Mathematics

This course builds on Stochastic Processes I and introduces an array of stochastic models with biomedical and other real world applications. Topics include Poisson process, compound Poisson process, marked Poisson process, point process, epidemic models, continuous time Markov chain, birth and death processes, martingale. The course requires students to take prerequisites.

This upper division course applies concepts in symbolic methods and analysis to solve a variety of problems in combinatorics. Course content includes: 1. Combinatorial Structures and Ordinary Generating Functions: symbolic enumeration methods, integer compositions and partitions, words and regular languages, tree structures 2. Labelled Structures and Exponential Generating Functions: labelled classes, surjections, set partitions, words, alignments, permutations, labelled trees, mapping and graphs 3. Complex Analysis, Rational and Meromorphic Asymptotics: generating functions as analytic objects, analytic functions and meromorphic functions, singularities and exponential growth of coefficients 4. Singularity Analysis of Generating Functions: coefficient asymptotics, process of singularity analysis. The course requires students to take prerequisites.

This course introduces advanced mathematical knowledge used in quantitative finance, including differential equations, numerical partial differential equations, optimization and dynamic programming, advanced probability, and neural network. Motivating examples in finance will be given as well. The course requires students to take prerequisites.

This course addresses current needs for the statistical modeling of random patterns and structures in spatial contexts, which arise in multiple fields ranging from geophysical, life and earth sciences, to communication engineering and social network analysis. The course approach relies on computational and statistical tools from stochastic geometry. The course requires students to take prerequisites.

The objective of this course is to work on optimisation problems which can be formulated as linear and network optimisation problems. The course covers formulating linear programming (LP) problems and solving them by the simplex method (algorithm); looking at the geometrical aspect and developing the mathematical theory of the simplex method; studying problems which may be formulated using graphs and networks. These optimisation problems can be solved by using linear or integer programming approaches. However, due to its graphical structure, it is easier to handle these problems by using network algorithmic approaches. Applications of LP and network optimisation are demonstrated. Major topics: Introduction to LP: solving 2-variable LP via graphical methods. Geometry of LP: polyhedron, extreme points, existence of optimal solution at extreme point. Development of simplex method: basic solution, reduced costs and optimality condition, iterative steps in a simplex method, 2-phase method and Big-M method. Duality: dual LP, duality theory, dual simplex method. Sensitivity Analysis. Network optimisation problems: minimal spanning tree problems, shortest path problems, maximal flow problems, minimum cost flow problems, salesman problems and postman problems. The course requires students to take prerequisites.

Study the fundamentals of real analysis, including Axioms of the real numbers, supremum and infimum; Countable sets; Sequences and series; Open and closed sets, compactness; Limits, continuity, differentiability; Sequences and series of functions, uniform convergence, power series; Integration. Please note that lectures alternate during the week so that students can take any of MAM2012S, MAM2013S and MAM2014S concurrently.

The course covers basic theory of analytic functions including elementary properties of analytic functions in one variable. Complex differentiability and Cauchy-Riemann equations. Calculation rules. Elementary examples of analytic functions: power series expansions, exponential functions, branches of logarithms, and functions defined by these calculation rules. Contour integrals in the complex plane. Cauchy’s integral theorem and integral formula. Existence of a primitive function and local power series expansion of analytic functions. Cauchy estimates, Liouville’s theorem, and the fundamental theorem of algebra. Theory of meromorphic functions, Laurent series expansion, and the residue theorem. Residue calculus. Further elements of the theory of holomorphic functions such as argument principle, Rouché’s theorem, and open mapping property. Harmonic functions. Regularity, existence of harmonic conjugate, mean value property, maximum principle, Poisson integrals.

This course gives knowledge of and familiarity with concepts and methods from the theory of dynamical systems which are important in applications within almost all subjects in science and technology. In addition, the course should develop the student's general ability to assimilate and communicate mathematical theory, to express problems from science and technology in mathematical terms and to solve problems using the theory of dynamical systems.