Master of Philosophy in Mathematics
Doctor of Philosophy in Mathematics
MPhil(MATH)
PhD(MATH)
Both full and parttime
MPhil
Fulltime: 2 years
Parttime: 4 years
PhD
Fulltime: 3 years (with a relevant research master’s degree), 4 years (without a relevant research master’s degree)
Parttime: 6 years
PG Programs Coordinator:
Prof Yang XIANG, Professor of Mathematics
The Master of Philosophy (MPhil) Program seeks to strengthen students’ general background in mathematics and mathematical sciences, and to expose students to the environment and scope of mathematical research. A candidate for an MPhil degree is expected to demonstrate knowledge in the discipline and to synthesize and create new knowledge, making a contribution to the field. It can be a terminal degree or a preliminary degree leading to the PhD.
The Doctor of Philosophy (PhD) Program aims to prepare students to become research scholars in an academic or industrial environment and enable students to do independent and original research. It provides a broad background in mathematics and mathematical sciences. Students can choose to focus their research in one of the three areas: Pure Mathematics, Applied Mathematics, and Probability and Statistics. A candidate for a PhD degree is expected to demonstrate mastery of knowledge in the chosen discipline and to synthesize and create new knowledge, making an original and substantial contribution to the discipline.
The programs are offered by the Department of Mathematics with the following research foci and stateoftheart facilities, strengthening students’ knowledge in mathematics and training them to carry out original research independently and innovatively.
Research Foci
Algebra and Number Theory
The theory of Lie groups, Lie algebras and their representations is an important branch in modern mathematics and it has interactions with many other branches in mathematics and physics. Our research includes representation theory of reductive groups, quantum groups, infinite dimensional Lie groups and Lie algebras and applications to mathematical physics. Number theory has a long and distinguished history, and the concepts and problems relating to the theory have been instrumental in the foundation of a large part of mathematics. Number theory has flourished in recent years, as made evident by the proof of Fermat's Last Theorem. Our research specializes in automorphic forms and applications of number theory.
Geometry and Topology
Geometry and topology provide an essential language describing all kinds of structures in nature. The subject has been vastly enriched by close interaction with other mathematical fields and with fields of science such as physics, astronomy and mechanics. The result has led to great advances in the subject, as highlighted by the proof of the Poincare conjecture. Active research areas in the department include algebraic geometry, differential geometry, lowdimensional topology, equivariant topology, combinatorics and combinatorial topology, and geometrical structures in mathematical physics.
Analysis and Differential Equations
The analysis of real and complex functions plays a fundamental role in mathematics. This is a classical yet still vibrant subject that has a wide range of applications. Differential equations are used to describe many scientific, engineering and economic problems. The theoretical and numerical study of such equations is crucial in understanding and solving problems. Our research areas include complex analysis, exponential asymptotics, functional analysis, harmonic analysis, wavelets analysis, nonlinear equations and dynamical systems, integrable systems, fluid dynamics, and inverse problems.
Applied and Computational Mathematics
The research focuses on development of mathematical models and advance numerical algorithms, as well as applications of mathematics to interdisciplinary science and engineering areas. The research areas include: modeling and simulation in fluid dynamics and materials science, computational fluid dynamics, multiscale modeling, kinetic theory, atomistic simulations, image processing, computational harmonic analysis, inverse problems and wave propagation, graph theory, optimization, evolutionary genetics, environmental science, numerical weather prediction, ocean and coastal modeling, numerical PDEs and numerical linear algebra, parallel algorithms, and numerical software.
Financial Mathematics
Traditionally, financial mathematics includes three areas: asset allocation/management, derivative modeling, pricing and hedging, and division or firmwise risk management. The mathematical tools include probability, stability, stochastic calculus and numerical analysis. The recent trend of financial mathematics research, however, has been data driven, parallel to the development in the financial market where both sellside and buyside firms are trying to adopt the methodologies of big data and machine learning, in order to gain advantages in proprietary trading, market making, hedging and investments. Some faculties of the department working in related fields have embraced the changes and adopted the new technologies in their teaching and research.
Probability and Statistics
Statistics, the science of collecting, analyzing, interpreting, and presenting data, is an essential tool in a wide variety of academic disciplines as well as for business, government, medicine and industry. Our research is conducted in four categories. Time Series and Dependent Data: inference from nonstationarity, nonlinearity, longmemory behavior, and continuous time models. Resampling Methodology: block bootstrap, bootstrap for censored data, and Edgeworth and saddle point approximations. Stochastic Processes and Stochastic Analysis: filtering, diffusion and Markov processes, and stochastic approximation and control. Survival Analysis: survival function and errors in variables for general linear models. Probability current research includes limit theory.
Data Science
Statistics and data science play a more and more important role in scientific research and applications, particularly in the era of big data. Some research interests include random matrix, survival analysis, financial statistics, econometrics, bioinformatics, statistical machine learning, statistical genetics and genomics, computational biology and bioinformatics, error in variables model, generalized linear models; biological and medical statistics, topological and geometric methods in data analysis.
We also develop algorithms that provably and significantly improve the stateoftheart for mathematical data analysis, in terms of accuracy, running time, memory efficiency, scalability, or other computational measures. This includes adapting existing algorithms or approaches to exploit contemporary computing architectures, new statistical paradigms, or heterogeneous information sources. We investigate the solution of specific practical data science problems that use novel methodology and are of broad interest. These problems include imaging, high dimensional data analysis, machine learning, social networks, genetics and genomics.
Facilities
The Department enjoys a range of uptodate facilities and equipment for teaching and research purposes. It has two computer laboratories and a Math Support Center equipped with 100 desktop computers for undergraduate and postgraduate students. The Department also provides an electronic homework system and a storage cloud system to enhance teaching and learning.
To assist computations that require a large amount of processing power, a High Performance Computing (HPC) laboratory was setup in 2000. The highspeed computers in this laboratory in which InfiniBand technology and GPU technology are applied are capable of delivering a processing power of 150 TFLOPS practically. By making use of these powerful computing facilities, our faculty and postgraduate students are able to solve computationally intensive problems in their innovative research projects so that they can stay at the forefront of their fields.

Minimum Credit Requirement
MPhil: 24 credits
PhD: 36 credits

Credit Transfer
PhD students who have obtained an MSc or MPhil degree from other institutions may be granted credit transfer of up to 18 credits, subject to departmental approval.

Required Courses
MPhil:
24 credits in mathematics or related fields, normally including at least 18 credits of mathematics courses at postgraduate level.
PhD:
36 credits in mathematics or related fields, including at least 24 credits of mathematics courses at postgraduate level.
PhD students can choose to focus their research in one of the three areas:

Pure Mathematics;

Applied Mathematics; and

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

Graduate Teaching Assistant Training
All fulltime RPg students are required to complete PDEV 6800. The course is composed of a 10hour training offered by the Center for Education Innovation (CEI), and session(s) of instructional delivery to be assigned by the respective departments. Upon satisfactory completion of the training conducted by CEI, MPhil students are required to give at least one 30minute session of instructional delivery in front of a group of students for one term. PhD students are required to give at least one such session each in two different terms. The instructional delivery will be formally assessed.

Professional Development Course Requirement
Students are required to complete PDEV 6770. The 1 credit earned from PDEV 6770 cannot be counted toward the credit requirements.
PhD students who are HKUST MPhil graduates and have completed PDEV 6770 or other professional development courses offered by the University before may be exempted from taking PDEV 6770, subject to prior approval of the School.
Students are required to complete MATH 6771. The 1 credit earned from MATH 6771 cannot be counted toward the credit requirements.
PhD students who are HKUST MPhil graduates in Mathematics and have completed MATH 6771 before may be exempted from taking MATH 6771, subject to prior approval of the Department Head or Department PG Coordinator.

English Language Requirement
Fulltime RPg students are required to take an English Language Proficiency Assessment (ELPA) Speaking Test administered by the Center for Language Education before the start of their first term of study. Students whose ELPA Speaking Test score is below Level 4, or who failed to take the test in their first term of study, are required to take LANG 5000 until they pass the course by attaining at least Level 4 in the ELPA Speaking Test before graduation. The 1 credit earned from LANG 5000 cannot be counted toward the credit requirements.
Students must complete and pass LANG 5010, which should be taken in the first year of study. The 1 credit earned from LANG 5010 cannot be counted toward the credit requirements.
PhD students who are HKUST MPhil graduates may be considered for exemption from this requirement, subject to prior approval from the Department Head and PG Coordinator.

Postgraduate Seminar
Fulltime students are required to complete MATH 6900 in their first four regular terms of study. The maximum number of credits to be earned from this course is 4.

PhD Qualifying Examination
PhD students are required to pass a qualifying examination.
Pure Mathematics
To become PhD candidates, students must first pass a written qualifying examination (normally at the end of the first year of study) on two of the three subject areas: analysis, algebra, and geometry. At a later date (normally no later than the end of the second year of study) an oral examination on a major area excluding the two areas covered in the written examination.
Applied Mathematics
To become PhD candidates, students must first pass a written preliminary examination (normally before the end of the first year of study) on two subjects: advanced calculus and linear algebra. Students must also submit a thesis proposal, and pass an oral examination on the thesis proposal and two minor subjects. The oral examination should normally take place before the end of the second year of study.
Probability and Statistics
To become PhD candidates, students must pass an oral qualifying examination on one major subject and two minor subjects (normally no later than the end of the second year of study). For Probability students, the major subject is Probability while the two minor subjects would be Statistics and an area in Mathematics or an appropriate area outside Mathematics. For Statistics students, the major subject is Statistics while the two minor subjects would be Probability and an area in Mathematics or an appropriate area outside Mathematics.

Thesis Research
MPhil:

Registration in MATH 6990; and

Presentation and oral defense of the MPhil thesis.
PhD:

Registration in MATH 7990; and

Presentation and oral defense of the PhD thesis.
Last update: 10 August 2020
Scientific Computation Concentration
In addition to the existing program requirements, students who opt for the Scientific Computation concentration are required to:

MPhil:
Complete a minimum of 7 credits from the following course lists.
PhD:
Complete a minimum of 10 credits from the following course lists.
The credits earned under the concentration will be counted toward the total credit requirements of the programs.
Core Courses
MPhil: at least 3 credits
PhD: at least 6 credits
All students must take MATH 6915, which can be repeated for up to 2 credits.
Elective Courses

Give a onehour seminar on computation related research within their first four regular terms of study.

Conduct research in the area of scientific computation.
Last update: 10 August 2020
To qualify for admission, applicants must meet all of the following requirements. Admission is selective and meeting these minimum requirements does not guarantee admission.

Applicants seeking admission to a master's degree program should have obtained a bachelor’s degree from a recognized institution, or an approved equivalent qualification;

Applicants seeking admission to a doctoral degree program should have obtained a bachelor’s degree with a proven record of outstanding performance from a recognized institution; or presented evidence of satisfactory work at the postgraduate level on a fulltime basis for at least one year, or on a parttime basis for at least two years.
Applicants have to fulfill English Language requirements with one of the following proficiency attainments:

TOEFLiBT: 80*

TOEFLpBT: 550

TOEFLRevised paperdelivered test: 60 (total scores for Reading, Listening and Writing sections)

IELTS (Academic Module): Overall score: 6.5 and All subscore: 5.5
* refers to the total score in one single attempt
Applicants are not required to present TOEFL or IELTS score if

their first language is English, or

they obtained the bachelor's degree (or equivalent) from an institution where the medium of instruction was English.