CY1601/RE1011  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/RE1012  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 GramSchmidt 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 RE1011.
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, firstderivative and secondderivative 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 GramSchmidt 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

 Basic of Counting, InclusionExclusion Principle
 Pigeonhole Principle, Permutations and Combinations
 Binomial Theorem, Generalized Permutations and Combinations
 Generalized Permutations
 Recurrence Relations, Linear Recurrence Relations
 Linear Recurrence Relations
 Review of Counting and Recurrence Relations
 Graphs: Examples, Definitions, Representation
 Traversing Graphs: Shortest, Euler, HamiltonPaths and Cycles
 Graph Properties: Coloring, Planarity
 Weighted Graphs
 Directed Graphs
 Review of Graphs
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 problemsolving, it develops the ability to think algorithmically, which is essential for any professional working in an increasingly computerdriven 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 divideandconquer 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, divideandconquer 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 computerbased 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 computerbased learning are included. Topics covered include:
 Differential equations; firstorder and secondorder 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.

MH1805  Calculus 
4 AU 
 Sets and functions
 Limits and continuity, onetoone and inverse functions
 Differentiation and optimization
 Definition of Riemann Integral, Fundamental Theorem of Calculus, applications
of integration.
 Methods of integration
 Series, Power Series, Taylor Series
 Elementary theory, and methods of Ordinary Differential
Equations
Mutually exclusive with CY1601, MH1100, MH1101, MH1802 and RE1011

MH2100  Calculus III

4 AU

Intermediate course in calculus.
 Parametric equations; polar coordinates.
 Vectorvalued functions, calculus of vectorvalued 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 15puzzle and Rubik's cube).
Prerequisite: MH1200 or CY1602

MH2401  Algorithms and Computing III

2 AU

Application of computing skills and previouslylearnt mathematical topics (Linear Algebra, Calculus, Discrete Mathematics, etc.) for solving realworld 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 computerbased learning. Topics covered include:
 Systems of linear equations.
 Matrices and determinants.
 Vectors in 2 and 3dimensional 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 CauchyRiemann 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, BolzanoWeierstrass theorem, Cauchy sequences, infinite series, double summations, products of infinite series.
 Limits of functions, continuity, uniform continuity, intermediate value theorem, extremevalue 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, CauchyRiemann 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 nonhomogeneous equations, variation of parameters, principle of superposition.
 Second order equations, Wronskian, Abel's formula, variation of parameters, exact equations, adjoint and selfadjoint 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.
 Numbertheoretic functions: τ, σ, Euler's φfunction, 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 ErdösStone theorem.
 Graph colouring and the fourcolour theorem.
 Ramsey theory.
 Probabilistic methods in graph theory.
 Use of software to solve graphtheoretic problems.
Prerequisites: MH1201 and MH1301 or MH2802.

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, subgameperfect equilibrium, finitely and infinitelyrepeated 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 nperson 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





 Review
of probability
 Random
samples, sample mean and sample variance, distributions derived from the normal
distribution, Central Limit Theorem and its significance for statistics
 Introduction
to parameter estimation, quality criteria for parameter estimators
 Constructing
good estimators: method of moments and maximum likelihood method
 Asymptotic
properties of estimators, CramerRao bound and efficient estimators
 Confidence
intervals for estimators
 Introduction
to hypothesis testing and Fishertype tests
 NeymanPearson
tests and NeymanPearson Lemma

Prerequisite: MH2500.

MH3510  Regression Analysis

4 AU

Introduction to regression analysis, one of the most widelyused statistical techniques.
 Simple and multiple linear regression, nonlinear regression, analysis of residuals and model selection.
 Oneway and twoway 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 pointset topology.
 Continuous functions.
 Surfaces.
 Euler characteristics.
 Maps and graphs on surfaces.
 Vector fields on surfaces.
 Knots.
 Fixedpoint 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 goodnessoffit tests.
Prerequisite: MH2500.

MH3512  Stochastic Processes

4 AU

 Introduction and motivations
 Probability background
 Gambling problems
 Random walks
 Discretetime Markov chains
 First step analysis (hitting probabilities)
 First step analysis (mean hitting times)
 Classification of states
 Longrun behavior of Markov chains
 Branching processes
 Continuoustime Markov chains (definitions and properties)
 Continuoustime 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 postoptimality analysis
 Lagrange duality and the KarushKuhnTucker 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

 Firstorder equations, quasilinear equations, general firstorder 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

Sums, products and radicals of ideals of communicative
rings
 Nilradical and Jacobson radical, Nakayama's lemma
Noetherian rings and Artinian rings, Hilbert's basis
theorem
Prime ideals, maximal ideals, Localization of rings and
Hilbert's Nullstellensatz
Grobner basis and Buchberger's algorithm
Modules, submodules, quotient modules
Isomorphism theorems for modules
Direct sum of modules, and fundamental theorem for
finitely generated abelian groups
Projective modules, injective modules and short exact
sequences
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

 Partiallyordered sets, wellorderings and ordertypes, 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, LöwenheimSkolem 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.
 ChurchTuring
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.
 Polynomialtime reductions and Cook’s Theorem.
 Satisfiability and other NPcomplete problems.
 CoNP space.
Prerequisites: MH1300 and MH1301 and (MH1402 or MH1403 or CZ001)

MH4310  Coding Theory

4 AU

 The definition of a linear code, its dimension and its length.
 The definition of generator matrix, parity check matrix and dual code.
 The definition of Hamming distance/weight, and how to compute it.
 How to encode and decode.
 Hamming codes
 Perfect codes
 Golay codes
 Maximum distance separable (MDS) codes
 ReedMueller codes
 BCH codes
 ReedSolomon codes
 Bounds on code parameters
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

 Extensiveform games
 Strategicform games and Domination
 Nash Equilibria and Maxmin Strategies
 Mixed Equilibria, Zerosum Games
 Computing Equilibria
 Computing Equilibria and Nash’s Theorem
 Social Choice Theory
 Auctions
 VCGMechanisms
 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
 KNearest Neighbors Methods, Linear Models for Regression
 Linear Models for Regression
 Generalized Linear Models for Classification
 CrossValidation 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; multistage 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, multicentre 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, KaplanMeier 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 and motivations
 DiscreteTime Martingales
 Assets, Portfolios, and Arbitrage
 DiscreteTime Model
 Pricing in Discrete Time
 Hedging in Discrete Time
 Brownian Motion
 Stochastic Calculus
 The BlackScholes PDE
 Martingale Approach to Pricing and Hedging
 Estimation of Volatility
 Basic Numerical Methods
 Revisions and additional tutorial exercises
Prerequisites: MH2500 and MH3512.

MH4600  Algebraic Topology

4 AU

 Review of pointset 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 secondorder 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

 Onedimensional optimization: sectioning methods, Newton’s method.
 Unconstrained optimization: optimality conditions, steepest descent method, Newton descent method.
 Setconstrained optimization: optimality conditions, conditional gradient method
 Constrained optimization: Lagrange multiplier theory, KarushKuhnTucker 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, supplierretailercustomer interaction, ebusiness.
 Inventory and materials management: economic order quantity model, Lot sizing models, models with uncertain demands, MRP/JIT.
 Facility location and transportation  singlesource capacitated facility location, vehicle routing problems with equal, unequal demands and timewindow 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:
 Modelbased data clustering.
 Maximum likelihood method.
 Hidden Markov models.
 Regression analysis.
Prerequisite: MH2500.

MH4900  Final Year Project

8 AU

Semesterlong 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. Click here for more information about the Final Year Project.
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

12week job placement for acquiring practical working experience and exposure to the workplace.
This course is offered on a Pass/Fail basis only. Click here for more information about the Professional Attachment.
Prerequisite: division approval.

MH4903  Professional Internship
For students who matriculated in AY16/17 to AY18/19

11 AU

22week job placement for acquiring practical working experience and exposure to the workplace.
This course is offered on a Pass/Fail basis only. Click here for more information about the Professional Internship.
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

12week job placement for acquiring practical working experience and exposure to the workplace.
This course is offered on a Pass/Fail basis only. Click here for more information about the Professional Attachment.
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.

MH4912  Professional Internship
For students who matriculated in AY19/20 and later. 
10 AU 
20week job placement for acquiring practical working experience and exposure to the workplace. This course is offered on a Pass/Fail basis only. Click here for more information about the Professional Internship.
Prerequisite: division approval. Not available to students who have taken/are taking MH4900 or MH4907. 
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 Reallife 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, reallife 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.

MH8301  Modern Cryptography: RealWorld Applications and Impact 
3 AU 
 Encryption algorithms
 Message authentication codes
 Authenticated encryptions
 Publickey schemes
 Random number generators
 Digital signatures
 Hash functions
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 ProblemSolving

2 AU

A course about solving challenging nonstandard 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: (MH1100, MH1101, MH1200, MH1201and MH1300) OR
(CY1601, CY1602, MH1201, MH1300) OR
Approval by the Division 
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 openended real life problems. The endproduct 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. For more information, visit the course website. For enquires, email PS9888@ntu.edu.sg.
