# MAM3000W - Mathematics 3000

72 credits at NQF level 7

## Entry Requirements:

MAM2000W and MAM1019H (with permission from the Head of Department, MAM1019H may be taken concurrently with MAM3000W. However, this permission will usually only be granted for students who decide after first year of study to major in mathematics).

## Course Outline:

This course aims to introduce students to advanced topics in mathematics. 3AL: MODERN ABSTRACT ALGEBRA Group Theory (Isomorphism Theorems, p-Groups, Sylow Theory, Direct Products and finitely generated Abelian Groups). Further Linear Algebra (Primary decomposition, Jordan normal forms, Bilinear forms). 3CA: COMPLEX ANALYSIS Field of complex numbers. Power series. Analytic functions. Complex integration. Liouville’s theorem, Fundamental Theorem of Algebra. Maximum Modulus Theorem. Index of a closed curve. Cauchy’s Integral Formula. Counting Zeros and Open Mapping Theorems. Goursat’s Theorem. Singularities. Laurent series. Residues. theory, recurrences, algorithms, applications. 3MS: METRIC SPACES Metric spaces and topology; applications 3TA: TOPICS IN ALGEBRA A selection from lattices and order, congruences, Boolean algebra, representation theory, naive set theory, universal algebra. 3TN: TOPICS IN ANALYSIS Compactness in metric spaces, normed spaces, linear continuous mappings between normed spaces, Hilbert spaces, orthogonal projection, differential calculus on normed spaces, review of the Riemann integral and its limitations.