• the Riemann integral

You will also learn about functions of complex variables and study topics including, analyticity, Laurent series, contour integrals and residue calculus and its applications.

This module teaches you the mathematical foundations of multidimensional differential and integral calculus of scalar and vector functions. This provides essential background for later study involving mathematical modelling with differential equations, such as fluid dynamics and mathematical physics. You will learn about vector differential operators, the divergence theorem and Stokes’ theorem, as well as meeting various curvilinear coordinate systems.

By studying this module, you’ll explore the abstract mathematical structures of different types of groups, including rings. Topics that you’ll study include:

• permutations
• Abelian groups
• quotient groups
• ring homomorphisms
• polynomial rings

This module teaches you how Newtonian mechanics can be developed into the more powerful formulations due to Lagrange and Hamilton. You will also be introduced to the basic structure of quantum mechanics.

The module provides the foundation for a wide range of more advanced modules in mathematical physics.

This module will provide you with tools to develop and analyse linear and nonlinear mathematical models based on ordinary and partial differential equations. You will also meet the fundamental mathematical concepts required to model the flow of liquids and gases. This will enable you to apply the resulting theory to model physical situations.

Number theory is a branch of pure mathematics that primarily studies the integers, and has applications, for example, in cryptography.

Topics that you will study include:

• Diophantine equations
• congruence equations
• Fermat’s little theorem

This module will develop your understanding of probability theory and random variables from Probability 1. There's particular attention paid to continuous random variables.

Fundamental concepts relating to probability will be discussed in detail, including limit theorems and the multivariate normal distribution. You will then progress onto more advanced topics such as transition matrices, one-dimensional random walks and absorption probabilities.

Most mathematical problems cannot be solved analytically or would take too long to solve by hand. Instead, computational algorithms must be used. 

Scientific Computation teaches you about algorithms for approximating functions, derivatives, and integrals, and for solving many types of equation.

The first part of this module provides you with an introduction to statistical concepts and methods. The second part introduces a wide range of techniques used in a variety of quantitative subjects. The key concepts of inference including estimation and hypothesis testing will be described, as well as practical data analysis and assessment of model adequacy.

This module involves the application of mathematics to a variety of practical, open-ended problems - typical of those that mathematicians encounter in industry and commerce.

Specific projects are tackled through workshops and student-led group activities. The real-life nature of the problems requires you to develop skills in model development and refinement, report writing and teamwork. There are various streams within the module, for example:

• Pure Mathematics
• Applied Mathematics
• Data Analysis
• Mathematical Physics

This ensures that you can work in the area that you find most interesting.

In this module you will apply the quantum mechanics that you learned in Year 2 to more general problems. New topics will be introduced such as the quantum theory of the hydrogen atom and aspects of angular momentum such as spin.

During this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will progress to enhance your understanding of statistical methodology including the analysis of discrete and survival data. You will also be trained in the use of a high-level statistical computer program.

The module introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics. We explore how classical concepts enter quantum mechanics, and how they can be used to find approximate semi-classical solutions.

This module provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. You’ll learn cryptography, including classical mono- and polyalphabetic ciphers.  There will also be a focus on modern public key cryptography and digital signatures, their uses and applications.

This is an excellent opportunity to gain first-hand experience of being involved with providing mathematical education. You will work in a local school alongside practising mathematics teachers in a classroom environment and improve your skills at communicating maths.

Typically, you will work within a class (or classes) for half a day a week for about sixteen weeks. You will be given a range of responsibilities, from classroom assistant to leading a self-originated mathematical activity or project.

The assessment is carried out in a variety of ways: an on-going reflective log, contribution to reflective seminar, oral presentation and a final written report.

This module introduces various analytical methods for the solution of ordinary and partial differential equations.

You will begin by studying asymptotic techniques, which can be used when the equations involve a small parameter, which is often the case. We will also study some aspects of dynamical systems theory, which has wide applicability to models of real world problems.

In this module, a graph consists of vertices and edges, each edge joining two vertices. Graph theory has become increasingly important recently through its connections with computer science and its ability to model many practical situations.

Topics covered include:

• paths and cycles
• the resolution of Euler’s Königsberg Bridge Problem
• Hamiltonian cycles and many others

Elliptic curves are useful tools in many areas of number theory and can be used in cryptography. You'll  study topics including:

• basic notions of projective geometry
• plane algebraic curves including elliptic curves
• addition of points on elliptic curves
• results on the group of rational points on an elliptic curve
• properties of elliptic curves and their applications

The module provides an introduction to electromagnetism and the electrodynamics of charged particles. You will meet Maxwell’s equations and learn how they describe a wide variety of phenomena, including electrostatic fields and electromagnetic waves.

You will extend your understanding of fluid flow by introducing the concept of viscosity and studying the fundamental governing equations for the motion of liquids and gases.

Methods for solution of these equations are introduced, including exact solutions and approximate solutions valid for thin layers. A further aim is to apply the theory to model fluid dynamical problems of physical relevance.

Number theory concerns the solution of polynomial equations in whole numbers, or fractions. We will establish the basic properties of the Riemann zeta-function to find out how evenly the prime numbers are distributed.

This module will also present several methods to solve Diophantine equations including analytical methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems. We will also cover an introduction to more general modular forms.

Game theory is relevant to many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The module starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner’s Dilemma. 

We look at tree-searching, including alpha-beta pruning, the ‘killer’ heuristic and its relatives. It then turns to the mathematical theory of games, exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

This module builds on the basic ideas of group theory. It covers a number of key results such as:

• the simplicity of the alternating groups
• the Sylow theorems (of fundamental importance in abstract group theory)
• the classification of finitely generated Abelian groups (required in algebraic number theory, combinatorial group theory and elsewhere).

Other topics to be covered are group actions, used to prove the Sylow theorems, and series for groups, including the notion of solvable groups.

You will study an introduction to the ideas of functional analysis with an emphasis on Hilbert spaces and operators on them. Many concepts from linear algebra in finite dimensional vector spaces, for example, writing a vector in terms of a basis, eigenvalues of a linear map, diagonalization, have generalisations in the setting of infinite dimensional spaces.

This makes this theory a powerful tool with many applications in pure and applied mathematics.

You will explore the concepts of discrete time Markov chains to understand how they used. We will also provide an introduction to probabilistic and stochastic modelling of investment strategies, and for the pricing of financial derivatives in risky markets.

You will gain well-rounded knowledge of contemporary issues which are of importance in research and workplace applications.

Mathematics can be usefully applied to a wide range of applications in medicine and biology.

Without assuming any prior biological knowledge, this module describes how mathematics helps us understand topics such as:

• population dynamics
• biological oscillations
• pattern formation
• nonlinear growth phenomena

There is considerable emphasis on model building and development.

A metric space generalises the concept of distance familiar from Euclidean space. It provides a notion of continuity for functions between quite general spaces. The module covers:

• metric spaces
• topological spaces
• compactness
• separation properties like Hausdorffness and normality
• Urysohn’s lemma
• quotient and product topologies, and connectedness

Finally, Borel sets and measurable spaces are introduced.

This module is concerned with the analysis of multivariate data, in which the response is a vector of random variables rather than a single random variable. A theme running through the module is that of dimension reduction.

Key topics to be covered include:

• principal components analysis
• modelling and inference for multivariate data
• classification of observation vectors into sub-populations using a training sample
• canonical correlation analysis
• factor analysis
• methods of clustering
• multidimensional scaling

In this module a variety of techniques of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming.

These techniques have a wide range of applications to real world problems, in which a process or system needs to be made to perform optimally.

You will be introduced to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain exciting new phenomena that occur in relativity.

Commutative rings and modules over them are the fundamental objects of what is often referred to as commutative algebra. Key examples of commutative rings are polynomials in one variable over a field and number rings such as the usual integers or the Gaussian integers. There are many close parallels between these two types of rings, for example the similarities between the prime factorization of integers and the factorization of polynomials into irreducibles.

In this module, these ideas are extended and generalized to cover polynomials in several variables and power series, and algebraic numbers.

You'll learn how to use numerical techniques for determining the approximate solution of ordinary and partial differential equations where a solution cannot be found through analytical methods alone. You will also cover topics in numerical linear algebra, discovering how to solve very large systems of equations and find their eigenvalues and eigenvectors using a computer.

This module is concerned with the two main theories of statistical inference, namely classical (frequentist) inference and Bayesian inference.

You will explore the following topics in detail:

• sufficiency
• estimating equations
• likelihood ratio tests
• best-unbiased estimators

There is special emphasis on the exponential family of distributions, which includes many standard distributions such as the normal, Poisson, binomial and gamma.

This module will develop your knowledge of discrete-time Markov chains by applying them to a range of stochastic models. You will be introduced to Poisson and birth-and-death processes. You will then move onto more extensive studies of epidemic models and queuing models, with introductions to component and system reliability.

This module will consist of individual, self-directed, but supervised, study of an appropriate area of mathematics during both semesters.

A list of possible topics will be supplied by the school and you will choose a topic of interest to you. You will be asked to produce a substantial piece of work, assessed by an interim report, an oral presentation and a dissertation.

You will develop your knowledge and skills relevant to the mathematical modelling of investment and finance. Also, research experience will be broadened by undertaking some independent reading, computer simulations, group work and summarising the material in a project report.

The development of techniques for the study of nonlinear differential equations is a major research activity to which members of the department here at Nottingham, have made important contributions.

The module covers a number of state-of-the-art methods, for example:

• Green’s function methods for the solution of linear partial differential equations
• characteristic methods
• classification and regularization of nonlinear partial differentiation equations
• bifurcation theory

These methods will be illustrated by applications in the biological and physical sciences.

This module presents the fundamental features of algebraic number theory, the theory in which numbers are viewed from an algebraic point of view. Numbers are often treated as elements of rings, fields and modules, and properties of numbers are reformulated in terms of the relevant algebraic structures.

This approach leads to understanding of certain arithmetical properties of numbers (in particular, integers) from a new point of view.

You’ll systematically study black holes and their properties, including astrophysical processes, horizons and singularities. You’ll have an introduction to black hole radiation to give you an insight into problems of research interest. You’ll gain knowledge to help you begin research into general relativity.

This module is largely concerned with infinite groups, especially free groups, although their role in describing and understanding finite groups is emphasized. Following the basic definitions of free groups and group presentations, the fundamental Nielson-Schreier Theorem is covered in some detail.

You will also learn methods for manipulating group presentations, and using them to read off properties of a given group.

This module introduces computational methods for solving problems in applied mathematics. You will develop knowledge and understanding to design, justify and implement relevant computational techniques and methodologies.

This module explores how computers allow the easy implementation of standard, but computationally intensive, statistical methods. You will explore their use in the solution of non-standard analytically intractable problems by innovative numerical methods. The material builds on the theory of the module to cover several topics that form the basis of some current research areas in computational statistics.

Particular topics to be covered include a selection from simulation methods such as:

• Markov chain
• Monte Carlo methods
• the bootstrap
• nonparametric statistics
• statistical image analysis
• wavelets

By studying this module, you’ll be equipped with the tools and knowledge to extend your understanding of general relativity. You’ll explore more abstract and powerful concepts using examples of curved space-times such as Lie groups and manifolds among others.

The first part of the module introduces the no-arbitrage pricing principle and financial instruments such as forward and futures contracts, bonds and swaps, and options.

The second part of the module considers the pricing and hedging of options and discrete-time, discrete-space stochastic processes.

The final part of the module focuses on the Black-Scholes formula for pricing European options and also introduces the Wiener process, Ito integrals and stochastic differential equations.

This module gives a mathematical introduction to quantum information theory. The aim is to provide you with a background in quantum information science. This will help with your further independent learning and allow you to understand the scope and nature of current research topics.

In this year-long module you’ll be introduced to the study of the quantum dynamics of relativistic particles. You’ll learn about the quantum description of electrons, photons and other elementary particles, leading to an understanding of the standard model of particle physics.

The purpose of this module is to introduce you to the concepts of scientific programming using the object-oriented language C++ It is used for applications arising in the mathematical modelling of physical processes. You will develop knowledge and understanding of a variety or relevant numerical techniques and learn how to efficiently implement them in C++.

Machine Learning is a topic at the interface between statistics and computer science that concerns models that can adapt to and make predictions based on data. This module builds on principles of statistical inference and linear regression. It introduces a variety of methods of regression and classification, trade-off, and on methods to measure and compensate for overfitting.

You will benefit with hands-on learning using computational methods to tackle challenging real world machine learning problems.

This module illustrates the applications of advanced techniques of mathematical modelling using ordinary and partial differential equations. A variety of medical and biological topics are covered bringing you closer to active fields of mathematical research.

This module will provide you with a general introduction to the analysis of data that arise sequentially in time. You will discuss several commonly-occurring models, including methods for model identification for real-time series data. You will develop techniques for estimating the parameters of a model, assessing its fit and forecasting future values.

You will learn to apply ideas from probability and statistics and various mathematical tools in traditional areas of applied and computational mathematics. There will be a focus on ordinary differential equation (ODE) and partial differential equation (PDE) models. The module will provide an introduction to a variety of techniques which are useful in Uncertainty Quantification and will concentrate on a more in-depth study of selected application areas.