Introduction and development of some basic algebraic structures - groups, rings, fields.

Introduction and development of some basic algebraic structures - groups, rings, fields including Galois theory, representations of finite groups, algebras.The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.

Selected topics concerning fields, including Galois theory.

Galois theory and related topics.

Real and complex numbers, vectors, limits, sequences, series, power series, functions, continuity, differentiation and integration in one variable

Differential and Integral calculus in many variables, vector analysis.

Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, conformal mappings, Riemann mapping theorem.

Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, special functions, conformal mappings, Riemann mapping theorem.

The course will focus on the study of fundamental functional analysis methods relevant to the analysis of Partial Differential Equations and harmonic analysis.

Introduction to linear algebra: vectors and matrices, solving systems of linear equations, vector spaces and subspaces, orthogonality and least squares, determinants, eigenvalues and eigenvectors, singular value decomposition and linear transformations. Applications in and links to computer science will be presented in parallel.

matrices and determinantsvector spaces and linear mapscalculus of observationseigenvalue problemapplications of the eigenvalue problemnormal formsnumerical treatment of the eigenvalue problemvector algebra, first order tensorsecond order tensorapplications of tensorshigher order tensor

This is an introduction to the theory of modular forms and itsapplications to number theory.

Introduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)