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Number Theory is the study of integers and their generalisations such as the rationals, algebraic integers or finite fields. The problem more or less defining Number Theory is to find integer solutions to equations, such as the famous Fermat equation x^n + y^n = z^n.

In this module we build on the group, ring and number theoretic foundations laid in MATH1001, MATH2003 and MATH3086.

We will first prove a structure theorem for the group of units modulo n. We then move on to the famous Gaussian Quadratic Reciprocity Law which yields an algorithm to decide solvability of quadratic equations over finite fields. Using geometric as well as algebraic methods, we will then characterise which integers can be written as the sum of two and four squares, respectively. The former leads us naturally to the study of binary quadratic forms, a central topic of this module.

In the final part of this module, we will explore rings of integers in algebraic number fields; they generalise the role the integers play within the rational numbers; the simplest new example is the ring of Gaussian integers, Z[i]. We will investigate to what extent certain central properties of the integers, such as unique prime power factorisation, generalises to these rings. The deviation from unique prime factorisation is measured by the so-called ideal class group, probably the most important invariant of algebraic number fields. It can be seen that it is finite and that its order for quadratic number fields is intimately related to the number of equivalence classes of quadratic forms introduced earlier in the module.

One of the primary domains where number theory finds applications is cryptography. We will study some of the famous cryptosystems where number theory has applications. In particular, Rabin cryptosystem, Goldwasser-Micali cryptosystem, lattice based cryptosystems, elliptic curve cryptography are among those.

This module provides a broad introduction to more advanced regression methods such as multilevel models, non-parametric and penalised regression and Generalized Additive models.
The module assumes that students are familiar with basic regression techniques such as Linear Regression and Logistic regression.

This module aims to expand your statistical toolbox by exposing you to a broad set of modelling techniques to employ with data that would not satisfy the assumptions of the mainstream Linear and Generalized Linear models. The first half of the module will follow an explanatory approach, introducing Multilevel (Mixed effects) and Marginal models to understand and deal with the type of correlation found in hierarchical and longitudinal data. The second half of the module will introduce a set of modelling techniques widely used in the predictive approach, such as non-parametric regression, Generalized Additive Models (GAM), Penalized Regression or Classification and Regression Trees (CART).

This module aims to expand your statistical toolbox by exposing you to a broad set of modelling techniques to employ with data that would not satisfy the assumptions of the mainstream Linear and Generalized Linear models. The first half of the module will follow an explanatory approach, introducing Multilevel (Mixed effects) and Marginal models to understand and deal with the type of correlation found in hierarchical and longitudinal data. The second half of the module will introduce a set of modelling techniques widely used in the predictive approach, such as non-parametric regression, Generalized Additive Models (GAM), Penalized Regression or Classification and Regression Trees (CART).

This course aims to introduce some advanced techniques that hold potential for applications in the future generations of wireless communication systems. Currently, research and development in wireless communications is focused on the sixth generation (6G), which is expected to significantly enhance 5G in both techniques and services. This course will cover several candidate techniques designed to enable 6G wireless systems.

The course begins by covering the principles of cooperative communications. Various relay/cooperation protocols are considered and analysed to demonstrate their advantages and challenges.

Next, it focuses on non-orthogonal multiple access (NOMA), a technique that allows densely deployed users (or devices) to simultaneously transmit their information.

Subsequently, the course addresses the principles of full-duplex communication, exploring the challenges of self-interference and corresponding self-interference cancellation techniques, as well as examining the potential of full-duplex for wireless system design.

Then, it introduces integrated sensing and communication (ISAC), providing several examples to explain the principles and illustrate the design trade-offs.

A review of the fundamentals of MIMO is then provided, followed by analysing the potential of MIMO for meeting the requirements of future wireless systems. A range of technical options for MIMO transceiver optimisation are discussed. Built on the above theoretical foundation, the course then covers the multi-user MIMO and massive MIMO, with the emphasis on their principles, characteristics, and implementation challenges.

Finally, the course covers millimeter wave (mmWave) communications. It begins with an overview of mmWave technology, then characterizes mmWave channels, highlighting key differences from conventional radio frequency (RF) communication channels. The course concludes with an introduction to several advanced techniques for the design and optimization of mmWave systems.

This course aims to introduce some advanced techniques that hold potential for applications in the future generations of wireless communication systems. Currently, research and development in wireless communications is focused on the sixth generation (6G), which is expected to significantly enhance 5G in both techniques and services. This course will cover several candidate techniques designed to enable 6G wireless systems.

The course begins by covering the principles of cooperative communications. Various relay/cooperation protocols are considered and analysed to demonstrate their advantages and challenges.

Next, it focuses on non-orthogonal multiple access (NOMA), a technique that allows densely deployed users (or devices) to simultaneously transmit their information.

Subsequently, the course addresses the principles of full-duplex communication, exploring the challenges of self-interference and corresponding self-interference cancellation techniques, as well as examining the potential of full-duplex for wireless system design.

Then, it introduces integrated sensing and communication (ISAC), providing several examples to explain the principles and illustrate the design trade-offs.

A review of the fundamentals of MIMO is then provided, followed by analysing the potential of MIMO for meeting the requirements of future wireless systems. A range of technical options for MIMO transceiver optimisation are discussed. Built on the above theoretical foundation, the course then covers the multi-user MIMO and massive MIMO, with the emphasis on their principles, characteristics, and implementation challenges.

Finally, the course covers millimeter wave (mmWave) communications. It begins with an overview of mmWave technology, then characterizes mmWave channels, highlighting key differences from conventional radio frequency (RF) communication channels. The course concludes with an introduction to several advanced techniques for the design and optimization of mmWave systems.

This course aims to introduce some advanced techniques that hold potential for applications in the future generations of wireless communication systems. Currently, research and development in wireless communications is focused on the sixth generation (6G), which is expected to significantly enhance 5G in both techniques and services. This course will cover several candidate techniques designed to enable 6G wireless systems.

The course begins by covering the principles of cooperative communications. Various relay/cooperation protocols are considered and analysed to demonstrate their advantages and challenges.

Next, it focuses on non-orthogonal multiple access (NOMA), a technique that allows densely deployed users (or devices) to simultaneously transmit their information.

Subsequently, the course addresses the principles of full-duplex communication, exploring the challenges of self-interference and corresponding self-interference cancellation techniques, as well as examining the potential of full-duplex for wireless system design.

Then, it introduces integrated sensing and communication (ISAC), providing several examples to explain the principles and illustrate the design trade-offs.

A review of the fundamentals of MIMO is then provided, followed by analysing the potential of MIMO for meeting the requirements of future wireless systems. A range of technical options for MIMO transceiver optimisation are discussed. Built on the above theoretical foundation, the course then covers the multi-user MIMO and massive MIMO, with the emphasis on their principles, characteristics, and implementation challenges.

Finally, the course covers millimeter wave (mmWave) communications. It begins with an overview of mmWave technology, then characterizes mmWave channels, highlighting key differences from conventional radio frequency (RF) communication channels. The course concludes with an introduction to several advanced techniques for the design and optimization of mmWave systems.

This course aims to introduce some advanced techniques that hold potential for applications in the future generations of wireless communication systems. Currently, research and development in wireless communications is focused on the sixth generation (6G), which is expected to significantly enhance 5G in both techniques and services. This course will cover several candidate techniques designed to enable 6G wireless systems.

The course begins by covering the principles of cooperative communications. Various relay/cooperation protocols are considered and analysed to demonstrate their advantages and challenges.

Next, it focuses on non-orthogonal multiple access (NOMA), a technique that allows densely deployed users (or devices) to simultaneously transmit their information.

Subsequently, the course addresses the principles of full-duplex communication, exploring the challenges of self-interference and corresponding self-interference cancellation techniques, as well as examining the potential of full-duplex for wireless system design.

Then, it introduces integrated sensing and communication (ISAC), providing several examples to explain the principles and illustrate the design trade-offs.

A review of the fundamentals of MIMO is then provided, followed by analysing the potential of MIMO for meeting the requirements of future wireless systems. A range of technical options for MIMO transceiver optimisation are discussed. Built on the above theoretical foundation, the course then covers the multi-user MIMO and massive MIMO, with the emphasis on their principles, characteristics, and implementation challenges.

Finally, the course covers millimeter wave (mmWave) communications. It begins with an overview of mmWave technology, then characterizes mmWave channels, highlighting key differences from conventional radio frequency (RF) communication channels. The course concludes with an introduction to several advanced techniques for the design and optimization of mmWave systems.

In the last 30 years derivatives have become increasingly important in finance and many different types of derivatives are actively traded on exchanges throughout the world. This module explores the pricing and use of forwards, futures and options with a particular focus on contracts where the underlying asset is a financial asset - for example, a stock index (i.e. stock index futures or stock index options). Students will learn how to price these derivatives using various techniques as well as understand how we can use them for (i) speculation, (ii) hedging strategies and (iii) arbitrage. The nature of the subject makes the module more suitable for students with a solid background in mathematics and familiarity with differential calculus and systems of equations.

In the last 30 years derivatives have become increasingly important in finance and many different types of derivatives are actively traded on exchanges throughout the world. This module explores the pricing and use of forwards, futures and options with a particular focus on contracts where the underlying asset is a financial asset - for example, a stock index (i.e. stock index futures or stock index options). Students will learn how to price these derivatives using various techniques as well as understand how we can use them for (i) speculation, (ii) hedging strategies and (iii) arbitrage. The nature of the subject makes the module more suitable for students with a solid background in mathematics and familiarity with differential calculus and systems of equations.