Mathematics

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.

This course takes a big-picture look at the nature of mathematics and its role in human thought, emphasizing its interactions with society, history, philosophy, science, and culture. The course involves studying key mathematical arguments in some detail, but the goal is not to develop a repertoire of technical skills. Instead, accessible mathematical topics are specifically selected for their rich interconnections with cultural context, and mathematical work is integrated with reflections on its broader meaning and implications. Instead of the drill and practice problems of a traditional mathematics class, mathematics is approached through seminar discussions, hands-on activities, and readings connecting it to broader issues. Thus, a selection of emblematic and important mathematical proofs are analyzed and used as a platform for reflecting on the nature of mathematics. In parallel, excerpts from seminal historical texts across the ages are read as well as modern scholarship from a wide range of academic disciplines that shed light on the interplay between mathematics and its societal and intellectual context. The focus is especially on geometry, from the origins of mathematical reasoning in early civilizations, to Euclid's Elements that was the gold standard of exact reasoning for millennia and the model for countless philosophical systems, to the projective geometry of Renaissance art, to the more modern non-Euclidean geometry that overturned conventional wisdom about the nature of human spatial perception and the shape of space.

This course provides an introduction to statistics: the study of collecting, analysing, and interpreting data, which is fundamental to doing any form of quantitative research. Students learn to recognize which analysis procedure is appropriate for a given research problem involving one or two variables; understand principles of study design; and apply probability theory to practical problems. They apply statistical procedures on a computer using RStudio/R; interpret computer output for statistical procedure; calculate confidence intervals and conduct hypothesis tests by hand for small datasets; and understand the usefulness of statistics in their professional area.

This course is a study of the general techniques and ideas found in professionally written numerical software, as well as the general concepts students need to know to apply suitable software to computational problems. The course is designed more for potential users of mathematical software than for potential creators of such software. At the completion of the course, students are able to choose an appropriate numerical method to solve a problem, and evaluate the numerical method with respect to potential accuracy, computational efficiency, and memory requirements. Students are also taught to perform the required computations using Matlab or similar systems, and then evaluate the quality of the solution with respect to the accuracy obtained and the sensitivity to model parameter variations. They then estimate whether the quality of the solution is adequate relative to the desired use of the model, and analyze the reasons behind a possible total failure of a method when applied to a concrete problem. This course also covers simple mathematical models from science and numerical analyses of them. Students learn about fundamental numerical methods for the solution of linear and nonlinear equations, linear and nonlinear optimization, eigenvalue problems, initial value problems for ordinary differential equations, partial differential equations, and the fast Fourier transform.

This course examines the core ideas and concepts and principles of limits, differentiation, and integration of functions in multiple variables, vector analysis. It teaches students to apply the rigorous, analytical and numeric approach to analyze and solve problems using concepts of multivariable calculus. Course topics include: differentiation in several variables, with applications in approximation, maximum and minimum, and geometry, integration in several variables, vector analysis.