## PhD Programme

The PhD in Mathematical Sciences programme is a four-year programme that provides graduate students with a stimulating environment for conducting original research in the mathematical sciences. Graduate students take advanced courses focusing on active research topics, and perform research under the supervision of a faculty member. Our students receive intensive support in their theoretical work, as well as in practical and computational aspects (where appropriate), including access to state-of-the-art computational resources. The programme culminates in writing and defending a doctoral research thesis before a panel of experts.

The PhD programme has a minimum candidature period of 2 years, and a maximum of 5 years. Most students complete the programme in 4 years.

# Graduation Requirements

The PhD programme has a minimum candidature period of 2 years, and a maximum of 5 years. Most students complete the programme in 4 years.

Applicants must satisfy the following criteria:

- BSc in Mathematics, Mathematical Sciences, or a very closely related discipline (for those with degrees in other fields, see below). If the university has an honours system, at least second-upper class honours, or the equivalent, is required.
- For the PhD programme, applicants may optionally have a MSc degree in Mathematics or a related discipline, but this is not mandatory. If the university has an honours system, at least second-upper class honours in the MSc degree, or the equivalent, is required.
- International applicants must have GRE General Test scores or GATE scores. A GRE subject score in mathematics is welcome but not required.
- International applicants who are not native English speakers must have TOEFL scores or IETLS scores.

There are two application periods each year: October to January (for admission in August), and June to July (for admission in January). Most students are admitted during the first period.

For more information about admission procedures (including the list of required supporting documents and application fees), please visit NTU Graduate Admissions page.

### Applicants with Bachelor Degrees in Other Fields

Applicants without a Bachelor degree in Mathematics may be considered for admission if they have a Bachelor degree (or equivalent) in a related area, such as Computer Engineering or Natural Sciences, and have sufficient mathematical background. In addition to the other requirements, such students must submit the following documents together with their application:

- A complete list of courses taken in Mathematics and related areas, with detailed description of the course contents.
- A list of any further material relevant for the mathematical background, e.g. an annotated list of mathematics textbooks used for self-study.
- An informal essay describing their motivation to enter graduate studies in the Mathematical Sciences.

Students must complete a total of 16 Academic Units (AU) of graduate-level coursework, consisting of:

- At least two MAS71X modules, 4 AU each
- At least one MAS79X (Graduate Seminar) module, 4 AU
- One additional MAS7XX module (apart from MAS79X), 4 AU

Alternatively, the student may take one or two courses offered by other Schools/Divisions, counting for at least 4 AU, subject to approval from the Division and the other Schools/Divisions.

Students are expected to maintain a minimum CGPA of 3.5. We recommend finishing the coursework by the end of the first year, in order to sit for the Qualifying Examinations on time.

### Communication Courses

Students are also required to complete the following courses on communication and related topics. These courses do not carry any AU.

- Courses from the Transferable Skills Programme:
- Residential Programme (during first 12 months)
- Scholarly Communication and Impact (during first 12 months)
- Any 3 elective modules
- Career Preparation Workshop (during 3rd or 4th year)
- Three Minute Thesis Presentation Symposium (for students admitted from Academic Year 2019/2020 onwards)

- HWG703 - Graduate English (during first 18 months; must be taken before HWG702; part-time students and students who qualify as native English speakers are exempted)
- HWG702 - University Teaching for Teaching Assistants (during first 18 months; part-time students are exempted)
- HWG704 - Research Communication for Graduate Studies (during first 24 months)

- Online NTU Epigeum Research Integrity Course (during first semester)
- SPMS Research Integrity Course (during first year)
- Seminar attendance of at least 5 seminars per semester.
- Completion of Graduate Assistantship Programme (GAP), if applicable
- Annual meeting with Thesis Advisory Committee (TAC) members
- Submission of regular progress reports, including list of publications, TAC reports, and degree audit

The Qualifying Examinations (QEs) test a PhD student's mastery of mathematical fundamentals, research progress, and ability to answer technical questions from other mathematicians. Unless special permission is granted by School, the QEs must be completed **within 18 months** from enrolment in the PhD programme.

Before being allowed to take the QEs, PhD students must have completed the required MAS7XX courses with a CGPA of at least 3.5, and must have completed the communication courses HWG703 and HWG702.

The QEs for the Division of Mathematical Sciences consist of two parts: a written QE and an oral QE. Each student is allowed to take each part of the QE up to two times. The written QE must be passed before taking the oral QE.

### Written QE

Written QEs are conducted twice a year, once in January and once in July. The duration is 3 hours.

Each written QE consists of the following 5 topics:

- Algebraic Methods
- Continuous Methods
- Discrete Methods
- Mathematical Statistics
- Algorithms and the Theory of Computation

Students are to choose **2 out of 5 topics** from the exam paper. Syllabus details are given below.

The passing grade for the written QE is 50%.

### Oral QE

Students are only allowed to take the oral QE after passing the written QE, and they must take it **within 2 months after passing the written QE**.

The Oral QE has a duration of 40 minutes, consisting of a 30 minute presentation about the student's research progress, and a 10 minute Q&A session. It is assessed by three faculty examiners.

The passing grade for the oral QE is 65%.

**1. Algebraic Methods (MAS712)**

Syllabus: Group action, the Sylow Theorems, applications of the Sylow Theorems, solvable and nilpotent groups, direct and semidirect products of abelian groups, ring homomorphisms, polynomial rings,
unique factorization domains, principal ideal domains, Euclidean domains, irreducibility criteria, splitting fields, normal extensions, separable extensions, algebraic closure, the fundamental theorem of Galois Theory, computing Galois group of polynomials.

Textbooks and References:

- D. S. Dummit and R. M. Foote,
*Abstract Algebra (3rd ed.)*, John Wiley & Sons, Inc., Hoboken NJ, 2004. Relevant Chapters: 1–9, 13,14.2 - T. W. Hungerford,
*Algebra*, Springer-Verlag, New York-Berlin, 1974. Relevant Chapters: I–III, V.3. - R. Ash,
*Abstract Algebra: The Basic Graduate Year*[Online Lecture Notes]. Relevant Chapters: 2, 3, 5, 6.

**2. Continuous Method****s (MAS710)**

Syllabus: The following chapters and sections from Rudin's textbook -

- Abstract integration, basic topology, measures and measurability (Chapter 1).
- Positive Borel measures, Lebesgue measure, Riesz representation theorem (Chapter 2).
- L
*p*-spaces, approximation by continuous functions (Chapter 3). - Differentiation, the fundamental theorem of calculus (FTC) (Chapter 7, up to and including the section on the FTC).
- Integration on product spaces, Fubini's theorem (Chapter 8, up to and including the section on Fubini's theorem).
- Holomorphic functions, Cauchy's theorem, power series, residues (Chapter 10).

Textbooks and References:

- W. Rudin,
*Real and Complex Analysis*(3rd ed.), McGraw-Hill 1976 (**main textbook**) - L. Ahlfors,
*Complex Analysis*(3rd ed.), McGraw-Hill, 1979. - B. P. Palka,
*An Introduction to Complex Function Theory*, Springer, 1991. - R. Wheeden and A. Zygmund,
*Measure and Integral: An Introduction to Real Analysis*, CRC, 1977.

**3. Discrete Methods**** (MAS711)**

Syllabus:

- Basic notions of graph theory, Euler circuits and Euler trails, Minimum spanning trees, Prim's and Kruskal's algorithms, Prüfer codes. Network flows, Ford-Fulkerson algorithm, Augmenting Path Theorem, Maximum Flow-Minimum Cut Theorem, Minimum cost flows, Network simplex algorithm.
- Linear programs, Duality Theorem, The Structure of Polyhedra, Extreme points, Simplex algorithm.

Textbooks and References:

- R. K. Ahuja, T. L. Magnanti, and J. B. Orlin,
*Network flows. Theory, algorithms, and applications*. Prentice Hall. - M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali,
*Linear programming and network flows (4th ed.)*, John Wiley & Sons. - R. Diestel,
*Graph Theory*, Graduate Texts in Mathematics. - A. Schrijver,
*Theory of Linear and Integer Programming*, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons.

**4. Mathematical Statistics**** (MAS713)**

Syllabus: Probability, random variables and their distributions, moments and inequalities, point estimation in parametric setting, point estimation in nonparametric setting, interval
estimation and hypothesis testing, asymptotic evaluation and robustness.

Textbooks and References:

- G. Casella and R. L. Berger,
*Statistical Inference*(2nd ed.), Duxbury Thomson Learning, 2001. - P. Bickel and K. A. Doksum,
*Mathematical Statistics*(vol. 1, 2nd ed.), Prentice-Hall, 2006. - S. Jun,
*Mathematical Statistics*(2nd edition), Springer, 2003. (Reference book)

**5. Algorithms and the Theory of Computing**** (MAS714)**

Syllabus: Turing machines, decidability, time complexity, space complexity, algorithm design and analysis (greedy, divide and conquer, dynamic programming), graph
algorithms, network flow, approximation algorithms.

Textbooks and References:

- M. Sipser,
*Introduction to the Theory of Computation*(2nd ed.), Thomson, 2005. - J. Kleinberg and E. Tardos,
*Algorithm Design*, Addison Wesley, 2005. - C. Papadimitriou,
*Computational Complexity*, Addison Wesley, 1993. (Reference book) - T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein,
*Introduction to Algorithms*(2nd ed.), MIT Press, 2001. (Reference books)

Each PhD student must write a research thesis. Once the student and supervisor have agreed that the thesis is ready, it should be submitted online (via GSLink → Academic → Thesis → Thesis Submission). After being endorsed by the supervisor, the thesis is sent to three independent examiners for evaluation. After this evaluation, the student must defend the thesis in an oral examination scheduled by the school.

Completed theses must be posted to the NTU Digital Repository.