Mathematics with a Year in Industry - U7UMATHIY

In 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 and polynomial rings. 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 and Fermat’s little theorem.

In this module you will learn how Newtonian mechanics can be developed into the more powerful formulations due to Lagrange and Hamilton and 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 is an introduction to methods of solving some of the most important ordinary and partial differential equations that occur in applied mathematics and mathematical physics. You will learn about finding series solutions to ordinary differential equations, Fourier series representation of functions, and Fourier and Laplace transforms, using them to solve ordinary and partial differential equations.

The purpose of this module is to provide a thorough grounding in a broad range of techniques required in the analysis of probabilistic models, and to introduce stochastic processes by studying techniques and concepts common in the analysis of discrete time Markov Chains.

In this module you will develop your understanding of probability theory and random variables, with 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 also meet some new statistical concepts and methods. The key concepts of inference including estimation and hypothesis testing will be described as well as practical data analysis and assessment of model adequacy.

Most mathematical problems cannot be solved analytically or would take too long to solve by hand. Instead, computational algorithms must be used. In this module, you’ll learn about algorithms for approximating functions, derivatives, and integrals, and for solving many types of algebraic and ordinary differential equations.

In this module, you will be introduced to a wide range of statistical concepts and methods fundamental to applications of statistics, and meet the key concepts and theory of linear models, illustrating their application via practical examples drawn from real-life situations.

This module provides a grounding in the techniques of vector calculus and illustrates their use by developing the theory of electromagnetism and Maxwell’s equations. You will be introduced to the vector differentiation operations of gradient, divergence and curl, integration methods for scalar and vector quantities over paths, surfaces and volumes, and the relationship of these operations to each other via the integral theorems of Green, Stokes and Gauss. These concepts will be illustrated through examples drawn from the theory of electromagnetism

This course builds on the foundations of quantum mechanics introduced in the module MATH2013. It further develops the fundamental theory so that it applies to more general problems, such as those involving spin, and introduces key calculational approaches, such as those underlying angular momentum, the hydrogen atom, scattering problems and approximation methods such as perturbation theory.

20 credits in the Autumn Semester

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.

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 module provides an opportunity for mathematics students to gain first-hand experience of being involved with providing mathematical education. Students will work at local schools alongside practising mathematics teachers in a classroom environment and will improve their skills at communicating mathematics. Typically, each student will work with a class (or classes) for half a day a week for about fifteen weeks. Students will be given a range of responsibilities from classroom assistant to leading mathematical activities. As well as half a day a week spent in local schools, students will typically attend 4 timetabled seminars during the year, as outlined in the “Method and frequency of class” section below.

The assessment is carried out by a variety of means: writing a collection of reflective logs (on-going throughout the year); contributing to a reflective group seminar; giving an individual oral presentation; and writing a research project report.

Please note that applications for the module are made during the Spring semester for the following academic year, and information about the application process will be sent to eligible students. Students who select this module during module enrolment without confirmation from the module convenor of their successful application will automatically have their choice rejected.

The aim of Discrete Mathematics is the study of discrete and finite rather than continuous quantities. This includes counting problems, graphs and other quantities parametrised by integers.

As such Discrete Mathematics is of great importance for various branches of Pure Mathematics, Mathematical Physics, Statistics and Computer Sciences.

The course will cover a range of Discrete Mathematics topics, including:

• an introduction to advanced aspects of combinations and permutations
• ordinary and exponential generating functions
• recurrence relations and applications to counting problems
• graphs
• digraphs
• Eulerian and Hamiltonian trails
• planar graphs
• graph colouring
• trees
• minimum spanning tree algorithms

This module will provide the basic knowledge of electricity and magnetism required for entry into the first year of degree courses in the Faculties of Science and Engineering. The aim of the module is to develop the student awareness of and ability to solve problems in the field of electricity and magnetism including dc and ac circuits. It will prepare students for future studies involving the study of electricity and magnetism.

Game theory contains 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 mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

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.

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. Key topics to be covered include:

• Vector and matrix algebra ideas relevant to multivariate statistics, including the eigen and singular value decompositions
• Methods for dimension reduction including principal components analysis, canonical correlation analysis, and factor analysis
• Multivariate regression methods.
• Classification methods, such as linear discriminant analysis.
• Multivariate normal distributions and their use in multivariate analysis of variance tests
• Exploratory data analysis methods such as clustering tools, and multi-dimensional scaling, as well as other data visualisation techniques.
• Statistical software for conducting analysis of multivariate data

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.

Scientific Computation and Numerical Analysis covers advanced numerical methods, and its analysis, for the approximate solution of important mathematical problems, such as solving partial differential equations (PDEs), finding solutions to overdetermined systems, rapidly solving large linear systems of equations, and finding matrix approximations. Methods discussed have important applications in modern-day society, including PDEs in engineering and applied sciences, Google's PageRank algorithm, model reduction and machine learning in data science. Specific techniques include:

• The finite element method (FEM) for PDEs, including weak formulations and its Galerkin discretization, and a priori error analysis
• Advanced methods for linear systems, including QR factorization, perturbation and conditioning theory, and the innovative Conjugate Gradient (CG) method
• Methods for low-rank approximation, including computation of eigenvalues, the singular value decomposition (SVD), and the fundamental Eckart-Young theory for best rank-k matrix approximation.'

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.

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.

General relativity predicts the existence of black holes which are regions of space-time into which objects can be sent but from which no classical objects can escape. This course uses techniques learnt in MATH4015 to systematically study black holes and their properties, including horizons and singularities. Astrophysical processes involving black holes are discussed, and there is a brief introduction to black hole radiation discovered by Hawking.

This course aims to introduce the physics of black holes and its mathematical description, giving insight into problems of research interest. It provides an opportunity to apply techniques and ideas learned in previous modules to important astrophysical problems. Students will acquire knowledge and skills to a level sufficient to begin research in general relativity.

20 credits in the Spring Semester

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 provides an opportunity for mathematics students to gain first-hand experience of being involved with providing mathematical education. Students will work at local schools alongside practising mathematics teachers in a classroom environment and will improve their skills at communicating mathematics. Typically, each student will work with a class (or classes) for half a day a week for about fifteen weeks. Students will be given a range of responsibilities from classroom assistant to leading mathematical activities. As well as half a day a week spent in local schools, students will typically attend 4 timetabled seminars during the year, as outlined in the “Method and frequency of class” section below.

The assessment is carried out by a variety of means: writing a collection of reflective logs (on-going throughout the year); contributing to a reflective group seminar; giving an individual oral presentation; and writing a research project report.

Please note that applications for the module are made during the Spring semester for the following academic year, and information about the application process will be sent to eligible students. Students who select this module during module enrolment without confirmation from the module convenor of their successful application will automatically have their choice rejected.

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

20 credits in the Spring Semester

The course introduces notions of topology and differential geometry which are required for modern research in relativity and other topics involving geometry. The course will be illustrated with a body of concrete geometrical examples drawn from general relativity. The modern study of general relativity requires familiarity with a number of tools of differential geometry, including manifolds, symmetries, Lie Groups, differentiation and integration on manifolds. These are introduced using examples of curved space-times whose context is familiar from the study of general relativity, the presentation of geometric concepts will be significantly more abstract and powerful than in Relativity MATH3018

20 credits in the Autumn Semester

The first part of the module introduces 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.

20 credits in the Autumn Semester

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 course is to introduce concepts of scientific programming using the object oriented language C++ for applications arising in the mathematical modelling of physical processes. Students taking this module will develop knowledge and understanding of a variety or relevant numerical techniques and how to efficiently implement them in C++.

20 credits in the Autumn Semester

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 will provide 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 gain experience of using a statistical package and interpreting its output.

20 credits in the Spring Semester

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 treated bringing students close to active fields of mathematical research.

20 credits in the Spring Semester