Lecture Course  Contents 

153 First Year Analysis,  Properties of the real number system. Inequalities.
Bounded and unbounded sets. Least upper bound and greatest lower bound. Sequences of real numbers. Subsequences. Limit of a sequence. Convergence and divergence. Convergent sequences are bounded. A monotonic increasing sequence which is bounded above is convergent. Sums, products and quotients of convergent sequences. The sequences {n^{k}} and {r^{n}}. Sandwich rule. Series of real numbers as sequences of partial sums. Sums of Geometric series. Divergence of series. Comparison and ratio tests for series of positive terms. Series with positive and negative terms. Alternating series test. Relative and absolute convergence. Convergence of power series. Radius of convergence. Cauchy's root test. Functions defined by power series (trigonometric and exponential functions). 
MATH10101 Sets, Numbers and Functions.  Numbers of injections and bijections. Numbers of subsets and Binomial Numbers, Pascal's Triangle, Binomial Theorem.
Division Theorem, Greatest Common Divisor, Euclid's Algorithm, Bezout's Lemma, Linear Diophantine Equations, Congruences, Modular Arithmetic, Solving Linear Congruences, Multiplicative inverses, Pairs of congruences, Triplets of congruences, Method of Successive Squaring, nonlinear Diophantine equations, Congruence Classes, Multiplication Tables, Invertible Elements, Reduced Systems of Classes. Partitions, Relations, generalizing Congruence Classes, from relations to partitions, from partitions to relations. Prime Numbers, Sieve of Eratosthenes, Infinitude of Primes, Conjectures about Primes, Euler's Theorem, Fermat's Little Theorem. Applications of Euler's and Fermat's Theorem. Permutations, Bijections, two row notation, Composition. Cycles, factoring, orders. Groups. 
1K1 Logic, Set Theory and Matrices  Propositions, Connectives (and, or, not), Truth Tables, The Boolean Laws of Logics. Equivalence of forms, tautology and contradiction.
Conditional and Biconditional, contrapositive, converse. Definition of a valid argument. Two ways to use a truth table to show an argument is Valid. Natural Deduction: A Rule of Assumption, MPP, MTT, DN. Natural Deduction: Eliminating the "and", Eliminating the "or", Introducing "or", Introducing "and". Natural Deduction: Conditional Proof, Proof by Contradiction. Subset, equality, denoting a set. Formal languages. Complement, union, intersection, difference, symmetric difference. The Boolean Laws for Sets. Universal and existential quantifiers. Symbolising English sentences. Negating quantified sentences. Proving validity of quantified arguments. Cartesian product, Power set. Cardinalities of a union of sets, of a produxt of sets and of a power set. Relations. Digraphs. Reflexive, symmetric, transitive relations. Equivalence Relations. Describing a function. Onto and onetoone functions. Composition and inverses. Defintion, addition and multiplication. Identity and inverses. Solving systems of linear equations. Gaussian Elimination. 
MATH10242 Sequences and Series, 

MATH20101 Real Analysis,  Limit of a realvalued function at a finite point, onesided limits, limits of a function at infinity.
Divergence. Limits Rules, including the Sum, Product and Quotient rules and the Sandwich Rule. Special Limits; the exponential and trigonometric functions. Definition of continuity and continuous functions. Continuity Rules, including the Sum, Product and Quotient rules and the Composite rule. The Intermediate Value Theorem, the Boundedness Theorem and the Inverse Function Theorem. Definition of differentiation Rules for differentiation, including the Sum, Product and Quotient rules, the Chain Rule and the Inverse Rule. Derivative Results including: Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem and L'Hôpital's Rule. Taylor Polynomials and Taylor's Theorem with Cauchy's and Lagrange's forms of the error. Taylor Series and a number of standard examples. Definition of integration with worked examples. Fundamental Theorem of Integration. 
MATH20132 Calculus of Several Variables, 
Functions of Several Variables: Notation; Limits of vectorvalued functions; Scalarvalued examples; Sandwich Rule; Directional limits; Limits along curves; Continuity of vectorvalued functions; Continuity Laws; Composition Laws.
Differentiation of functions of several variables: Directional derivatives; Partial Derivatives; Differentiable implies Continuous; Vectorvalued Linear Functions; Fréchet Derivative.
Fréchet Differentiable implies Continuous; Derivative exists implies directional derivative exists.
Jacobian Matrix; Special cases: functions of one variable and scalar valued functions.
When is a function differentiable? C^{1} functions are Fréchet differentiable.
Examples; Product and Quotient functions; The Chain Rule; Rules for differentiation; Special cases; the Inverse function. Surfaces and the Implicit Function Theorem: A surface as a graph; a surface as an Image set, i.e. given parametrically; a surface as a Level set, i.e. given Implicitly; a graph is a level set; converses.
Proof of the Implicit Function Theorem: by Induction. Extremal Values: Extremal Values and Lagrange multipliers;
Differential forms and their integration: Differential 1forms; Exact 1forms; higherorder derivatives; Closed 1forms; line integrals; Differential 2forms; Surface integrals; Products of 1forms; Derivatives of 1forms; Stoke's Theorem (statement of); Green's Theorem (statement of).
Differentiable kforms; products and derivatives of kforms; integration of a kform over a manifold. Higher Derivatives: The Hessian Matrix; Taylor's Theorem; Surfaces; examples of local minima, maxima and saddle points; Tests for definiteness or otherwise. 
341 Measure Theory  Classes of subsets: Topology, Rings and Fields. Cardinality, Topological Space results.
Set functions: (finitely) additive functions, sigmaadditive functions, Extending a sigmaadditive function, Measure and Outer measure. Outer measure and Measurable sets, Lebesgue Measurable sets, Nonmeasurable sets, Sets of measure zero. Measurable functions: Sequences of functions. Simple functions. An interesting simple function. Integration: Integration of nonnegative simple functions. Chebychev's Theorem and an application to Normal Numbers. Interchanging integrals with other operations. Extended version of Monotonic Convergence Theorem with yet another proof that ∑(1/n^{2})=π^{2}/6. Integration of measurable functions. Comparison of the Riemann and Lebesgue integrals, Measure Preserving Transformations including Poincare's Recurrence Theorem. Spaces of integrable functions, showing that these spaces are complete. Product Spaces and Fubini's Theorem. Completion of product spaces. 
MATH6\41022 Analytic Number Theory.  Two proofs of the infinitude of primes. Definition of the Riemann zeta function, infinite products. Elementary Prime Number Theory. von Mangoldt's Function; Partial Summation; Replacing sums by integrals.
Arithmetic functions. Cauchy Products; Convolutions; Dirichlet Series; multiplicative and additive functions.
Sums of Convolutions. Convolution method for 1*g; Various summations  of Q_{2} , 2^{ ω}, d(n^{2}), d^{ 2}(n) and d^{ 3}(n).
Sums of Additive Functions. Turán's inequality; HardyRamanujan Theorem. Appendix TuránKubilius inequality; statement of the ErdosKac Theorem. The Prime Number Theorem. Introduction
Actual, though unproved in this course, & conjectured results on the zeros of ζ(s). 