# Courses in the Mathematical Sciences

Course Code & Title AU Course Contents
CY1601/RE1001 - Mathematics I 4 AU Introductory mathematics course for CN Yang scholars and Renaissance Engineering students.
• Limits, continuity, the min/max theorem, and the intermediate value theorem.
• Differentiability and differentiation rules.
• Critical points, the mean value theorem, and l'Hospital's rule.
• Inverse functions and derivatives of inverse functions.
• Trigonometric, logarithm and exponential functions.
• The Riemann integral, the Fundamental Theorems of Calculus.
• Techniques of integration. Using integrals to calculate volume, mass, work, etc.
• Infinite sequences and infinite series, power series and convergence criteria, Taylor series.
• Ordinary differential equations.

Only offered to CN Yang scholars and Renaissance Engineering students.
Mutually exclusive with MH1100, MH1101, MH1800, and M1801.
CY1602/RE1021 - Mathematics II 4 AU Second mathematics course for CN Yang scholars and Renaissance Engineering students.
• Systems of linear equations, and the Gaussian elimination algorithm.
• Matrices, and their inverses and determinants.
• Vector spaces, subspaces, linear independence, basis, dimension, row and column spaces, rank, linear transformations, eigenvectors, eigenvalues, and diagonalization.
• Inner products, inner product spaces, orthonormal sets, the Gram-Schmidt process, and Fourier series.
• Calculus of several variables: Partial derivatives, limits and continuity, chain rule, directional derivatives, gradients, and Lagrange multipliers.
• Double integrals, and the calculation of the area of a surface; triple integrals.
• Vector calculus, line integrals, Green's Theorem, surface integrals, Gauss's divergence theorem, and Stokes' Theorem.

Prerequisite: CY1601 or RE1001.
Not available to students who have taken/are taking MH1200 or MH2100.
MH1100 - Calculus I​ 4 AU Introductory course on differential and integral calculus.
• Real numbers, functions, their inverses and graphs.
• Trigonometric and inverse trigonometric functions, logarithms and exponentials, and hyperbolic functions.
• Limits of functions, continuity at a point, and continuity on an interval.
• Differentiability, derivatives of functions, the chain rule, implicit differentiation, derivatives of higher order.
• Local maxima and local minima, Rolle's Theorem and the Mean Value Theorem, points of inflection, first-derivative and second-derivative tests, L'Hospital's Rule.
• Antidifferentiation, indefinite integrals, substitution rule, and integration by parts.

Prerequisite: A level Mathematics or equivalent.
Not available to students who have taken/are taking MH1800.
MH1101 -​ Calculus II 4 AU Further topics in calculus.
• Definite integrals; the Fundamental Theorems of Calculus.
• Area of plane regions, volumes of solids, length of arcs.
• Mean Value Theorem for integrals, and other applications of the definite integral.
• Techniques of integration, numerical integration, and improper integrals.
• Sequences: monotonic and bounded sequences, Newton's method, infinite series, tests for convergence and divergence, alternating series, and absolute/conditional convergence criteria.
• Power series: differentiation and integration of power series, Taylor series, binomial series, and Fourier series.

Prerequisite: MH1100
Not available to students who have taken/are taking MH1801.
MH1200 - Linear Algebra I 4 AU Introductory course on linear algebra.
• Systems of linear equations; Gaussian elimination.
• Matrices, inverses, and determinants.
• Vectors, dot products, and cross products.
• Vector spaces, subspaces, linear independence, basis, dimension, row and column spaces, and rank.

Prerequisite: A or H2 level Mathematics or equivalent.
Not available to students who have taken/are taking MH2800.
MH1201 - Linear Algebra II​ 4 AU Further topics in linear algebra.
• Linear transformations, kernels and images.
• Inner products, inner product spaces, orthonormal sets, and the Gram-Schmidt process.
• Eigenvectors and eigenvalues; matrix diagonalization and its applications.
• Symmetric and Hermitian matrices.
• Quandratic forms and bilinear forms; Jordan normal form and other canonical forms.

Prerequisite: MH1200.
Not available to students who have taken/are taking MH2800.
MH1300 - Foundations of Mathematics 4 AU Introductory course on core mathematical concepts, including logic and the theory of sets.
• Elementary logic, mathematical statements, and quantified statements.
• Sets, operations on sets, Cartesian products, and properties of sets.
• Natural numbers, integers, rational numbers, real numbers, and complex numbers.
• Relations, equivalence relations, and equivalence classes.
• Functions, injective and surjective functions, inverse functions, and composition of functions.
• Division algorithm, greatest common divisor, Euclidean algorithm, fundamental theorem of arithmetic, modulo arithmetic.

Prerequisite: A or H2 level Mathematics or equivalent.
MH1301 - Discrete Mathematics 3 AU Introductory course on discrete mathematics.
• Counting, permutations and combinations; the binomial theorem.
• Recurrence relations.
• Graphs, paths and circuits, and isomorphisms.
• Trees and spanning trees.
• Graph algorithms (e.g., shortest path, maximum flow) and their computational complexity; big-O notation.

Prerequisite: A or H2 level Mathematics or equivalent.
MH1401 - Algorithms and Computing I 2 AU Core course introducing fundamentals of programming (including variables, data types, control statements, iteration, and recursion), using the Python programing language. By emphasizing applications to problem-solving, it develops the ability to think algorithmically, which is essential for any professional working in an increasingly computer-driven world. This course is required for future computing courses and for courses using Python as a supporting tool. No prior programming experience is required. The topics covered include:
• Python Basics (operators, variables, types, ...)
• Lists
• Matrices with NumPy module
• Strings
• Input/Output
• Selection statements
• Loop statements
• Functions
• File I/O
• Errors and debugging
• Recursion
• Algorithm complexity
• Sorting algorithms
• Plotting with Python

Prerequisite: A or H2 level Mathematics or equivalent.
MH1402 - Algorithms and Computing II 2 AU Further topics in algorithms and computing.
• The concept of an algorithm.
• The basic structures of a C/C++ program: variables, types, declarations, expressions, input/output structures, flow control structures.
• Debugging and good programming style.
• Vectors and arrays.
• Algorithms for searching and sorting vectors and arrays.
• Basic concepts of algorithm efficiency.
• Functions, classes, and libraries in C/C++.
• Recursion and the divide-and-conquer paradigm.

Prerequisite: MH1401.
MH1403 ​- Algorithms and Computing​​ ​ 3 AU Systematic introduction to data structures and algorithms for efficient computer programs.
• Data abstraction in the program development process.
• Design of efficient algorithms.
• Simple algorithmic paradigms such as greedy algorithms, divide-and-conquer algorithms and dynamic programming.
• Elementary analyses of algorithmic complexity

Prerequisite: PS0001
MH1800 - Calculus for the Sciences I​ 3 AU First of two courses on calculus for students in the sciences. Applications and computer-based learning are included. Topics covered include:
• Functions and graphs, real numbers
• Differentiation of functions of one variable, derivative as rate of change, chain rule, implicit functions, and inverse functions.
• Local maxima and minima.
• Indefinite and definite integrals, and applications of integration.
• Methods of integration.
• Fundamental theorem of calculus.

Prerequisite: A or H2 level Mathematics or equivalent.
Not available to students who have taken/are taking MH1100 and MH1101.
MH1801 - Calculus for the Sciences II 3 AU Second of two courses on calculus for students in the sciences. Applications and computer-based learning are included. Topics covered include:
• Differential equations; first-order and second-order linear differential equations.
• Techniques of solving differential equations, and applications.
• Series and power series.
• Taylor's series.
• Fourier series.

Prerequisite: MH1800 or equivalent.
Not available to students who have taken/are taking MH1101.
MH1802 - Calculus for the Sciences 4 AU Introductory course in calculus, for students majoring in the physical sciences.
• Types of Numbers; Functions and Graphs
• Algebraic, trigonometric, logarithmic and exponential functions and identities
• Basic Complex Numbers
• Limits & Continuity
• Derivatives & Techniques of Differentiation
• Applications of Differentiation; Numerical Approximation of differentiation
• Indefinite Integrals and Definite Integral, Fundamental Theorem of Calculus, Techniques of Integration 1
• Techniques of Integration 2; Applications of Integration
• Applications of Integration in Science; Numerical Approximation of integration
• Introduction to Differential Equations; First Order Ordinary Differential Equations
• Second Order Linear Differential Equations with constant coefficients
• Power Series Method

Prerequisite: A or H2 level Mathematics or equivalent.
Not available to students who have taken/are taking MH1800 and MH1801.
MH1803 - Calculus for Physics 4 AU Additional topics in calculus, for students majoring in physics.
• Vectors and multivariable calculus.
• Vector analysis.
• Ordinary differential equations.
• Partial differential equations.

Prerequisite: A or H2 level Mathematics or equivalent.
Not available to students who have taken/are taking MH1800 and MH1801.
MH1804 - Mathematics for Chemistry 2 AU Additional topics in calculus, for students majoring in chemistry.
• Cartesian and spherical coordinates.
• Complex numbers.
• Vectors; linear algebra and matrices.
• Summation, series, and expansions of functions.
• Fourier series & Fourier transforms.

Prerequisite: MH1802.
MH2100 - Calculus III 4 AU Intermediate course in calculus.
• Parametric equations; polar coordinates.
• Vector-valued functions, calculus of vector-valued functions, and analytic geometry.
• Functions of more than one variable, limits, continuity, partial derivatives, differentiability, total differentials, the chain rule, and the implicit function theorem.
• Directional derivatives, gradients, and Lagrange multipliers.
• Double integrals; the area of a surface; triple integrals.
• Line integrals, Green's Theorem, surface integrals, the Gauss divergence theorem, and Stokes' Theorem.

Prerequisite: MH1101 or  MH1802 or MH1805
Not available to students who have taken/are taking MH2800.
MH2200 - Groups and Symmetry​ 3 AU Introductory course on group theory, with emphasis on symmetry groups of geometric structures.
• Symmetries of 2D and 3D objects (e.g. quadrangles, tetrahedrons).
• Group axioms.
• Cyclic and dihedral groups.
• Permutation groups.
• Representation of rotations and reflections by matrices.
• Wallpaper groups.
• Puzzles (especially 15-puzzle and Rubik's cube).

Prerequisite: MH1200 or CY1602
MH2401 - Algorithms and Computing III 2 AU Application of computing skills and previously-learnt mathematical topics (Linear Algebra, Calculus, Discrete Mathematics, etc.) for solving real-world problems. This course emphasizes group project work, and assessments are based substantially on a term project.

Prerequisites: MH1100, MH1101, MH1200, MH1401, and MH1402.
MH2500 - Probability & Introduction to Statistics 4 AU Introductory course on probability and statistics.
• Discrete distributions (binomial, hypergeometric and Poisson).
• Continuous distributions (normal, exponential) and densities.
• Random variables, expectation, independence, conditional probability.
• The law of large numbers and the central limit theorem.
• Sampling distributions.
• Elementary statistical inference (confidence intervals and hypothesis tests).

Prerequisite: (MH1100 & MH1101) or (MH1800 & MH1801) or (MH1101 & MH110S) or (MH1100 & MH111S) or MH1802 or CY1601 or MH1805.
MH2800 - Linear Algebra and Multivariable Calculus 4 AU Techniques in linear algebra and multivariable calculus, and their applications. This course includes computer-based learning. Topics covered include:
• Systems of linear equations.
• Matrices and determinants.
• Vectors in 2- and 3-dimensional Euclidean spaces; Vector spaces, linear independence, basis, and dimension.
• Linear transformations.
• Eigenvectors and eigenvalues.
• Calculus of functions of several variables; partial derivatives.
• Constrained and unconstrained optimization.

Prerequisite: MH1800 or equivalent.
Not available to students who have taken/are taking MH1200, MH1201, or MH2100.
MH2801 - Complex Methods for the Sciences 3 AU Introduction to complex numbers and their applications in physics and the other sciences.
• Complex numbers, the argand diagram, modulus and argument.
• Complex representations of waves and oscillations.
• Functions of a complex variable, analyticity, and the Cauchy-Riemann equations.
• Contour integration, Cauchy's integral formula, and the residue theorem.
• Fourier series and Fourier transformations, and their applications.
• Green's functions methods.

Prerequisites: (MH1801 & MH2800) or (MH1101 & MH1200) or (MH1802 & MH1803 & MH1200) or (MH1802 & MH1803 & MH2802) or (CY1601 & CY1602).
Not available to students who have taken/are taking MH3101.
MH2802 - Linear Algebra for Scientists​ 3 AU Introduction to linear algebra and its applications in physics and the other sciences.
• Vector algebra and analytical geometry.
• Linear spaces.
• Linear transformations and matrices.
• Eigenvalues and eigenvectors.
• Applications of linear algebra to problems in physics and computing.

Prerequisite: A or H2 level Mathematics or equivalent.
MH3100 - Real Analysis I 4 AU
• Basic properties of real numbers, supremum and infimum, completeness axiom, open and closed sets, compact sets, countable sets.
• Limits and convergence of sequences, subsequences, Bolzano-Weierstrass theorem, Cauchy sequences, infinite series, double summations, products of infinite series.
• Limits of functions, continuity, uniform continuity, intermediate value theorem, extreme-value theorem.
• Differentiability, derivatives, intermediate value property, Cauchy mean value theorem, Taylor's theorem, Lagrange's form of the remainder.
• Sequence and series of functions, uniform convergence and differentiation.
• Power series, radius of convergence, local uniform convergence of power series.

Prerequisites: (MH1100 & MH1101) or CY1601 or MH1802.
MH3101 - Complex Analysis 4 AU
• Analytic functions of one complex variable, Cauchy-Riemann equations.
• Contour integrals, Cauchy's theorem and Cauchy's integral formula, maximum modulus theorem, Liouville's theorem, fundamental theorem of algrebra, Morera's theorem.
• Taylor series, Laurent series, singularities of analytic functions.
• Residue theorem, calculus of residues.
• Fourier transforms, inversion formula, convolution, Parseval's formula.

Prerequisite: MH2100.
Not available to students who have taken/are taking MH2801.
MH3110 - Ordinary Differential Equations​ 4 AU
• First order equations, exact equations, integrating factors, separable equations, linear homogeneous and non-homogeneous equations, variation of parameters, principle of superposition.
• Second order equations, Wronskian, Abel's formula, variation of parameters, exact equations, adjoint and self-adjoint equations, Lagrange and Green's identities, Sturm's comparison and separation theorems.
• First order linear systems, Wronskian, Abel's formula, variation of parameters, systems with constant coefficients.
• First order nonlinear systems, initial value problem.
• Use of ODEs in simple modeling problems.

Prerequisite: MH2100.
MH3200 - Abstract Algebra I 3 AU Introduction to modern algebra, including basic algebraic structures such as groups, rings and fields. Topics covered include:
• Groups, subgroups, cyclic groups, groups of permutations, cosets, Lagrange's Theorem, homomorphism, and factor groups.
• Rings and fields, ideals, integral domains, quotient fields, rings of polynomials, factorization of polynomials over a field.

Prerequisites: MH1201, MH1300, and MH2200.
MH3210 - Number Theory​ 4 AU Introduction to basic number theory, including modern applications. Topics covered include:
• Review of modular arithmetic; the Chinese remainder theorem, Fermat's little theorem, and Wilson's theorem.
• Number-theoretic functions: τ, σ, Euler's φ-function, Möbius inversion formula, applications to cryptography.
• Primitive roots and indices.
• Legendre's symbols; the quadratic reciprocity law.
• Continued fractions and Pell's equations.
• Primality tests and factorization of integers, and the RSA cryptosystem.

Prerequisite: MH1300.
MH3300 - Graph Theory 4 AU
• Connectivity and matchings, Hall's theorem, Menger's theorem, network flows.
• Paths and cycles, complete subgraphs and Turán's theorem, and the Erdös-Stone theorem.
• Graph colouring and the four-colour theorem.
• Ramsey theory.
• Probabilistic methods in graph theory.
• Use of software to solve graph-theoretic problems.

Prerequisites: MH1301 and MH2500.
MH3310 - Mathematical Foundations of Game Theory 4 AU
• Games of normal form and extensive form, and their applications in economics, relations between game theory and decision making.
• Games of complete information: static games with finite or infinite strategy spaces, Nash equilibrium of pure and mixed strategy, dynamic games, backward induction solutions, information sets, subgame-perfect equilibrium, finitely and infinitely-repeated games.
• Games of incomplete information: Bayesian equilibrium, first price sealed auction, second price sealed auction, and other auctions, dynamic Bayesian games, perfect Bayesian equilibrium, signaling games.
• Cooperative games: bargaining theory, cores of n-person cooperative games, the Shapley value and its applications in voting, cost sharing, etc.

Prerequisite: MH2500.
Not available to students who have taken/are taking HE302 / HE3002.
MH3400 - Algorithms for the Real World​ 4 AU
• Mathematical concepts for analysis of algorithms.
• Fundamental algorithm design techniques, with applications to various problems: network algorithms, matrix algorithms, optimization algorithms, and algorithms for data analysis and machine learning.
• Applications to problems in combinatorial optimization, networks, operations research, data analysis and machine learning.

Prerequisites: (MH1201, MH1301, MH1402, and MH2500) or (MH1201, MH1301, MH1403 and MH2500).
MH3500 - Statistics 4 AU
• Probability distributions of functions of random variables, the law of large numbers and the central limit theorem.
• Point and interval estimation, optimal estimation, maximum likelihood methods.
• More on tests of hypotheses; the Neyman-Pearson lemma, likelihood ratio tests, large sample theory, Chi-square tests and contingency tables.

Prerequisite: MH2500.
MH3510 - Regression Analysis 4 AU Introduction to regression analysis, one of the most widely-used statistical techniques.
• Simple and multiple linear regression, nonlinear regression, analysis of residuals and model selection.
• One-way and two-way factorial experiments, random and fixed effects models.

Prerequistes: MH2500 and MH3500.
MH3600 - Knots and Surfaces: Introduction to Topology 4 AU Introductory course on the main ideas and results of topology. The emphasis is on beautiful, intuitive material such as surfaces, graphs and knots, rather than technicalities of point-set topology.
• Continuous functions.
• Surfaces.
• Euler characteristics.
• Maps and graphs on surfaces.
• Vector fields on surfaces.
• Knots.
• Fixed-point theorems and their applications to economics.
• Basics of homotopy theory.

Prerequisites: (MH2100 & MH2200) or (MH1803 & MH2200).
MH3511 - Data Analysis with Computer​ 3 AU
• Data collection and analysis processes; graphical and numerical methods for describing data.
• Summarizing bivariate data; probability and population distributions; estimation and hypothesis testing using a single sample; comparing two population or treatments.
• Analysis of categorical data and goodness-of-fit tests.

Prerequisite: MH2500.
MH3512 - Stochastic Processes 4 AU
• Introduction and motivations
• Probability background
• Gambling problems
• Random walks
• Discrete-time Markov chains
• First step analysis (hitting probabilities)
• First step analysis (mean hitting times)
• Classification of states
• Long-run behavior of Markov chains
• Branching processes
• Continuous-time Markov chains (definitions and properties)
• Continuous-time Markov chains (examples and first step analysis)

Prerequisite: MH2500.
MH3700 - Numerical Analysis I​ 3 AU Introduction to the theory and applications of numerical approximation techniques.
• Commonly used numerical algorithms, and how to implement them.
• Computational errors.
• Numerical methods for solving systems of linear equations.
• Iterative methods for systems of linear equations.
• Polynomial interpolation.
• Numerical integration.
• Numerical solutions of nonlinear equations.

Prerequisites: (MH1200 & MH1201) or (MH1800 & MH2800) or CY1602 or MH2802.
MH3701 - Basic Optimization​ 4 AU
• Geometric simplex method
• Algebraic simplex method in tabular form
• Implementing the simplex method
• Revised simplex method
• Fundamental Theorem of the network simplex method
• The network simplex method
• Linear programming duality
• Sensitivity and post-optimality analysis
• Lagrange duality and the Karush-Kuhn-Tucker conditions

Prerequisite: MH1201 or MH2800 or MH2802.
MH4100 - Real Analysis II​ 4 AU

• Basic topology on the real line and extended real line
• Measurable sets and measurable functions
• Lebesgue integration
• Differentiation, bounded variation, absolute continuity, and convex functions
• Classical Banach spaces​​

Prerequisites: (MH2100 & MH3100) or (CY1602 & MH3100) or (MH1803 & MH3100).
MH4110 - Partial Differential Equations​ 4 AU
• First-order equations, quasi-linear equations, general first-order equation for a function of two variables, Cauchy problem.
• Wave equation, wave equation in two independent variables, Cauchy problem for hyperbolic equations in two independent variables.
• Heat equation, the weak maximum principle for parabolic equations, Cauchy problem for heat equation, regularity of solutions to heat equation.
• Laplace equation, Green's formulas, harmonic functions, maximum principle for Laplace equation, Dirichlet problem, Green's function and Poisson's formula.

Prerequisite: (MH3100 and MH3110) or (MH1803 and MH3100). (MH4100 is useful but not required.)
MH4200 - Abstract Algebra II 4 AU
• Unique factorization domains, Euclidean domains, principal ideal domains.
• Modules, submodules, homomorphisms, quotient modules, modules over principal ideal domains.
• Field extensions, automorphisms of fields, spilitting fields, normal and separable extensions.
• Galois extensions, Galois groups, Galois correspondence, finite fields.

Prerequisites: MH1201 and MH3200.
MH4300 - Combinatorics 4 AU
• Recursions and generating functions.
• Partitions and tableaux.
• Designs, Latin squares, combinatorial designs and projective geometries.
• Extremal combinatorics, asymptotic analysis.

Prerequisite: (MH1101 & MH1201 & MH1301) or (MH1802 & MH1201 & MH1301).
MH4301 - Set Theory and Logic 4 AU
• Partially-ordered sets, well-orderings and order-types, induction and recursion on ordinals, ordinal arithmetic, cardinals, cardinal arithmetic.
• Axiom of choice and its equivalences, axiom of determinacy.
• Propositional calculus, truth tables, validity and contradictions.
• Predicate calculus with equality, completeness and compactness theorems, Löwenheim-Skolem theorem.

Prerequisites: MH1300 and MH1301.
​MH4302 - Theory of Computing ​4 AU
• ​Models of computation and finitary representations.
• ​Formal languages and Chomsky's grammars.
• Finite automata, regular expressions, regular grammars, and their equivalence.
• Properties of regular languages: pumping lemma for regular languages and its applications.
• Pushdown automata, context free languages and context free grammars.
• Properties of context free languages: pumping lemma for context free languages.
• Turing machines: definition and construction for simple problems.
• Church-Turing thesis and computability.
• Uncountable numbers and the diagonalization argument.
• Computably enumerable sets and Post’s problem
• Nondeterministic Turing machines and the classes P and NP.
• Polynomial-time reductions and Cook’s Theorem.
• Satisfiability and other NP-complete problems.
• Co-NP space.

Prerequisites: MH1300 and MH1301 and (MH1402 or MH1403 or CZ001)

MH4310 - Coding Theory 4 AU Introduction to the theory of error-correcting codes, and their applications in data storage and telecommunication.
• Error detection, correction and decoding, Hamming distance.
• Basic facts about finite fields.
• Linear codes, Hamming weight, generator and parity-check matrices, encoding, and decoding.
• Bounds, Hamming codes, Golay codes, perfect codes, MDS codes.
• Construction of codes, Reed-Muller codes.
• Cyclic codes, generator polynomials, BCH codes, Reed-Solomon codes.
• Computer implementation of efficient coding and decoding.

Prerequisite: MH2200 ​and MH1301.
MH4311 - Cryptography 4 AU
• Classical ciphers, cryptanalysis, linear complexity.
• The Data Encryption Standard (DES).
• The RSA cryptosystem, primality testing and factorization of integers.
• Discrete logarithms.
• Signatures; the Digital Signature Standard.

Prerequisites: MH1301 and MH2200.
MH4312 - Topics in Mathematics of Information and Communication 4 AU Introduction to specialized advanced topics related to information theory, coding theory and cryptography. The choice of the topic depends on the instructor.

Prerequisite: division approval.
MH4320 - Computational Economics 4 AU
• Extensive-form games
• Strategic-form games and Domination
• Nash Equilibria and Maxmin Strategies
• Mixed Equilibria, Zero-sum Games
• Computing Equilibria
• Computing Equilibria and Nash’s Theorem
• Social Choice Theory
• Auctions
• VCG-Mechanisms
• Games of Incomplete Information
• Bayesian Equilibria
• Revenue Equivalence
• Stable Matchings​

Prerequisite: MH1200 and MH2500.
MH4500 - Time Series Analysis 4 AU Introduction to time series models and their applications in economics, engineering and finance.
• Trend fitting, autoregressive and moving average models, spectral analysis.
• Seasonality, forecasting and estimation.
• Use of computer package to analyze real data sets.

Prerequisites: MH2500, MH3500, and MH3510.
MH4501 - Multivariate Analysis 4 AU
• Distribution theory: multivariate normal distribution, Hotelling's T2 and Wishart distributions.
• Inference on the mean and covariance, principal components and canonical correlation.
• Factor analysis, discrimination and classification.

Prerequisites: MH2500, MH3500, and MH3510.
MH4510 - Statistical Learning and Data Mining​​ 4 AU
• Introduction to Data Analytics, Optimal Decision Rules
• K-Nearest Neighbors Methods, Linear Models for Regression
• Linear Models for Regression
• Generalized Linear Models for Classification
• Cross-Validation and Bootstrap Methods
• Subset selection, Ridge Regression and LASSO
• Artificial Neural Networks
• Classification and Regression Trees
• Ensemble Methods
• Support Vector Machines
• Association Analysis
• Further Topics
• Group Project Presentation​

Prerequisites: MH2500, MH3500, MH3510, and MH3511.
MH4511 - Sampling & Survey 4 AU
• Ratio and regression estimators under simple random sampling, separate and combined estimators for stratified random sampling.
• Systematic sampling and its relationship with stratified and cluster sampling.
• Further aspects of stratified sampling, cluster sampling with clusters of unequal sizes.
• Subsampling; multi-stage sampling.
• Complex sample designs.

Prerequisites: MH2500 and MH3500.
MH4512 - Clinical Trials​ 4 AU Introduction to the design and analysis of clinical trials, with emphasis on the statistical aspects.
• Phases of clinical trials.
• Objectives and endpoints, the study cohort, controls, randomization and blinding, sample size determination, treatment allocation.
• Monitoring trial progress: compliance effects, ethical issues, quality of life assessment.
• Data analysis involving multiple treatment groups and endpoints, stratification and subgroup analysis, intent to treat analysis, analysis of compliance data, surrogate endpoints, multi-centre trials.
• Good practice versus misconduct.

Prerequisites: MH3510 and MH3500.
MH4513 - Survival Analysis 4 AU
• Examples of survival data analysis.
• Types of censoring, parametric survival distributions (exponential, Weibull, lognormal), nonparametric methods, Kaplan-Meier estimator, tests of hypotheses.
• Graphical methods of survival distribution fitting, goodness of fit tests.

Prerequisites: MH2500, MH3500, and MH3510.
MH4514 - Financial Mathematics​ 4 AU Introduction to the mathematical concepts underlying pricing and hedging for financial derivatives.
• Assets, portfolios and arbitrage.
• Discrete-time models.
• Brownian motion and stochastic calculus.
• Option pricing and hedging.
• Relation with market data.

Prerequisites: MH2500 and MH3512.
Not available to students who have taken/are taking BA2202.
MH4600 - Algebraic Topology 4 AU
• Review of point-set topology.
• Homotopy.
• CW- or simplicial complexes.
• Fundamental group.
• Homology groups.

Prerequisites: MH3200 and MH3600.
MH4601 - Differential Geometry 4 AU
• Curves.
• Surfaces and curvature.
• Manifolds.
• Differential forms.

Prerequisites: MH3100 and MH3600.
MH4700 - Numerical Analysis II 4 AU
• Finite difference formulae, consistency of difference schemes, finite difference methods for ordinary differential equations.
• Classification of second-order partial differential equations, first and second order characteristics.
• Matrix method and von Neumann method for stability analysis, Lax's equivalence theorem for convergence, method of characteristics.
• Application to heat equation, wave equation and Poisson's equation.

Prerequisites: MH3700 and MH3110. (MH4110 is useful but not required.)
MH4701 - Mathematical Programming​ 4 AU
• One-dimensional optimization: sectioning methods, Newton’s method.
• Unconstrained optimization: optimality conditions, steepest descent method, Newton descent method.
• Set-constrained optimization: optimality conditions, conditional gradient method
• Constrained optimization: Lagrange multiplier theory, Karush-Kuhn-Tucker theory, augmented Lagrangian method, barrier method.

Prerequisites: (MH2100 & MH3701) or (MH1803 & MH3701).
MH4702 - Probabilistic Methods in OR​​ 4 AU Introduction to probabilistic methods used in operations research and statistics.
• Queueing: basic models, performance analysis, simulation of queueing systems.
• Stochastic optimization: Stochastic programming, modeling and algorithms, Markov decision process, stochastic approximation.

Prerequisites: (MH2500 & MH3701 & MH3512).
MH4710 - Topics in Scientific Computing 4 AU Specialized advanced topics in scientific computation and continuous applied mathematics. The choice of the topic depends on the instructor.

Prerequisite: division approval.
MH4711 - Mathematical Modeling in Imaging, Vision and Graphics 4 AU
• Calculus of variations.
• Convexity and differential geometry, level set method; phase field method.
• Image denoising and debluring.
• Image segmentation.
• Image inpainting and registration.
• Curve reconstruction and smoothing.
• Surface reconstruction and smoothing.

Prerequisites: (MH2100 & MH3100) or (MH1803 & MH3100).
MH4720 - Logistics and Supply Chain Management 4 AU
• Overview of supply chains: components of a supply chain, material and information flow, supplier-retailer-customer interaction, e-business.
• Inventory and materials management: economic order quantity model, Lot sizing models, models with uncertain demands, MRP/JIT.
• Facility location and transportation - single-source capacitated facility location, vehicle routing problems with equal, unequal demands and time-window constraints.
Prerequisite: MH2500 and MH3701.
MH4730 - Mathematics in Biology and Medicine 4 AU Mathematical models and methods often used in bioinformatics, computational biology, and medicine. Topics covered include:
• Model-based data clustering.
• Maximum likelihood method.
• Hidden Markov models.
• Regression analysis.

Prerequisite: MH2500.
MH4900 - Final Year Project​ 8 AU Semester-long research course on an advanced topic, under the supervision of a faculty member, leading to a research thesis. Must be taken over two consecutive semesters.

Prerequisite: division approval.
Not available to students who have taken/are taking MH4903.
MH4901 - Professional Attachment

For students who matriculated in AY15/16 and earlier.
8 AU 12-week job placement for acquiring practical working experience and exposure to the workplace.
This course is offered on a Pass/Fail basis only.

Prerequisite: division approval.
MH4903 - Professional Internship

For students who matriculated in AY16/17 and later.
11 AU 22-week job placement for acquiring practical working experience and exposure to the workplace.
This course is offered on a Pass/Fail basis only.

Prerequisite: division approval.

Not available to students who have taken/are taking MH4900 or MH4907.
MH4907 - Professional Attachment

For students who matriculated in AY16/17 and later.
6 AU 12-week job placement for acquiring practical working experience and exposure to the workplace.
This course is offered on a Pass/Fail basis only.

Prerequisite: division approval.

Not available to students who have taken/are taking MH4903.
MH4910 - Undergraduate Research Experience in Mathematical Sciences I 4 AU Research on a specific mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval.
MH4911 - Undergraduate Research Experience in Mathematical Sciences II 4 AU Further research on a specific mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval and MH4910.
MH4920 - Supervised Independent Study I 4 AU Independent reading on a mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval.
MH4921 - Supervised Independent Study II 4 AU Further independent reading on a mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval and MH4920.
MH4930 - Special Topics in Mathematics 4 AU Advanced topics in mathematics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

Prerequisite: division approval.
MH4931 - Special Topics in Applied Mathematics 4 AU Advanced topics in mathematics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

Prerequisite: division approval.
MH4932 - Special Topics in Statistics 4 AU Advanced topics in statistics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

Prerequisite: division approval.
MH8300 - It’s a Discreetly Discrete World: Mathematics in Real-life Applications 3 AU Introduction to some simple and yet useful mathematics. Important applications of mathematics are discussed, which demonstrate the influence of mathematics on our everyday life. Topics covered include:
• Coding Theory: detecting and correcting errors in data, basic modular arithmetic used in the design of codes, basic issues in theory and applications, real-life applications such as NRIC numbers, ISBN, CD, telecommunications, etc.
• Cryptography: ensuring security of information, basic issues and use in applications such as electronic transactions and communication, and the RSA cryptosystem.
• Graph Theory: basic notions and algorithms, the travelling salesman problem, computational complexity, brute force methods, tour construction heurists, and applications.
• Probability and statistics: examples, visualization, counterintuitive results, coincidences, paradoxes.
• Searching for information on the web: applications of probability and linear algebra, especially eigenvalues, underlying search engines such as Google.

Prerequisite: AO or H1 level Mathematics or equivalent.
MH8500 - Tackling the Odds: Inside Statistics 3 AU Overview of statistics and its applications in other disciplines, with emphasis on statistics methodology and how to evaluate statistical studies that students may encounter in some other courses, their future career, or everyday life. Topics covered include:
• Overview of statistics.
• Measurement.
• Visual displays.
• Data descriptions.
• Probability and risk.
• Correlation and causality.
• Statistical methodologies.
• Statistical modeling.

Prerequisite: AO or H1 level Mathematics or equivalent.
MH9000 - Mathematical Problem-Solving 2 AU A course about solving challenging non-standard problems from various areas, including calculus, linear algebra, algebra, differential equations, probability, discrete mathematics, etc., with the aim of developing creative thinking and exposition skills.

Prerequisites: MH1101, MH1200, and MH1301.
MH9100 - Advanced Investigations in Calculus I 1 AU A course where students are given challenging problems in calculus. Supplement to MH1100 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1100.
MH9101 - Advanced Investigations in Calculus II 1 AU A course where students are given challenging problems in calculus. Supplement to MH1101 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1101.
MH9102 - Advanced Investigations in Calculus III 1 AU A course where students are given challenging problems in calculus. Supplement to MH2100 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH2100.
MH9200 - Advanced Investigations in Linear Algebra I 1 AU A course where students are given challenging problems in linear algebra. Supplement to MH1200 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1200.
MH9201 - Advanced Investigations in Linear Algebra II 1 AU A course where students are given challenging problems in linear algebra. Supplement to MH1201 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1201.
MH9300 - Advanced Investigations in Discrete Mathematics 1 AU A course where students are given challenging problems in discrete mathematics and number theory. Supplement to MH1301 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1301.
PS9888 - Making and Tinkering 4 AU A course in which students apply their scientific knowledge to identify and solve open-ended real life problems. The end-product of the investigation is a novel object designed and created by the students.

Offered as an Unrestricted Elective during Special Terms.
Prerequisites: approval by the Chair's Office. The course must be done in groups, containing students from more than 1 School.