Graduation Requirement for PhD students
Students admitted up until AY2022/23: Click here
Students admitted from AY2023/24 onwards: Click here
Continuation Requirement (PhD)
- A student will be issued a warning for any semester in which his/her GPA falls below 3.50.
- If in the following semester, the student’s GPA again falls below 3.50 but above 3.00, he/she will be placed on probation.
- The candidature of a student may be terminated if he/she obtains the following:
- A GPA of less than 3.00 for two consecutive semesters;
- A GPA of less than 3.50 for three consecutive semesters.
- Research scholars who do not meet the continuation requirements, their monthly stipend and/or scholarship may be terminated without notice. Reinstatement will be considered after he/she meets the continuation requirement.
Objective and Important Dates
Objective
The QE serves to ensure that a PhD candidate has sufficient background and breadth in mathematics for him/her to embark on his/her research. New graduate students are admitted to the Graduate Programme (by default). Students who are interested in the PhD degree must pass the QE by the end of their 24th month candidature before they are allowed to convert or upgrade from Graduate Programme to PhD programme.
Graduate students who fail the QE or students who choose not to sit for the QE will complete the requirements for an MSc (by Research) degree.
Important Dates
| To sit for exam in | Submit application by | |
| Semester 1, 2025/26 | 15 Jun 2025 | |
| Semester 2, 2024/25 | 15 Nov 2024 | |
| Exam | Exam date in Semester 1 | Exam date in Semester 2 |
| Paper 1 | 5 Aug 2024 | 6 Jan 2025 |
| Paper 2 | 7 Aug 2024 | 8 Jan 2025 |
| Paper 3 | 7 Aug 2024 | 12 Jan 2025 |
| Paper 4 | 6 Aug 2024 | 10 Jan 2025 |
Papers 1, 4, 2 and 3 are always held on Monday, Tuesday, Wednesday and Friday respectively, in the week before week 1 of a semester. Examination will be rescheduled if it falls on a public holiday.
Examination Format & Registration
Format
There are two components:
- Comprehensive (Written) Examination
- Qualifying (Oral) Examination
Student must pass both components in order to be converted/upgraded to read the PhD programme.
Registration
Registration is done online through a survey form, which will be sent out via email, at least one month before the closing date of respective semester.
Topics for Comprehensive Exam
Paper 1 – Algebra
Syllabus
- Sets: Cardinals, ordinals. Countability. Zorn’s Lemma.
- Linear algebra: Finite-dimensional vector spaces, bases. Tensor product. Isomorphism ofMn(F) and End(Fn). Orthogonality, examples of classical groups. Diagonalization, Cayley-Hamilton theorem, spectral theorem, Jordan canonical form.
- Group theory: Significance of classification of finite simple groups. Central and derived series. Structure of finitely generated abelian groups. Group presentations. Representations of finite groups.
- Ring and module theory: Euclidean domain, principal ideal domain, unique factorization domain. Polynomial rings, reducibility. Noetherian rings, Hilbert basis theorem. Some noncommutative rings, e.g. matrix rings. Free and projective modules. Exactness.
- Field theory: Fundamental theorem of algebra, algebraic closure. Classification of finite fields. Examples of Galois groups.
- Category theory: Examples of categories, functors, natural transformations, adjoint functors.
Suitable textbooks
- G D Crown, M H Fenrick & R J Valenza,Abstract Algebra, Marcel Dekker (NY, 1986)
- A I Kostrikin,Introduction to Algebra, Springer Universitext (NY, 1982)
Suitable reference books
- T W Hungerford,Algebra, Springer Graduate Texts in Math 73 (NY, 1974)
- S Lang,Algebra, Springer (NY, 2002), rev. 3rd ed.
AY2023/2024:
Student who scores A in both MA5203 Graduate Algebra I and MA5204 Commutative Algebra or MA5218 Representation Theory shall be deemed to have passed paper 1.
Student who scores A in both MA5204 Commutative and Homological Algebra and MA5218 Representation Theory shall be deemed to have passed paper 1.
From AY2024/2025 onwards:
Student who scores A in both MA5204 Commutative and Homological Algebra and MA5218 Representation Theory shall be deemed to have passed paper 1.
Paper 2 – Analysis
Syllabus
Part I: Advanced Calculus
- Properties of the reals such as Bolzano-Weierstrass, Heine-Borel and equivalent of norms inRn.
- Differential calculus ofRmvalued functions on subsets ofRn. Continuity and uniform continuity, differentiability, partial derivatives, Jacobians, implicit and inverse function theorems.
- Differential equations: existence and uniqueness theorem of initial value problems.
Infinite sequences and series of numbers and functions. Absolute and uniform convergence, equi-continuity, Arzela-Ascoli theorem, Weierstrass approximation theorem. - Riemann and Riemann-Stieltjes integrals, fundamental theorem of Calculus.
- Line integrals, surface integrals, differential forms. The theorems of Stokes and Green and the divergence theorem. Change of variables in multiple integrals.
- Metric spaces, completeness, limit and continuity.
Part II: Real Analysis
- Functions of bounded variation and absolutely continuous function.
- Definition and elementary properties of Lebesgue measure.
- Borel measures, measurable functions and simple functions.
- Lebesgue integral and its elementary properties.
- Convergence theorems.
- Various types of convergence such as almost everywhere, in measure, in mean.
- Multiple integrals and changing the order of integration (Fubini’s theorem).
- Lebesgue’s differentiation theorem, Vitali’s covering lemma.
- Basic properties ofLpspaces, such as density ofC¥functions, approximation identities, Riesz representation theorem.
- Hilbert space and its basic properties [AY2024/25 semester 1]
- Hilbert space and its basic properties [AY2024/25 semester 2]
Part III: Complex Variables
- Cauchy-Riemann equations. Analytic functions. Contour integration. Cauchy integral formula. Taylor series. Residues and poles. Laurent series. Isolated singular points, removable/essential singularities, poles, residues, residue theorem, improper real integrals and their evaluation using the residue theorem. The argument principle. The open mapping theorem and the maximum modulus principle. Conformal mapping and linear fractional transformations. Harmonic functions.
Part IV: Functional Analysis
- Normed linear spaces and their dual spaces, Banach spaces, Hilbert spaces;
- Hahn-Banach theorems and applications;
- The open mapping theorem, The uniform boundedness principle and Banach-Alaoglu theorem.
Remarks
The paper will test more on content instead of tricks. Students are usually expected to score at least 60% of the total mark in order to pass the paper.
Recommended textbooks
- (For Part I of this syllabus) Walter Rudin:Principles of Mathematical Analysis
(first 10 chapters), 3rd edition, McGraw Hill - (For Part II of this syllabus) H.L. Royden:Real Analysis, 3rd edition, Macmillan
- (For Part II of this syllabus) R. Wheeden, A. Zygmund:Measure and Integral: An Introduction to Real Analysis, Marcel Dekker
- (For Part III of this syllabus) James W. Brown, Ruel V. Churchill:Complex Variables and Applications, 7th edition, McGraw Hill
- (For Part IV of this syllabus) Introductory functional analysis with applications by Erwin Kreyszig
*From January 2025 onwards.
Student who scores A in both MA5205 Graduate Analysis I and MA5206 Graduate Analysis II or MA5213 Advanced Partial Differential Equations shall be deemed to have passed paper 2.
Paper 3 – Computational Mathematics
Syllabus
- Fundamentals of Computational Mathematics
- Approximation theory: polynomial interpolation; piecewise polynomial interpolation; orthogonal polynomials; least squares approximation
- Numerical integration: trapezoidal rule, Simpson’s rule and Newton-Cotes formulas; composite trapezoidal rule and Simpson’s rule; Richardson extrapolation and Romberg integration
- Matrix computation: matrix norms and vectors norms; direct and iterative methods for linear system (basic and Krylov subspace iterative methods); eigenvalue problem; QR factorization; singular value decomposition; linear least squares problem; Iterative methods for nonlinear systems: fixed point methods, Newton’s method;
- Numerical Solution of ODEs and PDEs
- For initial value problems: Runge-Kutta methods; one-step methods; multi-step methods; consistency, stability and convergence
- For two-point boundary value problems: shooting and finite difference methods
- Finite difference discretization for Poisson’s equation, heat equation and wave equation; consistency and stability analysis
- Optimization Theory and Computation
- Optimization theory: convex sets, hulls, cones and polar cones; convex functions; subgradients; KKT-optimality conditions; Lagrangian dual problems; weak and strong duality theorems
- Numerical optimization: Newton’s method and conjugate gradient methods for unconstrained optimisation; exterior penality function methods; Barrier function methods; Methods of feasible directions
Recommended textbooks
- Richard L. Burden and J. Douglas Faires, Numerical Analysis, 8th edition, Thomson/Brooks/Cole, 2005.
- L.N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, 1997.
- G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edition, Oxford University Press, 1985.
- J. Nocedal and S.J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.
- M. Bazaraa. H. Sherali and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd edition (1993) or 3rd edition (2006), Wiley.
Student who scores A in both MA5233 Computational Mathematics and MA5243 Advanced Mathematical Programming shall be deemed to have passed paper 3.
Paper 4 – Stochastic Processes and Machine Learning
Syllabus
- Fundamentals of Stochastic Processes
- Basic probability theory, including properties of fundamental discrete and continuous distributions
- Borel-Cantelli lemma,
- modes of convergence of random variables and distributions,
- weak and strong law of large numbers
- central limit theorem,
- large deviation,
- Markov chain theory including recurrence, transience, convergence
- martingale theory including Doob’s inequality, martingale convergence, Hoeffding’s inequality
- Fundamentals of Machine Learning
- The PAC Learning Framework: PAC model for learning, generalizations to stochastic cases
- Basic learning theory: Rademacher Complexity and VC Dimension
- Model Selection: cross-validation, regularization, surrogate losses
- Support Vector Machines: primal and dual formulations, separable and non-separable cases, multi-class SVMs
- Kernel methods: positive definite symmetric kernels, kernel ridge regression and kernel SVM
- Boosting: Adaboost, gradient boosting
Recommended textbooks
- Grimmett and D. Stirzaker, Probability and Random Processes, 4th ed.
- R. Durrett. Probability: Theory and Examples. 5th ed.
- Mohri, Mehryar, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of Machine Learning. MIT press, 2018.
- Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The Elements of Statistical Learning. Vol. 1. 10. Springer series in statistics New York, 2001.
Remarks
-
- Two courses are offered to help students with preparing for this paper – DSA5105 Principles of Machine Learning (offered in semester 1 per year) and MA5260 Advanced Probability.
- From AY2024/2025 onwards: (1) Student who scores A in both DSA5105 and MA5249 shall be deemed to have pass paper 4 OR (2) Students who have scored A/A+ in MA5260 Advanced Probability and DSA5105 Principles of Machine Learning shall be deemed to have passed QE paper 4.
- From AY2025/26 onwards: Students who have scored A/A+ in MA5260 Advanced Probability and DSA5105 Principles of Machine Learning shall be deemed to have passed QE paper 4.
Sample Papers
AY21/22-semester 1
paper 1
paper 2
paper 3
paper 4
AY21/22-semester 2
paper 1
paper 2
paper 3
paper 4
AY22/23-semester 1
paper 1
paper 2
paper 3
paper 4
AY22/23-semester 2
paper 1
paper 2
paper 3
paper 4
AY23/24-semester 1
paper 1
paper 2
paper 3
paper 4
AY23/24-semester 2
paper 1
paper 2
paper 3
paper 4
PhD QE Exemption Request Form
Students can raise for PhD QE Exemption, if they meet the requirements for exemption stated under “Topics for Comprehensive Exam”.
Please submit your request and your unofficial transcript in the e-form.
Administrators will notify you on the status of your exemption request within 2 working weeks.
- Unless otherwise stated, Graduate students are expected to register for courses themselves at Course Registration at Education Records System (CourseReg@EduRec). Students are also strongly encouraged to refer to the CourseReg User Guide for Students, which provides screenshots on how to navigate with CourseReg.
- Students can check the ‘Courses & Timetables‘ section to find out about curriculum update, courses offered, courses & tutorial registration and class and examination timetables.
Courses Offered
- Students may refer to the ‘Courses Offered‘ page to find out what courses are offered in the Academic Year.
Subject to departmental approval, students are allowed to substitute up to two MA-courses with approved level 5000 courses offered by other departments to fulfill the graduation requirements.
Important Notes:
- Graduation requirements for programme: 4 MA courses in the basic elective list must be completed and cannot be substituted.
- Departmental Approval: While students do not need to seek internal approval to take courses coded as QF5xxx, DSA5xxx, ST5xxx or CS5xxx, please inform department about your intention to substitute the non MA coded course with a corresponding MA-coded elective. Submit your intention via the e-form.
- For other courses not stated in point (2), students must seek permission by writing to the Department before the start of course registration in any semester.
- For QF5xxx, DSA5xxx, ST5xxx or CS5xxx, while students do not require the permission of the department to apply for these courses, the approval of these courses are determined by the MDSML/MQF programme managers, Department of Statistics and Data Science or the School of Computing.
Procedure:
- Go to NUSMODS to retrieve the course description of the course that you wish to read as a substitute for MA-coded course.
- Submit your request for course substitution via this link at the start of the semester. Deadline for course substitution request: End of 1st instructional Week of Semester.
- Department Administrator will share the requests with Associate Prof Cai Zhenning to seek his approval by Week 2, Wednesday.
- If you are a newly admitted student requesting for credit transfers together with course substitution, then the transcript that you should send is your last undergraduate transcript.
- Once the request is supported, proceed to do a “submit course request” through CourseReg. Please refer to the CourseReg schedule to find out the date where ‘Submit Couse Request’ function is available in the system.
- Final approval can only be granted by the course host. For e.g. ST5214 is owned by Department of Statistics and Data Science (DSDS). If student submits request to read the course, DSDS will make a decision on your request.
- Students are strongly encouraged to read up more information on CourseReg since all course registration must be done through this system.
- Students are required to submit a bi-annual research progress report via EduRec (Academics > Graduate Research > Submit Research Progress Report).
- Students will need to provide a summary of the work they did in the previous semester and what they will be doing in the current semester.
- The Research Progress Report submission periods are September (for assessment of research progress in Semester 2 of the previous Academic Year) and February (for assessment of research progress in Semester 1 of the current Academic Year).
- Students will be notified via email of the submission period. You will also be informed via email to collect your GAP summary sheet from the department. Upon receiving both emails, you should complete the submission online promptly to allow sufficient time for their supervisor(s), Head of Department/Programme and Faculty to complete their assessments and recommendations for continuation of candidature.
- For scholars, scholarship renewal will be tied in with the semesteral progress report. Students are to note that your scholarship will be suspended if you do not complete the research progress report by the deadline.
- Students who are in their first term (semester) of study are not required to submit the Research Progress Report.
- Please refer to http://www.askstudentservice.nus.edu.sg/ for the FAQs on the Research Progress Report Workflow.
Supervision
- The Department requires all PhD students to confirm their thesis advisor (previously known as supervisor) by the end of their first year in NUS.
Thesis Matters
- Information on thesis submission may be obtained from the Student Portal.
Leave Application
- There are various type of leave schemes available to Graduate Students. Please click here for more information.
Students who met their graduation requirements are not required to file for graduation as Department will do this for you. Please click the respective link below to find out information about:
2) Conversion of candidature (full-time to part-time or to PhD to MSc)
3) Change Thesis Title
4) Change Thesis Advisor
Please submit an e-form on Edurec.
Steps and Guide to submit an e-form for conversion of candidature.
Steps and Guide to submit an e-form for extension of candidature.
Steps and Guide to submit an e-form for Change of Thesis Title.
Steps and Guide to submit an e-form for Change of Thesis Advisor.
- Department will complete the assignment of GAP duties and hours to all PhD students on the myGAP system.
- All students need to submit their GAP hours near the end of the semester on myGAP: https://wws.nus.edu.sg/mygap/auth/login
- User Guide available here.