First Year Fall Semester
COMP 105 R Pascal Programming
H&SS E Humanities and Social Science (1) LANG 001 Language Skills Enhancement I
MATH 101 C Multivariable Calculus MATH 11 1 C Linear Algebra
PHYS 121 R Electricity and Magnetism
16 credits
(1) Students excused from this course by the Language Centre will replace it with a Humanities and Social Science course.
Spring Semester
ELEC 112 R Signals, Circuits and Systems H&SS E Humanities and Social Science MATH 102 C Introduction to Analysis MATH 151 C Differential Equations and
Applications
PHYS 124 R Optics, Waves and Particles
Fall Semester H&SS MATH 201 MATH 211 MATH 221 SClE
Humanities and Social Science Real Analysis
Abstract Algebra Differential Geometry Physics or Biology
19 credits Second Year
18 credits Spring Semester
COMP Computer Science
MATH 202 Integration Theory
MATH 212 Group Representation Theory MATH 223 Topology
SB&M Business & Management
Third Year Fall Semester
ENGG MATH MATH SB&M (1) SClE
Engineering Elective Mathematics Mathematics
Business and Management Science Elective
18 credits
17 credits
(1) Students will choose a course offered by any department in the School of Science other than the Department of Mathematics.
Spring Semester Spring Semester ENGG
( 1 ) FREE ( 1 ) FREE H&SS MATH
Engineering Open Elective Open Elective
Humanities and Social Science Mathematics
ELEC 11 2 R Signals, Circuits and Systems H&SS E Humanities and Social Science MATH 102 C Introduction to Analysis MATH 151 C Differential Equations and
Applications
PHYS 124 R Optics, Waves and Particles 16 credits
19 credits
( 1 ) Students will choose a course offered by any department not in the School of Science.
A minimum of 104 credits is required for the Pure Mathematics Option. A student's choice of electives may result in this minimum being exceeded.
MATHEMATICAL APPLICATIONS (Physical Science) OPTION First Year
Fall Semester
COMP 105 R Pascal Programming [O-1-2:1]
H&SS E Humanities and Social Science (3-0-0131 ( 1 ) LANG 001 Language Skills Enhancement I [O-3-2101
MATH 101 C Multivariable Calculus [3-1-0:4]
MATH 1 1 1 C Linear Algebra [3-1-0141
PHYS 121 R Electricity and Magnetism [3-0-3141 16 credits
( 1 ) Students excusedfrom this course by the Language Centre will replace itwith a Humanities and Social Science course.
Second Year Fall Semester
H&SS Humanities and Social Science [3-0-0131
MATH 201 Real Analysis [3-1-0:4]
MATH 251 Function of a Complex Variable [3- 1 -0 :4]
& Applications
( 1 ) PHSC Physical Sciences [3-0-0131
PHSC Physical Sciences [3-0-0:3]
17 credits ( 1 ) The courses identified as Physical Sciences (PHSC) will be selected in
consultation with the student's academic adviser.
Spring Semester
ENGG Engineering [3-0-0131
MATH 202 Integration Theory [3-1-0:4]
MATH 252 Applied Partial Differential Equations [3-1-0141
PHSC Physical Sciences [3-0-0131
SB&M Business & Management [3-0-0131
17 credits
Third Year MATHEMATICAL APPLICATIONS (Computer Science) OPTION Fall Semester
ENGG (1) FREE
MATH PHSC SB&M
Engineering Open Elective Mathematics Elective Physical Sciences Business & Management
16 credits (1) Students will choose acourse offered by any department not in the School of
Science.
Spring Semester
(1) FREE Open Elective [3-0-0:3]
H&SS Humanities and Social Science [3-0-0131
MATH Mathematics Elective [3-1-0:4]
PHSC Physical Sciences [3-0-0131
PHSC Physical Sciences [3-0-0 :3]
16 credits (1) Students will choose acourse offered by any department not in the School of
Science.
A minimum of 101 credits is required for the physical sciences option. A student's choice of electives may result in this minimum being exceeded.
First Year Fall Semester
COMP 105 C Pascal Programming [O-1-2:1]
COMP 102 C Programming Techniques [3-0-1:3]
H&SS E Humanities and Social Science [3-0-0131 (1) LANG 001 Language Skills Enhancement I [0-3-2:0]
MATH 101 C Multivariable Calculus [3-1-0141
MATH 111 C Linear Algebra [3-1-0141
15 credits
(1) Students excused from this course by the Language Centre will replace it with a course in Humanities and Social Science.
Spring Semester
COMP 171 C Data Structures COMP 191 C Computer Organization H&SS E Humanities and Social Science MATH 102 C Introduction to Analysis MATH 132 C Discrete Structures
Second Year Fall Semester
COMP Computer Science
COMP Computer Science
H&SS Humanities and Social Science MATH 201 Real Analysis
MATH 211 Abstract Algebra
19 credits
-
17 credits
Spring Semester MATHEMATICAL APPLICATIONS (Business & Management) OPTION
COMP Computer Science
COMP Computer Science
MATH 202 Integration Theory MATH 231 Numerical Analysis SB&M Business & Management
Third Ywr Fall Semester
COMP Computer Science
(1) ENGG Engineering Elective
MATH Mathematics Elective
SB&M Business & Management
SClE Science Elective
17 credits
--
16 credits (1) Students will choose a course offered by any department in the School of
Engineering other than the Department of Computer Science.
Spring Semester
COMP Computer Science [3-0-0:3]
(1) ENGG Engineering Elective [3-0-0:3]
H&SS Humanities and Social Science [3-0-0:3]
MATH Mathematics Elective (3-1-0:4]
SClE Science Elective [3-0-0131
16 credits (1) Students will choose a course offered by any department in the School of
Engineering other than the Department of Computer Science.
A minimum of 100 credits is required for the computer science option. The choice of electives may result in students exceeding this minimum.
First Year Fall Semester
ACCT 101 C Financial Accounting [2-2-0:3]
ECON 1 1 1 C Microeconomics [4-0-0141
H&SS E Humanities and Social Science [3-0-0131 (1) LANG 001 Language Skills Enhancement I [0-3-2:0]
MATH 101 C Multivariable Calculus [3-1-0141
MATH 1 1 1 C Linear Algebra [3-1-0:4]
18 credits (1) Students excused from this course by the Language Centre will replace itwith
a Humanities and Social Science course.
Spring Semester
ACCT 122 C Managerial Accounting ECON 121 C Macroeconomics
H&SS E Humanities and Social Science MATH 102 C Introduction to Analysis MATH 244 C Applied Statistics
Second Year Fall Semester
COMP 105 Pascal Programming
H&SS Humanities and Social Science MATH 201 Real Analysis
MATH Mathematics Elective
SB&M Business and Management SB&M Business and Management
18 credits
18 credits
Spring Semester Postgraduate Programmes and Research MATH 202 Integration Theory
MATH Mathematics Elective
SB&M Business and Management SB&M Business and Management
SClE Science Elective
Fall Semester ENGG FREE H&SS MATH SB&M
Spring Semester ENGG FREE MATH SB&M SClE
Third Year
Engineering Elective Open Elective
Humanities and Social Science Mathematics Elective
Business and Management
Engineering Elective Open Elective Mathematics Elective Business and Management Science Elective
17 credits
16 credits
Major research areas being planned include almost all the major pure and applied mathematical branches : analysis, algebra, geometry, differential equations, probability and statistics, operations research, numerical analysis, fluid and solid mechanics, mathematical physics, and scientific computation. During the initial years, three major areas of research are being emphasised: analysis, mechanics, and scientific computation.
1. MECHANICS. Research in this area is being developed in close collabora- tion with the Mechanical Engineering and Computer Science Departments, and to a lesser extent with the Chemical Engineering and Physics Depart- ments.
2. ANALYSIS. Most activities in applicable mathematics are in the area of analysis or related to analysis. Study of theoretical sciences and engineering relies heavily on applied analysis. At the same time, the field of analysis also includes pure mathematical studies.
3. SCIENTIFIC COMPUTATION. This area of mathematics cuts across every discipline in science and engineering.
The Department of Mathematics offers postgraduate programmes leading to the degrees of Master of Science (MSG), Master of Philosophy (MPhil), and Doctor of Philosophy (PhD) in Mathematics.
In general, qualified students with a bachelor's degree in mathematics, or a bachelor's degree in science or engineering with a strong mathematical background, can apply for admission to the postgraduate programmes in the Department of Mathematics.
16 credits
Master of Science (MSc) in Mathematics A minimum of 104 credits is required for the business and management option. The
choice of electives may result in this minimum being exceeded. The MSc programme emphasises course work to strengthen the students' general background in mathematics and mathematical sciences. It can be a terminal degree or a preliminary degree leading to the PhD, and requires a research project in addition to a programme of courses.
Master of Philosophy (MPhil) in Mathematics Professor Chung-Chun YANG The MPhil programme aims to strengthen the general background of the
student in mathematics and mathematical sciences, and to expose the student to the environment and scope of mathematical research. It can be a terminal degree or a preliminary degree leading to the PhD, and requires research leading to a thesis as well as a course programme.
Doctor of Philosophy (PhD) in Mathematics
The aim of the PhD degree programme is to prepare the student to become a research scholar either in an academic or industrial environment. The programme, besides providing broad background in mathematics and mathematical sciences, aims to equip the student to do independent and original research. Students have three options from which to choose their major concentration of study: Pure Mathematics, Applicable Mathematics, and Mathematical Science. A doctoral thesis representing an original contribution to the field is a requirement forthe degree as are courses.
Research Interests Professor Din-Yu HSlEH Head of Department
Waves and stability, asymptotic methods, two phase flows.
Professor Wu-Chung HSIANG
Algebraic topology, differential topology, algebraic K-theory.
Professor W.H. HUI
Computational fluid mechanics using the new Langrangian method applied to three dimensional, subsonic andviscousflow;water wavetheory, particularly wind- wave generation, propagation and decays, and applications to ocean wave predic- tion.
Factorization theory of entire and meromorphic functions using Nevanlinna value - distribution theory and classical complex analysis techniques; functional and differential equations of analytic functions; properties of differential polynomials of meromorphic functions.
Dr Kun-Rui YU Reader
Transcendental number theory, diophantine equations and approximations, simultaneous rational approximation, and linear forms in complex logarithms.
Dr Jimmy Chi-Hung FUNG Lecturer
Computational fluid mechanics, turbulence, environmental studies.
Dr Yue-Kuen KWOK Lecturer
Computational fluid mechanics, numerical analysis, geophysics. Analysis of computer extended series, in particular applications in fluid mechanics; inverse theory in geophysics, theoretical and algorithmicdevelopments; numerical simulation of hydrodynamic models of fluidization.
Dr Kin-Yin LI Lecturer
Complex analytic problems using functional analytical techniques with em- phasis on Hilbert space operator methods, in particular some fundamental questions about interpolation, such as characterizing coefficients for bounded Riemann map- pings and determining density for interpolating Blaschke products.
Dr Allanus Hak-Man TSOl Lecturer
Stochastic analysis, jump processes, Malliavin calculus, time reversal and semimartingale theory, probability in finance. Investigation of the problem of the existence of densities for functionals of single jump processes; analysis of the time reversal of infinite dimensional point processes and their associated duality equa- tions.
Undergraduate Courses MATH 103 Ordinary Differential Equations [3-1-0:4]
Permission of the Head of Department is an alternative to the stated prerequisite, and this is a requirement for all courses for which prerequisites are not stated.
MATH 001 Beginning Calculus [3-1-0:4]
Calculus of one variable including limits, continuity, differentiation; mean- value theorem; L'Hospital rule; maxima and minima; implicit differentiation;
elementary transcendental functions; introduction of integration with applica- tions to physical sciences, economics and business.
Prerequisite : None.
Textbook : Boyce & Di Prima, Calculus, Wiley.
MATH 002 Intermediate Calculus [3-1-0:4]
Further development of integration; inverse trigonometric and logarithmic functions; techniques of integration; improper integrals; infinite series, Taylor's series; coordinates systems, parametric equations; introduction to differen- tial equations.
Prerequisite : MATH 001.
Textbook : Same as MATH 001.
MATH 101 Multivariable Calculus [3-1-0:4]
Three-dimensional analytical geometry; continuous functions and funda- mental theorems of differentiation; differential and integral calculus for functions of two or three variables; line integrals, multiple integrals; Green's theorem; Stokes' theorem; divergence theorem; implicit function theorem;
maxima and minima of functions of several variables; constraints and Lagrangian multipliers.
Prerequisite : A-level Mathematics or MATH 002.
Textbook : Marsden & Tromba, Vector Calculus, W. H. Freeman.
MATH 102 Introduction to Analysis p-1-0:4]
Real number system; topology of real line, plane and Euclidean space;
fundamental theorems in differentiation and integration; infinite series; topics from differential equations and Fourier series.
Prerequisite : MATH 101.
Textbook : Ross, ElementaryAnalysis:TheTheoryof Calculus, Springer- Verlag, or Rosenlicht, lntroduction to Analysis, Dover.
Existence and uniqueness theorems of ordinary differential equations; theory of linear systems; stability theory; study of singularities; boundary value problems.
Prerequisites : MATH 101 and MATH 1 1 1.
Textbook : Hirsch & Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press.
MATH 11 1 Linear Algebra [3-1-0:4]
Vector spaces, linear transformations, matrices, system of linear equations, bases, determinants, inner products, eigenvalues, bilinear forms, decom- positions of matrices.
Prerequisite : A-level Mathematics or MATH 002.
Textbook : Anton, Elementary Linear Algebra, Wiley.
MATH 132 Discrete Structures 13-1 -0 :4]
Fundamentals of set theory, graph theory, enumeration, and algebraic structures. Basic properties and fundamental algorithms concerning integers (including induction, Euclidean algorithm, modular arithmetic). Propositional and predicate calculus and simple formal theories. Application to topics such as program correctness, formal program verification, algorithms from graph theory, and elementary set theory.
Textbook : C. L. Liu, Elements of Discrete Mathematics, Second Edition, 1 985, McGraw Hill
MATH 151 Differential Equations and Applications 13-1 -0:4]
First and second order diff erential equations, higher order equations, Laplace transform method, systems of first order linear equations; method of series solutions; nonlinear differential equations and stability criteria; partial differ- ential equations and Fourierseries; Sturm-Liouvilleequation; Besselfunctions and Legendre polynomials.
Prerequisite : MATH 002 or equivalent.
Textbook : Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems, Wiley.
MATH 201 Real Analysis [3-1-0:4]
Calculus on manifolds: differential forms, infinite series of functions, and Stokes' formulaon manifolds; elementary approximation theory, Weierstrass theorem; applications to geometrical and physical problems.
Prerequisites : MATH 101 and MATH 11 1. MATH 102 is also desirable.
Textbook : Rudin, Principles of Mathematical Analysis, McGraw-Hill.
MATH 202 Integration Theory 13-1 -0:4]
Topological spaces, compactness, connectivity, convexity and differentiability, elementary measure theory, Lebesgue integration, generalized integration and applications.
Prerequisite : MATH 201.
Textbook : Same as MATH 201.
MATH 204 Complex Analysis [3-1-0:4]
Complex differentiability; Cauchy-Riemann equations; analytic functions;
Cauchy's theorem; contour integration; residue theorem; geometric properties of complex mappings; analyticcontinuation; Hadamard's factorisation theorem;
canonical products; Picard's theorem.
Prerequisite : MATH 201.
Textbook : Ahlfors, Complex Analysis, McGraw-Hill.
MATH 21 1 Abstract Algebra 13-1 -0:4]
An introduction to the principles and concepts of modern abstract algebra.
Topics include groups, rings, and fields with applications to number theory, theory of equations, and combinatorics.
Prerequisite : MATH 11 1.
Textbook : Herstein, Topics in Algebra, Wiley.
MATH 212 Group Representation Theory [3-1-0:4]
Galois theory; Cayley's theorem; representation of groups: Maschke's theo- rem; Schur's lemma; representation of Abelian groups; the character of a group representation; the group algebra and the regular character;
orthogonality relations.
Prerequisite : MATH 21 1.
Textbook : Jacobson, Basic Algebra I, W.H. Freeman & Co.
MATH 221 Differential Geometry [3-1-0:4]
Curvature and torsion of curves; Frenet-Serret frames; global properties of closed curves; Gaussian curvature and mean curvature; geodesics; minimal surfaces; Gauss-Bonnet theorem.
Prerequisite : MATH 101.
Textbook : Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall.
MATH 223 Topology 13-1 -0:4]
Topology of Euclidean spaces; winding number; knot theory; fundamental group and covering spaces; Euler characteristic; simplicia1 complexes; clas- sification of two-dimensional manifolds;vectorfields; Poincare-Hopf theorem.
Prerequisites : MATH 101 and MATH 1 1 1.
Textbook : Blackett, Elementary Topology: A Combinatorial and Alge- braic Approach, Academic Press.
MATH 225 Mathematical Logic [3-1-0:4]
Propositional and predicate calculus; consequence and deduction; truth and satisfaction; Godel completeness theorem; Lowenheim-Skolem theorem;
Boolean algebra; axiomatic theories.
Prerequisite : MATH 11 1.
MATH 231 Numerical Analysis [3-1-0:4]
Basic numerical analysis. Topics include stability of computation, linear systems, eigenvalues and eigenvectors, nonlinear equations, interpolation and approximation, numerical integration and solution of ordinary differential equations, optimization. A short course on Fortran may also be given.
Prerequisite : MATH 11 1.
Textbook : Kahaner, Moler and Nash, Numerical Methods and Software, Prentice Hall.
MATH 241 Probability 13-1 -0:4]
Basic probability theory. Sample spaces; random variables; normal, Poisson and related distributions; expectation; correlation; limit theorems. Applications to biology, physics, communication sciences and other fields.
Prerequisite : MATH 101.
Textbook : Ross, A First Course in Probability, Macmillan.
MATH 243 Mathematical Statistics [3-1-0:4]
Central limit theorem; point estimation; interval estimation; multivariate normal distributions; testing of hypotheses; linear models.
Prerequisite : MATH 241.
Textbook : Hoel, Port and Stone, introduction to Statistical Theory, Houghton Mifflin.
MATH 244 Applied Statistics [3-1-0:4]
A systematic introduction to statistical inference, including the necessary probabilistic background; point and interval estimation; hypothesis testing.
Prerequisite : MATH 002.
Textbook : Milton and Arnhold, lntroduction to Probability and Statistics, McGraw-Hill.
Hines and Montgomery, Probability and Statistics in Engi neering and Management Science, Wiiey.
MATH 251 Function of a Complex Variable and Applications [3-1-0:4]
Differentiation and integration in the complex plane; Cauchy's integral for- mula;Taylorseries; Laurentseries; ana1yticcontinuation;contour integration;
conformal mapping;special functions; integral transforms; asymptoticmethods.
Prerequisite : MATH 151.
Textbook : Carrier, Krook and Pearson,Functions ofa Complex Variable, Hod Books.
MATH 252 Applied Partial Differential Equations 13-1 -0:4]
Methods to solve the Laplace equation, the wave equation and the diffusion equation; separation of variables; spherical harmonics; integral transforms;
Green's tunction; eigenfunction expansion; characteristics; retarded poten- tia1;variational method;similaritytransformations; brief discussion on nonlinear partial differential equations.
Prerequisite : MATH 251.
Textbook : Sneddon, ElementsofPartialDifferentialEquations, McGraw- Hill.
MATH 281 Introduction to Operations Research [3-1-0:4]
Linear programming; simplex method; duality theory; network analysis;
dynamic programming; game theory; integer programming; stochastic processes; queueing theory; inventory theory; forcasting; decision analysis.
Prerequisites : MATH 241 or MATH 244.
Textbook : Hillier and Lieberman, lntroduction to Operations Research, Holden-Day.
MATH 301 Introduction to Functional Analysis [3-1-0:4]
Normed space; inner product space; topological vector spaces; closed graph theorem; Hahn-Banach theorem; principleof uniform boundedness; LP-space;
elementary Banach space theory; contraction principle and its applications to differential and integral equations and numerical analysis.
Prerequisites : MATH 201 and MATH 202.
Textbook : Epstein, Linear Functional Analysis, Saunders.
MATH 302 Partial Differential Equations [3-1-0:4]
Classification of partial differential equations; first order equations; second order linearequations;Green'sfunctions; maximum principles;characteristics;
Riemann's method; well-posed problems.
Prerequisites : MATH 101 and MATH 1 11.
Textbook : Copson, Partial Differential Equations, Cambridge University Press.
MATH 303 Dynamical Systems [3-1-0:4]
Modern development of dynamic systems. Hamiltonian systems; dissipative systems; bifurcations; strange attractors; chaoticsystems;fractals; Hausdorff dimension; Lyapunov exponents.
Prerequisite : MATH 151, MATH 201 and MATH 252 are desirable.
MATH 312 Number Theory and Applications [3-1-0:4]
Prime numbers; unique factorization; modular arithmetic; quadratic number fields; finite fields; p-adic numbers; coding theory; computational complexity.
Prerequisite : MATH 21 1.
Textbook : Ireland & Rosen, A Classical lntroduction to Modern Number Theory, Springer-Verlag.
MATH 321 Algebraic Topology [3-1-0:4]
Homotopy theory; covering spaces and vibrations; simplicia1 and CW com- plexes; manifolds; homology theories; universal coefficients and Kunneth formulas; Hurewicz theorem; applications to fixed point theory and other topics.
Prerequisite : MATH 223.
MATH 331 Numerical Solutions of Partial Differential Equations [3-1-0:4]
An introduction to finite difference and finite element methodsforthesolutions of elliptic, parabolic and hyperbolic partial differential equations. The course will include the use of software for solving differential equations.
Prerequisites : MATH 151 and MATH 231.
Textbook : Sewell, The Numerical Solution of Ordinary and Partial Dif- ferential Equations, Academic Press.
MATH 333 Introduction to Scientific Computation [3-1-0:4]
A variety of projects drawn from different areas of scientific application to demonstrate the use of computers as a tool to solve real problems. Math- ematical concepts, scientific content, numerical techniques, computer pro- gramming and graphics are integrated.
Prerequisite : MATH 231.
Postgraduate Courses
All 500 levels courses have the course vector : [3-0-0:3]. Permission of the Head of Department is an alternative to the stated prerequisite, and this is a requirement for all courses for which prerequisites are not stated.
MATH 501 Real Function Theory
Abstract integration theory; measure theory; differentiation; convex functions and inequalities; Fourier analysis; complete metric spaces.
Prerequisites : MATH 201 and MATH 202.
Textbook : Rudin, Real and Complex Analysis, McGraw-Hill.
MATH 502 Functional Analysis
General theory of topological vector spaces; Banach space; Hilbert space;
Banach algebra; spectral theory ; Riesz representation theorem; distribution theory.
Prerequisites : MATH 201, MATH 202 and MATH 501.
Textbcok : Rudin, FunctionalAnalysis, McGraw-Hill and Rickart, General theory of Banach Algebra, van Nostrand.
MATH 503 Complex Function Theory
Functions of a complex variable and the fundamental theorem for complex integrals; hyperbolic geometry; open mapping theorem; Abel's theorem and converse; sequence and series of analytic functions; harmonic functions, subharmonic functions; Riemann mapping theorem; canonical representa- tion; Schwarz reflection principle and analytic continuation, monodromy theorem, Phragmen-Lindelof principle, generalization of Picard's theorem.
Prerequisite : MATH 204.
Textbook : Rudin, Real and Complex Analysis, McGraw-Hill.
MATH 505 Theory of Ordinary Differential Equations
Existence and general properties of solutions; linear systems; Floquet theory;
stability; Lyapunov's method; nonlinear systems; two-dimensional systems;
Poincare-Bendixson theory; nonlinear oscillations.
Textbook : Hale, Ordinary Differential Equations, Krieger.
MATH 507 Theory of Partial Differential Equations
Distributions; Fourier transforms; Sobolev spaces; Cauchy problem; Hilbert space methods; elliptical problems and regularity; hyperbolic and parabolic systems.
MATH 51 1, MATH 51 2 Advanced Algebra I, II
Finite groups; representation of groups; rings with minimum condition; Galois
Finite groups; representation of groups; rings with minimum condition; Galois