Many optimisation problems in business and industry can be expressed in the form of a linear programming problem and this is even more apparent with increasing reliance on Artificial Intelligence and Machine Learning. Businesses and industry use linear programming to determine what to make in order to maximise their profits, Amazon use it to schedule your parcels for delivery and it is also used widely in Game Theory: you can use it to beat your friends at rock-paper-scissors and other games!

In this module, we will study the theoretical background behind the linear programming methods, learn how to express real-world questions as linear programming problems and solve them by hand and using computer programs. We will also explore some other optimisation methods used in AI and Machine Learning. Topics may include:

• Canonical forms of linear programming problems.
• Theoretical considerations: relevant results from set theory and geometry.
• Integer linear programming.
• Solutions of linear programming models: Simplex and dual simplex methods, Pivot algorithm, and computer-based techniques.
• Degeneracy, cycling, duality.
• Use of a mathematical computer software package, for example, Python, Matlab, Excel etc.
• Application to logistics in transportation and assignment problems, VAM and the Hungarian algorithm.
• Game theory: zero-sum matrix games, multi-phase games.
• Non-linear optimisation techniques in Machine Learning

Stochastic processes serve as essential mathematical models for systems and phenomena exhibiting apparent randomness. Examples encompass diverse scenarios, such as the growth of a bacterial population, fluctuations in electrical current due to thermal noise, or the motion of gas molecules. The applications of stochastic processes span various disciplines, including biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, and financial markets. In order to understand such random behaviour, we introduce and study Markov chains, random walks, Brownian motion and stochastic differential equations. Through these topics, students will not only establish a robust foundation in the principles of stochastic processes but will also gain valuable insights into their diverse applications across numerous domains. The module's goal is to equip students with analytical tools essential for comprehending and modelling complex uncertainties, thereby enhancing their capacity to address real-world challenges in mathematics, statistics, and related fields.

Topics covered include:

• Brief review of Probability Theory via a Measure Theory approach.
• Martingales: Basic definitions, filtrations, stopping times.
• Doob's Martingale inequalities and Convergence Theorem.
• Markov Chains.
• Moment generating functions.
• Characteristic functions.
• Probability generating functions.

Although geometric intuition allows us to understand some simple calculus (differentiation and integration), when things become more complicated we need a more rigorous foundation. Mathematical analysis provides this framework and indispensable to understanding applied techniques such as numerical methods and solving differential equations. By treating the notions of distance and limit formally, this course develops a rigourous theory of convergence, differentiation and integration of real-valued functions.

Geometry is probably the most accessible branch of mathematics, and can provide an easy route to understanding some of the more complex ideas that mathematics can present. This module intends to introduce students to the major geometrical topics taught at undergraduate level, in a manner that is both accessible and rigorous. The course will consider Euclidean geometry and solid geometry, covering topics such as the equation of lines and planes, inner product, cross product, and triple product. Spherical geometry, featuring concepts including great circles, geodesics, and spherical trigonometry, will be explored for its significant applications in fields like astronomy, navigation and understanding the Earth. Additionally, the study will extend to quadratic curves, quadratic surfaces, and planetary motion.  Through this comprehensive study, students will understand the various geometric concepts, laying a foundation that combines accessibility with the necessary mathematical rigour.

Topics in Geometry are drawn from the following:

• Geometry of numbers (including the golden ratio and continued fractions).
• Coordinate geometry: lines, planes and hyperplanes.
• Euclidian geometry.
• Solid geometry: vector products, polyhedra.
• Projective geometry: basic concepts & applications.
• Spherical geometry.
• Use of appropriate software to visualise problems (such as Autograph or Matlab, for example).
• Real-life applications of geometry: examples drawn from a wide range of contexts (e.g. cartography, forensic science, navigation, architecture).

Students have the option to choose to take part in either a:

• Professional work placement
• Language module
• Term abroad