## Qualifying Examination

The Qualifying Examination (QE) tests 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, QE must be completed **within 18 months** from enrolment in the PhD programme.

Before being allowed to take QE, 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 QE 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 written QE is 50%.

**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)

### 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**.

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 oral QE is 65%.