Mathematics

This course introduces exciting new developments in advanced mathematics. The barriers between fields are being broken, many new unexpected applications are continually found, and out of this cross-fertilization, new kinds of mathematics are born. Topics are subject to change but may include various new advances of pure mathematics and logic, computational science and numerical analysis, fluid mechanics and geophysics, wavelets and signal processing, cryptology, quantum computation, mathematical biology (including bioinformatics, proteomics and neuroscience), intelligence science, financial mathematics and mathematical economics, and probability theory with various applications.  

This course introduces concepts and theories of mathematical analysis. Topics include limits of continuous functions and differentiable series of functions, uniform convergence of series of functions, Arzela-Ascoli theorem, Weierstrass theorem, power series, analytic functions, trigonometric series, Fourier series, etc. 

This course examines how to apply deterministic differential and difference equation models to real world examples, and how to solve them using numerical methods. it also covers how to quantify system uncertainties with the help of statistical and probabilistic methods. Students will be taught a range of methods that are employed in industry, research, consultancies and government to model complex natural resource problems. In the process, students will learn how certain fundamental mathematical concepts such as critical points, orthogonality, eigenvalues and singularity recur in different mathematical frameworks with different but, invariably, vitally important physical interpretations.

Algebraic topology is concerned with the construction of algebraic invariants associated to topological spaces which serve to distinguish between them. This course focuses on the concept of the fundamental group of a topological space, and discusses its relation to other important notions in topology such as homotopy, covering space, etc.  

Topics include homotopy of paths, covering spaces, the fundamental group of the circle, retractions and fixed points, the Borsuk-Ulam theorem, deformation retracts and homotopy type, the Jordan curve theorem, imbedding graphs in the plane, the winding number of a simple closed curve, the Cauchy integral formula, the Seifert-van Kampen theorem, the fundamental group of a wedge of circles, adjoining a two-cell, the fundamental group of the torus and the dunce cap, the classification theorem, equivalence of covering spaces, and existence of covering spaces. 

Prerequisite: Topology 1 

This is an independent research course with research arranged between the student and faculty member. The specific research topics vary each term and are described on a special project form for each student. A substantial paper is required. The number of units varies with the student’s project, contact hours, and method of assessment, as defined on the student’s special study project form.

This course is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, it develops the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. 

This course is a first introduction to algebraic topology, the area of mathematics in which algebra is used to study topological spaces. It defines the fundamental group and singular homology and studies their basic properties and applications. The course introduces foundational competencies in algebraic topology. Important concepts include homotopy, homotopy equivalence, fundamental group, covering space, chain complex, and homology. Prerequisites include knowledge about general topology and abelian groups, as obtained through courses such as Topology and Algebra 2, and Advanced Vector Spaces.