Poetically, “Group theory is the branch of mathematics that answers the question “What is symmetry?” (N. C. Carter).  Various physical systems, such as crystals and the hydrogen atom, can be modeled by symmetry groups. Group theory and the closely related representation theory have many applications in physics and chemistry. We first start by consider some main classes of groups such as permutation groups, matrix groups, transformation groups, topological and algebraic groups. We will study representation of groups and symmetry and the last part of the course is devoted to some applications of group theory such as Lie groups and the study of differential equations, and the Heisenberg, Lorentz, and Poincare groups, which play a fundamental role in modern theoretical physics.

This course aims to present a historical development of the subject area, leading to a significant partial proof of Fermat’s Last Theorem.

Geometry is a branch of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. This course aims to introduce students to the study of the above questions in curved manifolds. To this purpose, basic concepts of differential geometry with a particular emphasis to Riemannian and pseudo-Riemannian manifolds will be introduced and applied to geometric objects such as black holes and the Robertson-Walker universe, which find their roots in the theory of general relativity, i.e. the geometric theory of gravitation.

The Finite Element Method (FEM) is a sophisticated numerical scheme for interpolating the approximate solution to common boundary-value problems of engineering and mathematical physics. It is widely used, as it is a computationally efficient method for modelling real-world problems which typically have unusual geometries and variable material properties. The purpose of this course is to provide graduate students of engineering or applied science with a concise introduction to FEM. Although the theory of finite elements is based on sophisticated elements of mathematical analysis, the main purpose of this course is to provide a concise introduction to the subject.

This course considers the limitations of the Riemann integral, and shows that it it necessary to develop a precise mathematical notion of ‘length’ and ‘area’ in order to overcome them. Thus we develop the concept of measure, and use it to construct the more powerful Lebesgue integral, and explore its properties. Finally we look at applications of measure and Lebesgue integration in modern probability theory.

This course examines metric and topological spaces, continuity, completeness, and compactness, providing a theoretical foundation for further work in differential equations, probability theory, stochastic processes, differential geometry and mathematical physics.