Mathematics

This course deals with the application of statistical methods to test hypotheses and draw inferences from data, using maximum likelihood methods. The course starts by developing general-purpose maximum likelihood methods, with interval estimation by means of the information matrix and the bootstrap. It goes on to develop generalized linear models, linear models and analysis of variance models as special cases of maximum likelihood methods. It covers diagnostic methods, including methods for selecting between models, checking assumptions and testing goodness-of-fit. It has an applied focus, with extensive use of R to give students practice in doing inference with real datasets, from problem formulation through to final conclusions.

This is a course in the rigorous treatment of Fourier series and related topics, including Fourier series, Fourier coefficients, trigonometric polynomials and orthogonality; properties of Fourier coefficients; Bessel's inequality, Parseval's identity and the Riemann-Lebesgue lemma; various notions of convergence of Fourier series, including pointwise, uniform and mean square convergence. Summability methods, convolution and Young's inequality; Fourier Analysis in broader contexts; for example, Fourier integrals, Fourier expansions in groups, Schwartz spaces and tempered distributions.

This course introduces students to essential notions in algebraic topology, such as compact surfaces, homotopies, fundamental groups, and covering spaces.

Monte Carlo simulations are a powerful computational technique for probabilistic and deterministic problems with applications to various fields including computer science, finance, economics, engineering, mathematics, and physics. The initial part of the course is about computer-simulated randomness and begins with pseudo-random generators and simulating one-dimensional random variables. From the moment that we can simulate one random variable, we can simulate a whole discrete process, such as Markov chains and use the simulations to extract statistical results of their equilibria. The course also explores applications in Physics via the Ising model and in Statistics via the goodness of fit tests. The course is a mixture of coding with probability theory, and students use the R software for the simulations.

This Vertically Integrated Project (VIP) develops sophisticated integrated software systems which would allow mathematicians to combine the latest algorithms to solve their problems without needing to understand the details of their implementation. The project includes the following pieces of work: graphs and digraphs (for example, is a graph planar, connected, biconnected, what is its chromatic polynomial), groups and semigroups (for instance, what is the size of a semigroup or group generated by a set of elements, how to compute a presentation of one of these objects), other mathematical algorithms relating to other modules undertaken at St Andrews, and how to represent problems to a computer so they are easy to use and the implementations are as efficient as possible.

This is a basic course in designing experiments and analyzing the resulting data. It is intended for engineers, physical/chemical scientists, and scientists from other fields such as biotechnology and biology. The course deals with the types of experiments that are frequently conducted in industrial settings. Its objective is to learn how to plan, design, and conduct experiments efficiently and effectively, and analyze the resulting data to obtain objective conclusions. Both design and statistical analysis issues are discussed. Opportunities to use the principles taught in the course arise in all phases of engineering and scientific work, including technology development, new product design and development, process development, and manufacturing process improvement. Applications from various fields of engineering (including chemical, mechanical, electrical, materials science, industrial, etc.) will be illustrated throughout the course. Topics include simple design with fixed and random effects. Simultaneous confidence intervals. Requirements for analysis of variance: transformations, model validation, residual analysis. Factorial design with fixed, random, and mixed effects. Additivity and interaction. Complete and incomplete designs. Randomized block designs, Latin squares and confounding. Regression and analysis of covariance. Admission requirements include FMAA20 Linear Algebra with Introduction to Computer Tools or FMAA21 Linear Algebra with Numerical Applications or FMAB20 Linear Algebra or FMAB22 Linear Algebra and FMAB30 Calculus in Several Variables or FMAB35 Calculus in Several Variables or FMSF20 Mathematical Statistics, Basic Course or FMSF25 Mathematical Statistics - Complementary Project or FMSF32 Mathematical Statistics or FMSF45 Mathematical Statistics, Basic Course or FMSF50 Mathematical Statistics, Basic Course or FMSF55 Mathematical Statistics, Basic Course or FMSF70 Mathematical Statistics or FMSF75 Mathematical Statistics, Basic Course or FMSF80 Mathematical Statistics, Basic Course. Assumed prior knowledge: Basic mathematical statistics and programming experience.

This course introduces the fundamental theory of linear algebra and its applications in the field of agronomy. It is designed for students with limited mathematical background who wish to apply linear algebra to practical problems. The course covers essential concepts such as matrices and systems of linear equations, vector spaces, and eigenvalues and eigenvectors. Through the use of applied examples and case studies, students learn how to use linear algebraic methods to analyze and solve real-world problems.

This course develops foundational computing skills in Python and sharpens these skills through practice with exploration and problem-solving within the contexts of Applied, Pure, and Statistical Mathematics.

This is an interdisciplinary, project‑based course designed to introduce the principles, methods, and communication practices of modern scientific research. Through a flipped‑classroom approach, the course actively explores how different disciplines—such as biology, informatics, mathematics, physics, chemistry, and computer science—intersect to address complex scientific questions. Throughout the course, students work in small subgroups to build and communicate a scientific project. They learn how to identify and evaluate scientific literature, analyze research methodologies across fields, and critically assess the validity, reproducibility, and interpretation of results. Students develop strong skills in teamwork, scientific reasoning, and oral communication as they prepare an interdisciplinary presentation aimed at both specialists and non‑specialists. A major component of the course is the construction of a final oral presentation based on recent scientific publications. Students progressively refine their project through guided tutorials led by instructors from multiple disciplines. They also practice writing concise research abstracts, critically reading scientific articles, and using research tools such as PubMed and AI‑assisted platforms—while assessing their benefits and limitations. By the end of the course, students gain practical experience in the entire scientific communication pipeline: exploring a topic, building a multidisciplinary understanding of its methods, and presenting their findings clearly and rigorously to a diverse scientific audience.