Mathematics

The content of this course contains first order logic up to proofs of the completeness and incompleteness theorems. Students become familiar with the syntax and semantics of first-order logic, completeness theorem of first-order logic, compactness theorem and basic model theory, and Gödel's first incompleteness theorem.

This course covers the ideas and methods of university level pure mathematics. In particular, the course shows the need for proofs, to encourage logical arguments and to convey the power of abstract methods. This is done by example and illustration within the context of a connected development of the following topics: real numbers, sequences, limits, and series.

The syllabus first covers abstract vector spaces and linear transformations. It then introduces rings and modules, their quotients, and the first isomorphism theorem. The multilinear algebra of determinants is studied, together with eigenvectors and eigenvalues, culminating in the Cayley-Hamilton theorem and the Perron Frobenius Theorem. This is followed by an introduction to inner product spaces and the Spectral Theorem. The course then moves on to normal forms for linear transformations and particularly the Jordan Normal Form. Throughout the course, we will also work with a computer algebra system (e.g. Maple) to learn about programming skills and data structures which are useful in Pure Mathematics and beyond. In lab sessions, we will use these skills to investigate topics that are relevant to the theory being developed in lectures and workshops. Students will also carry out a group project which will require some computer algebra work and a short group presentation. Linear Algebra 1. Basic concepts in abstract linear algebra, abstract vector spaces, bases, linear maps, dimension, images and kernels. 2. Linear transformations, choice of basis, Smith normal form. Rings and Modules 1. Basic definitions and examples of rings, homomorphisms, kernels, images. 2. Polynomials, their Euclidean algorithm, roots and algebraically closed fields. 3. Basic definitions and examples of modules, homomorphisms, kernels, images. 4. Quotient rings, modules and vector spaces; the first isomorphism theorem. Determinants and Eigenvalues 1. Multilinear forms; characterizations of determinant. 2. Eigenvalues and eigenvectors; diagonalizable and triangularizable linear mappings; Cayley-Hamilton Theorem. 3. Perron-Frobenius Theorem and applications. Inner Product Spaces and Quadratic Forms 1. Basic definitions and examples of inner product spaces. 2. Orthogonal projection; Gram-Schmidt; 3. Adjoints of linear transformations; spectral theorem for finite dimensional inner product spaces. Jordan Normal Form 1. The Jordan Normal Form. 2. Applications of the Jordan Normal Form.

In this course, students learn about mathematical notation and concepts concerning vectors, matrices, and calculus. This includes important concepts of linear systems, linear independence, real convergence, power series representation, and transform representation.

This course covers: -Parametric families and likelihood -Sufficiency, Neyman factorisation, minimal sufficiency, joint sufficiency -Elements of statistical decision theory -Estimation, minimum variance unbiased estimators, Cramer-Rao lower bound, Bayes and minimax estimators -Hypothesis testing, pure significance tests, optimal tests, power, Neyman-Pearson lemma, uniformly most powerful tests -Confidence intervals, relationship to hypothesis testing -Selected topics in modern statistics.