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This course is designed to develop key mathematical and analytical chemistry skills.
The course will introduce students to applications of key mathematical concepts to chemical problems.
This will be taught alongside an introduction to analytical chemistry that can provide qualitative or quantitative information about the structure and chemical composition of a sample.

Biology is undergoing a quantitative revolution, generating vast quantities of data that are analysed using bioinformatics techniques and modelled using mathematics to give insight into the underlying biological processes. This module aims to give a flavour of how mathematical modelling can be used in different areas of biology.

Typically the models that are used in biology cannot be solved analytically. Nonetheless they give very useful information about the behaviour of the system. We will start by studying what we can say about differential equations that we cannot solve. For example, we cannot solve the equation of a simple pendulum analytically, but we can still say under what conditions it has periodic solutions. For biological oscillators this is usually what matters: it is important that your heart beats regularly, but whether your pulse rate is 68 or 71 beats per minute is less critical.

Having introduced the mathematical tools needed to study ordinary differential equations, we will apply them to simple models of population dynamics, epidemics and biochemical reaction networks.

One of the pre-requisites for MATH6149

Biology is undergoing a quantitative revolution, generating vast quantities of data that are analysed using bioinformatics techniques and modelled using mathematics to give insight into the underlying biological processes. This module aims to give a flavour of how mathematical modelling can be used in different areas of biology.

Typically the models that are used in biology cannot be solved analytically. Nonetheless they give very useful information about the behaviour of the system. We will start by studying what we can say about differential equations that we cannot solve. For example, we cannot solve the equation of a simple pendulum analytically, but we can still say under what conditions it has periodic solutions. For biological oscillators this is usually what matters: it is important that your heart beats regularly, but whether your pulse rate is 68 or 71 beats per minute is less critical.

Having introduced the mathematical tools needed to study ordinary differential equations, we will apply them to simple models of population dynamics, epidemics and biochemical reaction networks.

One of the pre-requisites for MATH6149

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk management of a portfolio of derivative securities is described.

Some practical aspect of modelling like risk management is studied via programing and simulation techniques.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk management of a portfolio of derivative securities is described.

Some practical aspect of modelling like risk management is studied via programing and simulation techniques.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk
management of a portfolio of derivative securities is described.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk
management of a portfolio of derivative securities is described.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk
management of a portfolio of derivative securities is described.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk
management of a portfolio of derivative securities is described.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

To provide students with the necessary skills and confidence to apply a range of mathematical methods to problems in the physical sciences. Both MATH1006 and MATH1008 cover essentially the same topics in calculus that are of relevance to applications in the physical sciences but MATH1006 is aimed at physics students. Students taking degrees related to other physical sciences such as chemistry, geology, and oceanography should take MATH1008. The module begins by looking at vectors in 2 and 3 dimensions, introducing the dot and cross products, and discussing some simple applications. This is followed by a section on matrices, determinants, and eigenvalue problems. The course then reviews polynomial equations and introduces complex numbers. After this, some basic abstract concepts related to functions and their inverses are discussed. The main part of the unit covers the basics of calculus, starting with limits, and going on to look at derivatives and Taylor series. The concept of integration is then defined, followed by an exploration (by means of examples) of various methods of integration.

One of the pre-requisites for MATH1007, MATH1049, MATH2015, MATH2038 and MATH2045