MAM3000W

Tutors for MAM3000W. This is UCT’s general mathematics third year course. for students majoring in Mathematics.

Topics include:

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).

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.

DISCRETE MATHEMATICS Graph theory, combinatorial counting, discrete probability theory, recurrences, algorithms, applications.

METRIC SPACES
Metric spaces and topology; applications

TOPICS IN ALGEBRA
Ring Theory (Isomorphism Theorems, Fields of Fractions of Domains, maximal, prime and principal ideals, Euclidean and Principal Ideal Domains, unique factorization, rings of algebraic integers). Field Theory (characteristic and prime subfields, extensions, finite fields, adjoining roots of polynomials). Further Group Theory (nilpotent and solvable groups, some finite simple groups).

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.