Each student will work on a mathematical project under the supervision of a faculty member. The project will culminate in an oral presentation to the Department of Mathematics. The topic of the project will agreed upon by the student and supervisor.

This course develops the ideas underlying modern, measure-theoretic probability theory, and introduces the various classes of stochastic process, including Markov chains, jump processes, Poisson processes, Brownian motion and diffusions. Their properties and applications are investigated.

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.

Functional analysis is mainly concerned with the study of vector spaces and operators acting upon them. It provides powerful tools in handling several problems in applied mathematics and theoretical physics. It is also basic for the understanding and development of very many other mathematical theories like the Theory of Partial Differential Equations and the Theory of Operators. The first part of the course is devoted to a short introduction in the theory of metric spaces and to a detailed study of normed and Banach spaces and in particular to the analysis of linear operators acting upon them. The second part of the course deals with Hilbert spaces and linear operators upon them, since they play a fundamental role in applied mathematics. Finally, we look at some  fundamental theorems for normed and Banach spaces such as the Hahn-Banach theorem for complex vector spaces and normed spaces and its application to bounded linear functionals;  the uniform boundedness theorem, and the  closed Graph theorem.