** szrch as locoi7zotion, evzotion, and control**

** DEPARTMENT OF MATHEMATICS**

Mathematics permeates almost every discipline of science and technology.

Modern research mathematicians, including even those specialising in the purest of mathematics, find themselves sought after by computer companies, biotechnology institutions and financial corporations This takes them beyond their traditional careers as facultv members in universities and staff scientists in research laboratories. For those specialising in mathematical sciences, scientific computation or statistics, the opportunity for exciting careers in education, industry and government is even wider.

The faculty in the Department of Mathematics consists of two overlapping groups;

one interested in pure and applicable mathematics, and the other in mathematical sciences and appli&tions. ~he'faculty in the first group are mathematicians. Whether their research activities be in pure or applicable mathematics, they are doing mathematics in the proper, narrow sense. They are mainly interested in the mathematical contents of the subject matter, and their work is judged mainly on mathematical merits. On the other hand, faculty in the second group are mathematical scientists, engineers or statisticians.

Thev are usuallv not mathematicians in the narrow sense. and are mainlv interested in the scie;lific conteh of the subject matter they are investigating. heir work is judged mainly by its contribution to science and engineering. The instructional programmes in the Department reflect these interests of the faculty. The complementary interactions of the two groups are manifested, for example, in the programme of scientific computation, which is supported by both theoretical numerical analysts and mathematical scientists.

It is evident that the two groups are quite different. However there are great advantages to combining these two groups of people in the same department. In a department which has a strong component of mathematical application, the mathemati- cians can uphold the integrity and traditional standards of the discipline and, forthose with a bent towards real world applications, the proximity with mathematically conversant scientists, enaineers and statisticians provides stimulus and inspiration for their explora- tions. It is a h beneficial to mathematical scientists, engineers and statisticians-to be exposed continually to new mathematical ideas. It is very likely that new mathematical tools for solvinavarious scientificand technological problems will be developed from these - . interactions a i d fermentations.

**Faculty **

Professor and Head of Department:

Din-Yu HSIEH, BSc National Taiwan; MSc Brown; PhD California lnst of Tech Professors:

John D. BUCKMASTER, BSc London; PhD Cornell Grafton Wai-How HUI, BSc Beijng, PhD, DSc Southampton

Ronnie LEE, BSc Chinese Univ of Hong Kong, PhD Univ of Michigan, Ann Arbor Chung-Chun YANG, BSc National Taiwan; MSc, PhD Univof Wisconsin, Madison Adjunct Professors:

Wu-Chung HSIANG, BSc National Taiwan; PhD Princeton James Sai-Wing WONG, BSc Baylor; PhD California lnst of Tech Readers:

Ngai-Hang CHAN, BSc Chinese Univ of Hong Kong; PhD Maryland Vladimir A. VLADIMIROV, BSc, PhD, DSc Novosibirsk

Kunrui YU, BSc Univ of Sc & Tech of China; Dr.rer.nat Bonn

**School of Science **

Senior Lecturers/Associate Professors:

Kwing-Lam CHAN, BA Univ of California, Berkeley; MA, PhD Princeton Yue-Kuen KWOK, BSc Hong Kong; MSc, PhD Brown

Lecturers/Assistant Professors:

Gopal K. BASAK, B.Stat, M.Stat Indian Statistical lnst; PhD Indiana

I Jeffrey R. CHASNOV, BA Univ of California, Berkeley; MA, MPhil, PhD Columbia Bei-Fang CHEN, BSc Huazhong Normal; MSc Huazhong Univ of Sc & Tech; MA,

PhD State Univ of New York, Buffalo Yik-Man CHIANG, BSc, PhD London

Kwai-Man FAN, BSc National Taiwan; MA Univof California, Santa Barbara; PhD Purdue

Jimmy Chi-Hung FUNG, BSc Durham; PhD Cambridge Walter G. GALL, MA, PhD State Univ of New York, Buffalo

Guo-Qiang GE, BSc Zhejiang; MA Wayne State; PhD Univof California, Berkeley Bi-Zhong HU, BSc Univ of Sc & Tech of China; MSc Academia Sinica; PhD State

Univ of New York, Stony Brook

Ji-Shan HU, BA, MA Jiao Tong Univ, Shanghai; PhD Princeton Jing Song HUANG, BSc Beijng; PhD Massachusetts lnst of Tech Bing-Yi JING, BSc Lanzhou; MSc Univ of Tech, Sydney; PhD Sydney Kin-Yin LI, BSc Univ of Washington; PhD Univ of California, Berkeley Wei-Ping LI, BA Nankai; MSc, PhD Columbia

Shiu-Hong LUI, BSc, MSc Toronto; PhD California lnst of Tech Jian-Min MAO, BSc East China; PhD Houston

Guo-Wu MENG, BSc Univ of Sc & Tech of China; MSc, PhD Brown Mo MU, BSc Southeast; MSc, PhD Academia Sinica

Yuan-Wei QI, BA Beijing; MSc Academia Sinica; PhD Oxford

Tai-Man TANG, BSSc Chinese Univof Hong Kong; PhD Univof California, Berkeley Charles H. TONG, BSc Univof California, Berkeley; MSc Univ of California, Davis;

PhD Univ of California, Los Angeles

Allanus Hak-Man TSOI, BSc Univ of Washington; MSc Univ of Illinois, Urbana- Champaign; PhD Alberta

Xiao-Ping WANG, BSc, MSc Beijng PhD New York Man-Yu WONG, BA Hong Kong; MSc, PhD London

Li-Xin WU, BSc, MSc Fudan; PhD Univ of California, Los Angeles Xiao-Ping XU, BSc Zhejiang Normal; MSc Xiamen; PhD Rutgers Min YAN, BSc Fudan; MSc, PhD Chicago

Visiting Scholar:

Guo-Chuang HUANG, Graduate Univ of Sc & Tech of China

**Undergraduate Programme **

There are four options within the first-degree programme of the Department of Mathematics: Pure Mathematics, Mathematical Sciences, Scientific Computation and Statistics. All courses of study lead to the Bachelor of Science degree in Mathematics.

Students in the Pure Mathematics option are interested mainly in the mathemati- cal contents of the subject matter, while students of Mathematical Sciences are more interested in the scientific content. The Mathematical Sciences option includes multidisciplinary study undertaken in conjunction with other departments of the University.

The Scientific Computation option is interdisciplinary and emphasises the study of large scale computational algorithms, that are reliable, accurate and economical, to solve

**School of Science **

complex problems in science and technology. The general theme of the Statistics option is to orovide students with statistical knowledge, helping them to develop problem-solving

### -

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^{. }. skills for real-life situations.

Students in all options take multivariable calculus and linear algebra in the first year. Students in Pure Mathematics, Mathematical Sciences and Statistics take a one- year course in real analysis during the second year of study. In addition, students in Pure Mathematics study subjects in abstract algebra, differential geometry and topology plus three subjects at a more advanced level as well as selected subjects in physical sciences and engineering. Forthose pursuing a programme in Mathematical Sciences, three areas of study, physical and engineering science, computer science, and business and manage- inent, have been designed. Students in Scientific Computation are required to undertake a nine-credit project in the third year of study. Students in Statistics are required to study courses in probability, statistics and stochastic modelling and selected subjects in application discipline.

Students pursuing the Mathematical Sciences option should note the University
Regulation governing joint options described on page **24. **

**Admission Requirements 1995-96 **

In addition to the general entrance requirements of the University, acceptable grades are required in two AL subjects (Pure Mathematics and Physics) plus one AUAS subject.

**Curriculum for BSc in Mathematics **

**Pure Mathematics Option **

**First Year ***Fall Semester *

MATH **101 ** C Multivariable Calculus **[3-1-0:4] **

MATH **1 1 1 ** C Linear Algebra **[3-1-0:4] **

**(1) ** MATH **151 ** E Differential Equations and Applications **[3-1-0141 **

**(1) ** COMP **101 ** E Computing Fundamentals **(2-0-2:3] **

**(2) ** LANG **001 ** Language Skills Enhancement I **[O-3-1 :O] **

Spring Semester

**(1 **) MATH **102 ** E Introduction to Analysis
**(1) ** MATH **204 ** E Complex Analysis

**15 **credits

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.COMP **102 ** R computer ~undamentals and Programming **[3-0-2:4] **

HLSS - - - E Humanities and Social Science Elective **13-0-0:31 **
SB&M E Business and Management Elective **i3-0-0:3j **

**School ****of ****Science **

**Second Year ***Fall Semester *

MATH **301 ** R Real Analysis **[3- 1 -0:4] **

MATH **31 1 ** R Abstract Algebra l **[3-1-0:4] **

**(3) ** FREE E Non-Mathematics Elective **[3-0-0:3] **

H&SS E Humanities and Social Science Elective **[3-0-0:3] **

PHYS **121 ** R Electricity and Magnetism **[3-0-3:4] **

**18 **credits

Spring Semester

MATH **302 ** R Integration Theory **[3- 1 -0:4] **

MATH **312 ** R Abstract Algebra ll **[3-1-0:4] **

H&SS E Humanities and Social Science Elective **[3-0-0:3] **

PHYS **124 ** R Vibrations and Waves **[3-0-3:4] **

SB&M E Business and Management Elective **[3-0-0:3] **

MATH **321 ** R Differential Geometry
MATH E Mathematics Elective
**(3) ** FREE E Non-Mathematics Elective

FREE E Free Elective FREE E Free Elective

**18 **credits

**Third Year ***Fall Semester *

**16 **credits

Spring Semester

MATH **323 ** R Topology **[3-1-0:4] **

MATH E Mathematics Elective **[3-0-0:3] **

FREE E Free Elective **[3-0-0:3] **

FREE E Free Elective **[3-0-0:3] **

H&SS E Humanities and Social Science Elective **[3-0-0:3] **

**16 **credits
**(1) ** The course shown is recommended, but may be replaced by a suitable elective

as approved by the Department.

**(2) ** Students may be exempted from this course by the Language Centre.

**(3) ** Any elective course may be chosen, except a course in the Department of
Mathematics.

**18 **credits

A minimum of **101 **credits is required for the Pure Mathematics Option.

**School of Science **

**Mathematical Sciences Option in Physical and Engineering Science ****First Year **

Fall *Semester *

MATH 101 C Multivariable Calculus [3-1-0:4]

MATH 111 C Linear Algebra [3-1-0:4]

(1) ENGG E Engineering Elective [3-1-0:3]

(3) LANG 001 Language Skills Enhancement I [O-3-1 :O]

PHYS 121 R Electricity and Magnetism [3-0-3141
15 credits
*Spring Semester *

(2) MATH 102 E Introduction to Analysis [3-1-0141 COMP 102 R Computer Fundamentals and Programming [3-0-2:4]

(1) ENGG E Engineering Elective [3-1-3:4]

H&SS E Humanities and Social Science Elective [3-0-0:3]

PHYS 124 R Vibrations and Waves [3-0-3:4]

Differential Equations and Applications [3-1-0141

Real Analysis [3-1-0:4]

Functions of a Complex Variable and [3-1-0:4]

Applications

Humanities and Social Science Elective [3-0-0:3]

Physical Science or Engineering Elective [3-0-0:3]

18 credits

Integration Theory [3-1-0:4]

Applied Partial Differential Equations [3-1-0:4]

Free Elective [3-0-0:3]

Physical Science or Engineering Elective [3-0-0:3]

Business and Management Elective [3-0-0:3]

17 credits
**Third Year **

Fall *Semester *

MATH E Mathematics Elective [3-1-0:4]

H&SS E Humanities and Social Science Elective [3-0-0:3]

(1) PHSCIENGG E Physical Science or Engineering Elective [3-0-0:3]

(1) PHSCIENGG E Physical Science or Engineering Elective [3-0-0:3]

SB&M E Business and Management Elective 13-0-0:3]

16 credits

**School of Science **

*Spring Semester *

MATH E Mathematics Elective [3-1-0:4]

FREE E Free Elective [4-0-0:4]

FREE E Free Elective [4-0-0:4]

H&SS E Humanities and Social Science Elective [3-0-0:3]

15 credits (1) Thecourse identified as Physical Science or Engineering Elective will beselected

in consultation with the student's academic advisor.

(2) The course shown is recommended, but may be replaced by a suitable elective as approved by the Department.

(3) Students may be exempted from this course by the Language Centre.

A minimum of 100 credits is required for the Mathematical Sciences Option in Physical and Engineering Science. Besides PHYS 121 and PHYS 124, at least six other Physical Science or Engineering courses are required. Sample detailed programmes will be provided by the departmental academic advisors.

**Mathematical Sciences Option in Computer Science *** First Year *
Fall

*Semester*

MATH 101 C Multivariable Calculus [3-1-0:4]

MATH 132 R Discrete Structures [3-1-0:4]

COMP 102 R Computer Fundamentals and Programming [3-0-2:4]

COMP 11 1 R Software Tools [2-0-2:3]

(1) LANG001 Language Skills Enhancement I [O-3-1:0]

15 credits
*Spring Semester *

(2) MATH 102 E Introduction to Analysis [3-1-0:4]

MATH 11 1 C Linear Algebra [3-1-0:4]

COMP 106 R C Programming [I -0-2:2]

COMP 171 R Data Structures and Algorithms [3-1-0:3]

COMP 180 R Computer Organisation [3-0-1:3]

H&SS E Humanities and Social Science Elective [3-0-0:3]

18 credits

**73 **

Spring Semester

**School of Science **

Spring Semester

MATH 302 R Integration Theory [3-1-0:4]

MATH E Mathematics Elective [3-1-0:4]

(3) COMP E Computer Science Elective [3-0-0131

HBSS E Humanities and Social Science Elective [3-0-0:3]

SBBM E Business and Management Elective [3-0-0:3]

17 credits

HBSS E Humanities and Social Science Elective [3-0-0:3]

SBBM E Business and Management Elective [3-0-0:3]

17 credits Spring Semester

MATH E Mathematics Elective [3-1-0141

(3) COMP E Computer Science Elective [3-0-0:3]

FREE E Free Elective [3-0-0:3]

FREE E Free Elective [4-0-0:4]

HBSS E Humanities and Social Science Elective [3-0-0:3]

17 credits (1) Students may be exempted from this course by the Language Centre.

(2) The course shown is recommended, but may be replaced by a suitable elective as approved by the Department.

(3) Students should seek departmental advice as to the choice of Computer Science electives.

A minimum of 100 credits is required for the Mathematical Sciences Option in
Computer Science. Besides COMP 102, COMP 106, COMP 11 **1 , **COMP 171 and COMP
180, at least four other Computer Science courses are required. Sample detailed
programmes will be provided by the departmental academic advisors.

**Mathematical Sciences Option in Business and Management ****First Year **

*Fall Semester *

MATH 101 C Multivariable Calculus [3-1-0:4]

(1) MATH **244 ** E Applied Statistics [3-1-0141

H&SS E Humanities and Social Science Elective [3-0-0:3]

(4) SB&M E Business and Management Elective [3-0-0:3]

18 credits

E Mathematics Elective [3-1-0:4]

E Computing Fundamentals [2-0-2:3]

E Business and Management Elective [3-0-0:3]

E Business and Management Elective [3-0-0:3]

17 credits

R Integration Theory [3-1-0141

R Computer Fundamentals and Programming [3-0-2:4]

E Humanities and Social Science Elective [3-0-0131 E Business and Management Elective 13-0-0:3]

E Business and Management Elective [3-0-0:3]

17 credits

E Mathematics Elective [3-1-0:4]

E Mathematics Elective [3-1-0:4]

E Free Elective [4-0-0:4]

E Free Elective [3-0-0:3]

E Humanities and Social Science Elective [3-0-0:3]

18 credits Spring Semester

MATH E Mathematics Elective [3-1-0:4]

FREE E Free Elective [3-0-0:3]

FREE E Free Elective [3-0-0:3]

H&SS E Humanities and Social Science Elective [3-0-0:3]

13 credits (1) The course shown is recommended, but may be replaced by a suitable elective

as approved by the Department.

(2) Students entering with a grade of B or above in AL Economics will take ECON

**School ****of ****Science **

191 ; those with a grade of C in AL Economics, or C or above in AL Mathematics will take ECON 11 1. All other students will take ECON 1 10.

(3) Students may be exempted from this course by the Language Centre.

(4) Students should seek departmental advice as to the choice of Business and Management electives.

A minimum of 101 credits is required for the Mathematical Sciences Option in Business and Management. Besides ACCT 101, ACCT 122, ECON 11 1 and ECON 11 2, at least five other Business and Management courses are required. Sample detailed programmes will be provided by the departmental academic advisors.

**Scientific CompUration Option **

**First Year ***Fall Semester *

MATH 101 C Multivariable Calculus [3-1-0:4]

MATH 11 **1 ** C Linear Algebra [3-1-0:4]

COMP 102 R Computer Fundamentals and Programming [3-0-2:4]

(1) LANG 001 Language Skills Enhancement I [0-3-1:0]

E lntroduction to Analysis

E Differential Equations and Applications E C Programming

R Data Structures and Algorithms E Computer Organisation

E Humanities and Social Science Elective

**Second Year **

**R Numerical Analysis **
**R ** Real Analysis

E Functions of a Complex Variable and Applications

**R Vibrations and Waves **

E Business and Management Elective

**R Numerical Solutions of Partial Differential **
Equations

R lntroduction to Scientific Computation I E Applied Partial Differential Equations E Design and Analysis of Algorithms E Humanities and Social Science Elective

16 credits

MATH 321 **R ** Differential Geometry [3-1-0141

MATH 334 **R Introduction to Scientific Computation II ** [3-1-0141
MATH 395 **R Scientific Computation Project I ** [0-0-9:3]

(3) FREE E Elective related to MATH 3951396 [3-0-0:3]

H&SS E Humanities and Social Science Elective [3-0-0131
17 credits
*Spring Semester *

MATH 396 R Scientific Computation Project II . [0-0-18:6]

FREE E Free Elective [4-0-0:4]

H&SS E Humanities and Social Science Elective [3-0-0:3]

SB&M E Business and Management Elective [3-0-0131 16 credits

(1) Students may be exempted from this course by the Language Centre.

(2) The course shown is recommended, but may be replaced by a suitable elective as approved by the Department.

(3) Course to be chosen in fluid mechanics, solid mechanics, theoretical physics, theoretical chemistry, statistics, etc.

A minimum of 105 credits is required for the Scientific Computation Option.

**Statistics Option **

**First Year ***Fall Semester *

MATH 101 C Multivariable Calculus [3-1-0:4]

MATH 111 C Linear Algebra [3-1-0:4]

H&SS E Humanities and Social Science Elective [3-0-0:3]

18 credits 18 credits

**School of Science **

HISS E Humanities and Social Science Elective [3-0-0:3]

SB&M E Business and Management Elective [3-0-0131 18 credits Spring Semester

MATH 302 R Integration Theory [3-1-0141

MATH 341 R Stochastic Modelling [3-1-0:4]

COMP 102 R Computer Fundamentals and Programming [3-0-2:4]

(4) FREE E Non-Mathematics Elective [3-0-0:3]

H&SS E Humanities and Social Science Elective [3-0-0:3]

18 credits

H&SS E Humanities and Social Science Elective [3-0-0:3]

SB&M E Business and Management Elective [3-0-0:3]

17 credits Spring Semester

(5) MATH E Statistics Elective MATH E Mathematics Elective FREE E Free Elective

(4) FREE E Non-Mathematics Elective

13 credits (1) Studentsentering with agrade of B or above in AL Economics will take ECON 191 ;

those with a grade of C in AL Economics, or C or above in AL Mathematics will take ECON 11 1. All other students will take ECON 110.

(2) Students may be exempted from this course by the Language Centre.

(3) The course shown is recommended, but may be replaced by a suitable elective as approved by the Department.

(4)

### .

^{, }Anv elective course may be chosen, except a course in the Department of

~ i h e m a t i c s .

(5) Courses must be chosen from MATH 346, MATH 347, MATH 541, MATH **542, **
**MATH 543, MATH 544, MATH 545, MATH 546, MATH 645, ISMT 354, **or ISMT

A minimum of 100 credits is required for the Statistics Option.

**School of Science **

**Postgraduate Programmes and Research **

Major research areas planned include almost all the major pure and applied mathematical branches. At present five major areas of research are emphasised:

analysis, algebra and geometry, scientific computation, fluid mechanics, and probability and statistics.

1. Analysis

Analysis includes harmonic analysis, real analysis, complex analysis, functional analysis, differential equations and other related fields, with emphasison complex analysis. Most activities in applicable mathematics are in the area of analysis or related toanalysis. Thestudy of theoreticalscienceand engineering relies heavily on applied analysis.

2. Aloebra and aeornetrv

~lgebraand geometry iiiclude number theory, Lie theory, algebraic and geometric topology, and algebraic geometry. Interactions between various areas are - - - ~

-emphasised. Current research inciudes linear forms in p-adic logarithms and the applications, self-dual codes and lattices, Lie algebras and vertex operator algebras, algebraic K-theory, intersection homology, low dimensional topology, group actions on manifolds, stratified spaces, stable vector bundles over alge- braic surfaces and Donaldson theory, fundamental groups of algebraic varieties, and knot theory.

3. Scientific computation

Over the past two decades, scientific computation has become an independent approach to studying science and technology, complementing the long-estab- lished theoretical and experimental approaches. With the advent of parallel computers and development of new algorithms, it plays an even more important role in future. Scientific computation in the Mathematics Department means not only large scale computation of solutions to problems in science, en~ineerina and business and management but also developing algorithms thatare relkble, accurate and economical. Current research areas include shock-cadurina schemes, parallel algorithms, symbolic computation, numerical linear aigebr<

numerical solutions to elliptic and hyperbolic partial differential equations, and computational quantum mechanics. There are also joint research projects with other departments in the University and other institutions in and outside Hong Kong .

4. Fluid mechanics

Fluid mechanics is the study of the motion of liquids and gases with direct application to industry and environmental research. It is particularly rich in nonlinear problems and is a major source of ideas and techniques in applied mathematics. Current research areas include two-phase flow, water wave motion, fluid dynamics of typhoon, fluid dynamics of combustion, rotating flow, high speed flow, bubble dynamics, flow instability, bifurcation, chaos and turbu- lence.

5. Probability and Statistics

Probability and statistics are subjects that study random phenomena. Random- ness arises frequently in diverse disciplines, ranging from economics tobiology.

Probability and statistics have direct applications in fields such as econometrics.

finance, medical sciences and controi engineering. Current faculty research interests include time series; spatial statistics; linear models and stochastic control, calculus, and differential equations.

**k **

**School of Science **

**Master of Science (MSc) in ~atheAatics **

**The MSc programme emphasises course work to strengthen 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. The duration of the programme normally rangesfrom 18 months **
to three years for full-time studies, and it may be extended to five years for part-time
studies.

In fulfilling the degree requirements, students are expected to attend and present seminars, undertake course work and complete an assigned project. The minimum number of credits needed to fulfil the degree requirements is 30, as follows:

Courses: 24 credits in mathematics or related fields, of which at least 18 credits are mathematics courses at the postgraduate level

Research: MSc project (6 credits)

The passing standard in a graded course is C and the overall average obtained
must be **6 **_{or above. }

**Master of Philosophy (MPhil) in Mathematics **

The MPhil programme aims to strengthen students' general background 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. The duration of the programme normally ranges from 18 months to three **
years for full-time studies, and may be extended to five years for part-time studies.

Students with a first degree in an area other than mathematics may be required to take additional courses.

In the final stage of the programme, students must submit their theses to the Department and, subsequently, to present and defend them. A student who has performed unsatisfactorily will be asked to re-submit the thesis. The result of the second

In the final stage of the programme, students must submit their theses to the Department and, subsequently, to present and defend them. A student who has performed unsatisfactorily will be asked to re-submit the thesis. The result of the second