## Course Information

# Mathematical Sciences Courses

- Ordinary Courses
- Projects and Internships
- Supervised Study and Special Topics
- General Education Requirement (GER) Courses
- Special Problem-Solving Modules
- Graduate-Level Courses

## Ordinary Courses

Introductory mathematics course for CN Yang scholars and Renaissance Engineering students. Topics include: limits and continuity; differentiability and differentiation rules; critical points, the mean value theorem, and l'Hospital's rule; inverse functions; trigonometric, logarithm and exponential functions; the Riemann integral; the Fundamental Theorems of Calculus; techniques of integration; infinite sequences and infinite series, power series and convergence criteria, and Taylor series; and ordinary differential equations.

*Only offered to CN Yang scholars and Renaissance Engineering students.**Mutually exclusive with MH1100, MH1101, MH1800, and MH1801.*

Second mathematics course for CN Yang scholars and Renaissance Engineering students. Topics include: 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; double and triple integrals; vector calculus, line integrals, Green's Theorem, surface integrals, Gauss's divergence theorem, and Stokes' Theorem.

*Prerequisite: CY1601 or RE1001.**Mutually exclusive with MH1200 or MH2100.*

Introductory course on differential and integral calculus. Topics include: 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, and L'Hospital's Rule; antidifferentiation, indefinite integrals, substitution rule, and integration by parts.

*Prerequisite: A level Mathematics or equivalent.**Mutually exclusive with MH1800.*

Further topics in calculus. Topics include: definite integrals; the Fundamental Theorems of Calculus; area of plane regions, volumes of solids, length of arcs; the Mean Value Theorem for integrals; techniques of integration, numerical integration, and improper integrals; monotonic and bounded sequences, Newton's method, infinite series, tests for convergence and divergence, alternating series, and absolute/conditional convergence criteria; differentiation and integration of power series, Taylor series, binomial series, and Fourier series.

*Prerequisite: MH1100**Mutually exclusive with MH1801.*

Introductory course on linear algebra. Topics include: 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.**Mutually exclusive with MH2800.*

Further topics in linear algebra. Topics include: 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.

Introductory course on core mathematical concepts, including logic and the theory of sets. Topics include: 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.*

Introduction to discrete mathematics, including: basics of counting; the inclusion-exclusion principle; the pigeonhole principle; permutations and combinations; the binomial theorem; recurrence relations and linear recurrence relations; graph concepts such as Shortest-, Euler-, Hamilton-Paths and Cycles, coloring, planarity, weighted graphs, and directed graphs.

*Prerequisite: A or H2 level Mathematics or equivalent.*

Core course introducing fundamentals of programming 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. Topics include: Python basics, lists, the NumPy module, strings, input/output, selection statements, loop statements, functions, errors and debugging, recursion, algorithm complexity, sorting algorithms, and plotting.

*Prerequisite: A or H2 level Mathematics or equivalent.*

Further topics in algorithms and computing. Topics include: the concept of an algorithm; the basic structures of a C/C++ program; 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.*

Systematic introduction to data structures and algorithms for efficient computer programs. Topics include: 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; and elementary analyses of algorithmic complexity.

*Prerequisite: PS0001*

First of two courses on calculus for students in the sciences. Applications and computer-based learning are included. Topics 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; and the Fundamental Theorem of Calculus.

*Prerequisite: A or H2 level Mathematics or equivalent.**Mutually exclusive with MH1100 and MH1101.*

Second of two courses on calculus for students in the sciences. Applications and computer-based learning are included. Topics include: differential equations; first-order and second-order linear differential equations; techniques of solving differential equations, and applications; series and power series; Taylor series; Fourier series.

*Prerequisite: MH1800 or equivalent.**Mutually exclusive with MH1101.*

Introductory course in calculus, for students majoring in the physical sciences. Topics include: types of Numbers; functions and graphs; algebraic, trigonometric, logarithmic and exponential functions and identities; complex numbers; limits and continuity; derivatives and techniques of differentiation; applications of differentiation; indefinite integrals and definite integrals; the Fundamental Theorem of Calculus; techniques of integration; applications of integration in science; differential equations; and power series.

*Prerequisite: A or H2 level Mathematics or equivalent.**Mutually exclusive with MH1800 and MH1801.*

Additional topics in calculus, for students majoring in chemistry. Topics include: Cartesian and spherical coordinates; complex numbers; vectors; linear algebra and matrices; summation, series, and expansions of functions; Fourier series and Fourier transforms.

*Prerequisite: MH1802.*

Further topics in calculus, including: sets and functions; limits and continuity; differentiation and optimization; the Riemann integral; the Fundamental Theorem of Calculus; applications of integration; methods of integration; series, power series, and Taylor series; and ordinary differential equations

*Mutually exclusive with CY1601, MH1100, MH1101, MH1802, and RE1011.*

Intermediate course in calculus. Topics include: 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 and triple integrals; line integrals, Green's theorem, surface integrals, the Gauss divergence theorem, and Stokes' theorem.

*Prerequisite: MH1101 or MH1802 or MH1805.**Mutually exclusive with MH2800.*

Introductory course on group theory, with emphasis on symmetry groups of geometric structures. Topics include: 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; and puzzles such as the 15-puzzle and Rubik's cube.

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.

Introductory course on probability and statistics. Topics include: 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; and 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.*

Techniques in linear algebra and multivariable calculus, and their applications. This course includes computer-based learning. Topics 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; and constrained and unconstrained optimization.

*Prerequisite: MH1800 or equivalent.**Mutually exclusive with MH1200, MH1201, and MH2100.*

Introduction to complex numbers and their applications in physics and the other sciences. Topics include: 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; and Green's functions methods.

*Prerequisites: (MH1801 and MH2800) or (MH1101 and MH1200) or (MH1802 and MH1803 and MH1200) or (MH1802 and MH1803 and MH2802) or (CY1601 and CY1602).**Mutually exclusive with MH3101.*

Introduction to linear algebra and its applications in physics and the other sciences. Topics include: vector algebra and analytical geometry; linear spaces; linear transformations and matrices; eigenvalues and eigenvectors; and applications of linear algebra to problems in physics and computing.

*Prerequisite: A or H2 level Mathematics or equivalent.*

Introduction to real analysis. Topics include: 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, and local uniform convergence of power series.

Introduction to complex analysis. Topics include: analytic functions of one complex variable, the 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, and singularities of analytic functions; the residue theorem and the calculus of residues; Fourier transforms, inversion formula, convolution, and Parseval's formula.

The course builds on Calculus and Linear Algebra. It aims to equip students with useful solution methods for solving various types of ordinary differential equations (ODEs), introduce the fundamental theory of ODEs, and develop skills for modeling real phenomena by ODEs.

*Prerequisite: MH2100 or CY1602*

Introduction to modern algebra, including basic algebraic structures such as groups, rings and fields. Topics 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, and factorization of polynomials over a field.

Introduction to basic number theory, including modern applications. Topics include: modular arithmetic; the Chinese remainder theorem, Fermat's little theorem, and Wilson's theorem; number-theoretic functions such as the τ, σ, Euler's φ-function; the 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, factorization of integers, and the RSA cryptosystem.

*Prerequisite: MH1300.*

Topics in graph theory, including: 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; and the use of software to solve graph-theoretic problems.

Introduction to game theory. Topics include: 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.**Mutually exclusive with HE302 / HE3002.*

Applications of algorithms. Topics include: 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; and 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).*

Further topics in statistics, including: random samples, sample mean and sample variance, distributions derived from the normal distribution, the central lmit theorem; parameter estimation and quality criteria for parameter estimators; the construction of good estimators; method of moments and maximum likelihood method; asymptotic properties of estimators, Cramer-Rao bound and efficient estimators; confidence intervals for estimators; hypothesis testing and Fisher-type tests; Neyman-Pearson tests and Neyman-Pearson Lemma.

*Prerequisite: MH2500.*

Introduction to regression analysis, one of the most widely-used statistical techniques. Topics include: 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.

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 or BS1008*

Introduction to the theory of stochastic processes, including: gambling problems; random walks; discrete-time Markov chains; first step analysis (hitting probabilities and mean hitting times); classification of states; branching processes; and continuous-time Markov chains.

*Prerequisite: MH2500.*

This course investigates deep learning from the perspectives of several mathematical theories: numerical optimisation, statistical learning, function approximation, and coding theory. The aim is to shed some light on why and under what circumstances deep learning can be expected to work well - or not.

This is a first introduction to topology and calculus on manifolds. The tools introduced in this course are the natural framework for the generalization of the ideas that you learnt in Calculus I, II, and III to infinite-dimensional and non-Euclidean spaces. These methods open the door to other fields in mathematics like algebraic topology, functional analysis, differential/Riemannian/symplectic/Poisson geometry, or Lie theory, to name a few. They also have strong ties with important applications in the physical sciences and engineering like dynamical systems, mechanics, symmetry analysis, or control theory.

The aim of this course is to enable you to formulate and solve mathematical problems using the ideas and the formalism coming from topology and global analysis.

Introduction to the theory and applications of numerical approximation techniques. Topics include: commonly used numerical algorithms; computational errors; numerical methods for solving systems of linear equations; iterative methods for systems of linear equations; polynomial interpolation; numerical integration; and numerical solutions of nonlinear equations.

*Prerequisites: (MH1200 & MH1201) or (MH1800 & MH2800) or CY1602 or MH2802.*

Introduction to the mathematics of optimization. Topics include: geometric simplex method; algebraic simplex method in tabular form; revised simplex method; the network simplex method; linear programming duality; sensitivity and post-optimality analysis;and Lagrange duality and the Karush-Kuhn-Tucker conditions.

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 and MH3100) or (CY1602 and MH3100) or (MH1803 and MH3100).*

Advanced course on partial differential equations. Topics include: 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; the heat equation, the weak maximum principle for parabolic equations, Cauchy problem for heat equation, regularity of solutions to heat equation; the 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.)*

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, and finite fields.

Partially-ordered sets, well-orderings and order-types, induction and recursion on ordinals, ordinal arithmetic, cardinals, cardinal arithmetic; the axiom of choice and its equivalences; axiom of determinacy; propositional calculus, truth tables, validity and contradictions; predicate calculus with equality, completeness and compactness theorems; the Löwenheim-Skolem theorem.

Models of computation and finitary representations. Topics include: 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; the 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; sSatisfiability and other NP-complete problems;and Co-NP space.

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

The definition of a linear code, its dimension and its length; generator matrix, parity check matrix and dual code; Hamming distance/weight; Hamming codes; perfect codes; Golay codes; maximum distance separable (MDS) codes; Reed-Mueller codes; BCH codes; Reed-Solomon codes; and bounds on code parameters.

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.*

Extensive-form games; strategic-form games and domination; Nash equilibria and maxmin strategies; mixed equilibria; zero-sum games; computing equilibria; Nash’s theorem; social choice theory; auctions; VCG mechanisms; games of incomplete information; Bayesian equilibria; revenue equivalence; and stable matchings.

Introduction to time series models and their applications in economics, engineering and finance. Topics include: trend fitting, autoregressive and moving average models, spectral analysis; seasonality, forecasting and estimation; and the use of computer package to analyze real data sets.

Introduction to data analytics; optimal decision rules; K-nearest neighbors methods; 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.

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.

Introduction to the design and analysis of clinical trials, with emphasis on the statistical aspects. Topics include: 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.

Introduction to survival 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.

Discrete-Time martingales; assets, portfolios, and arbitrage; discrete-time models; pricing in discrete time; hedging in discrete time; Brownian motion; stochastic calculus; the Black-Scholes equation; Martingale approach to pricing and hedging; estimation of volatility; basic numerical methods.

Topological data analysis models, including simplicial complex, nerve theorem, homology, cohomology, filtration, persistent homology, Morse theory, Hodge-Laplacian, Reeb graph. Geometric data analysis models, including multidimensional scaling, isomap, diffusion map, spectral graph, manifold learning, differential forms. Geometry and topology based learning, including data representation, feature engineering, molecular/chemical descriptors, graph neural network.

Generating random numbers and random variables, generating Brownian motion and other diffusion processes, variance reduction techniques, introduction to futures, options, and other derivatives, pricing exotic options with simulations, estimating sensitivities of derivatives with simulations, and applications in risk management

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.)*

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, and barrier method.

Introduction to probabilistic methods used in operations research and statistics. Topics include: basic models of queueing, performance analysis, simulation of queueing systems; stochastic programming, modeling and algorithms for stochastic optimization, Markov decision process, and stochastic approximation.

Specialized advanced topics in scientific computation and continuous applied mathematics. The choice of the topic depends on the instructor.

*Prerequisite: division approval.*

Calculus of variations; convexity and differential geometry, level set method; phase field method; image denoising and deblurring; image segmentation; image inpainting and registration; curve reconstruction and smoothing; and surface reconstruction and smoothing.

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.

Mathematical models and methods often used in bioinformatics, computational biology, and medicine. Topics include: model-based data clustering; maximum likelihood method; hidden Markov models; regression analysis.

*Prerequisite: MH2500.*

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.**Mutually exclusive with MH4903.*

12-week job placement for acquiring practical working experience and exposure to the workplace.

*Prerequisite: division approval.*

###### MH4905

Overseas Entrepreneurship Programme

10 AU | For students who matriculated in AY19/20 and after

###### MH4906

Overseas Entrepreneurship Programme

11 AU | For students who matriculated in AY16/17 to AY18/19

Detailed Course Information6-month internship with a startup company, located in a major global innovation hub (remote internship can be arranged if there are travel restrictions).

The Overseas Entrepreneurship Programme (OEP) allows entrepreneurially inclined students to intern at a startup company, so as to experience the process and challenges that entrepreneurs face in building and growing their companies. The startup companies are located in major global innovation hubs such as Silicon Valley, New York, Shanghai and Berlin. Students can also opt to work for a startup company that is based in Singapore, but with ambitions to grow their business globally.

12-week job placement for acquiring practical working experience and exposure to the workplace.

*Prerequisite: division approval.**Mutually exclusive with MH4903.*

Research on a specific mathematical topic, under the supervision of a faculty member.

*Prerequisite: division approval.*

Further research on a specific mathematical topic, under the supervision of a faculty member.

*Prerequisite: division approval and MH4910.*

###### MH4913

Professional Attachment

5 AU | Pass/Fail | For students who matriculated in AY21/22 and later.

Independent reading on a mathematical topic, under the supervision of a faculty member.

*Prerequisite: division approval.*

Further independent reading on a mathematical topic, under the supervision of a faculty member.

*Prerequisite: division approval and MH4920.*

Advanced topics in mathematics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

*Prerequisite: division approval.*

Advanced topics in mathematics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

*Prerequisite: division approval.*

Advanced topics in statistics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

*Prerequisite: division approval.*

Introduction to the key concepts of cryptography. Topics include: encryption algorithms; message authentication codes; authenticated encryptions; public-key schemes; random number generators; digital signatures; and hash functions.

*Prerequisite: AO or H1 level Mathematics or equivalent.*

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 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.*

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 include: measurement; visual displays; data descriptions; probability and risk; correlation and causality; statistical methodologies; statistical modeling.

*Prerequisite: AO or H1 level Mathematics or equivalent.*

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: (MH1100, MH1101, MH1200, MH1201 and MH1300) OR (CY1601, CY1602, MH1201, and MH1300) OR division approval.*

This course provides an introduction to foundational theory and algorithms of discrete mathematics. Instead of covering a variety of approaches and algorithms, the course focusses on those methods that have proved to be superior to alternative techniques and have become "workhorses" in modern applied mathematics. For instance, the theory of the Simplex Algorithm is worked out carefully, while other less practical or less efficient algorithms for linear programming are not covered. Aside from the algorithmic aspect, the course has a substantial theoretical component: graph theory, proofs of correctness of algorithms, and the structure of polyhedra.