# Postgraduate Courses

MATH

Mathematics

For undergraduate courses, those coded from 1000 to 1600 are designed to standardize the varying range of mathematics background of students. Students should consult with their own school advisor which MATH course they should take.

- MATH 5011Advanced Real Analysis I[3-0-0:3]Previous Course Code(s)MATH 501BackgroundMATH 3033DescriptionBasic topology, continuous function spaces, abstract measure and integration theory, Lp spaces, convexity and inequalities, Hilbert spaces, Banach spaces, Complex measure.
- MATH 5030Complex Function Theory[3-0-0:3]Previous Course Code(s)MATH 503BackgroundMATH 3033 and MATH 4023DescriptionReview of basic properties of analytic functions. Phragmen-Lindelof principle, normal family, Riemann mapping theorem. Weierstrass factorization theorem, Schwarz reflection principle, analytic continuation, harmonic function, entire function, Hadamard factorization theorem, Picard theorem.
- MATH 5111Advanced Algebra I[3-0-0:3]Previous Course Code(s)MATH 511BackgroundMATH 3121 and MATH 4121 (prior to 2014-15)DescriptionAdvanced theory of groups, linear algebra, rings, modules, and fields, including Galois theory.
- MATH 5112Advanced Algebra II[3-0-0:3]Previous Course Code(s)MATH 512BackgroundMATH 5111DescriptionAdvanced topics in algebra: group representations, associative algebras, commutative algebra, homological algebra, algebraic number theory.
- MATH 5143Introduction to Lie Algebras[3-0-0:3]Previous Course Code(s)MATH 514Prerequisite(s)MATH 2131 and MATH 3131DescriptionLie algebras. Nilpotent, solvable and semisimple Lie algebras. Universal enveloping algebras and PBW-theorem. Cartan subalgebras. Roots system, Weyl group, and Dynkin diagram. Classification of semisimple Lie algebras. Representations of semisimple algebras. Weyl character formula. Harish-Chandra isomorphism theorem.
- MATH 5145Introduction to Lie Groups[3-0-0:3]Previous Course Code(s)MATH 515Prerequisite(s)MATH 5111 and MATH 5143DescriptionThis course is an introduction to the structure and representation theory of compact and noncompact reductive Lie groups. Topics include general properties of Lie groups and Lie algebras, Peter-Weyl Theorems, representations of compact Lie groups, theorems of the highest weight, Harish-Chandra isomorphism, Weyl character formula, the structure theory of noncompact semisimple and reductive Lie groups, classification of simple real Lie algebras, induced representation and Frobenius reciprocity, classical branching theorems.
- MATH 5230Differential Topology[3-0-0:3]Previous Course Code(s)MATH 523BackgroundMATH 4225DescriptionManifolds, embedding and immersion, Sard's theorem, transversality, degree, vector fields, Euler number, Euler-Poincare theorem, Morse functions.
- MATH 5240Algebraic Topology[3-0-0:3]Previous Course Code(s)MATH 524DescriptionFundamental group, covering space, Van Kampen theorem, (relative) homology, exact sequences of homology, Mayer-Vietoris sequence, excision theorem, Betti numbers and Euler characteristic.
- MATH 5251Algebraic Geometry I[3-0-0:3]BackgroundMATH 5111 or equivalent postgraduate algebraDescriptionProjective spaces, algebraic curves, divisors, line bundles, algebraic varieties, coherent sheaves, schemes. Some commumative algebra and homological algebra such as notherian ring, regular ring, valuation ring, kahler differentials.
- MATH 5261Algebraic Geometry II[3-0-0:3]Prerequisite(s)MATH 5251BackgroundMATH 5111 or equivalent postgraduate algebraDescriptionDerived functors, cohomology of coherent sheaves on schemes, extension groups of sheaves, higher direct image of sheaves, Serre duality, flat morphisms, smooth morphisms, and semi-continuity, basics of curves and surfaces.
- MATH 5311Advanced Numerical Methods I[3-0-0:3]Previous Course Code(s)MATH 531DescriptionNumerical solution of differential equations, finite difference method, finite element methods, spectral methods and boundary integral methods. Basic theory of convergence, stability and error estimates.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Recognize and use appropriately numerical techniques in computation.
- 2.Develop numerical scheme to discretize partial differential equations.
- 3.Apply numerical analysis to examine the consistency, stability and convergence of the numerical methods.
- 4.Apply appropriate numerical schemes to solve real and hypothetical problems.

- MATH 5312Advanced Numerical Methods II[3-0-0:3]Previous Course Code(s)MATH 532Prerequisite(s)MATH 5311DescriptionDirect and iterative methods. Programming techniques and softwares libraries. Sparse solvers, Fast algorithms, multi-grid and domain decomposition techniques.
- MATH 5350Computational Fluid Dynamics for Inviscid Flows[3-0-0:3]Previous Course Code(s)MATH 535DescriptionDerivation of the Navier-Strokes equations; the Euler equations; Lagriangian vs. Eulerian methods of description; nonlinear hyperbolic conservation laws; characteristics and Riemann invariants; classification of discontinuity; weak solutions and entropy condition; Riemann problem; CFL condition; Godunov method; artificial dissipation; TVD methods; and random choice method.
- MATH 5351Mathematical Methods in Science and Engineering I[3-0-0:3]Previous Course Code(s)MATH 551DescriptionPerturbation Methods: Regular and singular perturbation. Boundary layer analysis. Calculus of variations, Hamiltonian theory. Stability and bifurcation, Hydrodynamic stability. Invariant variational problems, Noether theorem, Invariant PDEs, Self-similar solutions.
- MATH 5352Mathematical Methods in Science and Engineering II[3-0-0:3]Previous Course Code(s)MATH 552Prerequisite(s)MATH 5351DescriptionPartial differential equations of mathematical physics. Series and transform methods, method of characteristics. Soboley spaces and energy methods. Burger's equations, KdV equation and NLS equation. Soliton and nonlinear waves.
- MATH 5360Weather, Climate and Pollution[3-0-0:3]Previous Course Code(s)MATH 560DescriptionComposition and structure of the atmosphere; atmospheric thermodynamics; radiation balance; basic atmospheric dynamics, energy, momentum and water cycle of the earth; weather and climate of the Asian Pacific region; weather and pollution; variability of the climatic system; current issues (El-Nino, greenhouse warming).Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Apply physics laws of mechanics and thermodynamics to describe weather system and climate.
- 2.Qualitatively describe the various weather system and their relevance to pollution transport.
- 3.Solve real and hypothetical problems of air pollution.
- 4.Communicate the problem of climate change by using correct terminology.

- MATH 5380Combinatorics[3-0-0:3]Previous Course Code(s)MATH 538Prerequisite(s)MATH 2343 or MATH 3343BackgroundLinear algebra; CalculusDescriptionEnumerative Combinatorics: bijective counting, permutation statistics, generating functions, partially ordered sets, Mobius inversions, Polya theory. Graph Theory: cycle space, bond space, spanning-tree formulas, matching theory, chromatic polynomials, network flows. Matroid Theory: matroid axioms, representations, duality, lattice of flats, transversals.
- MATH 5411Advanced Probability Theory I[3-0-0:3]Previous Course Code(s)MATH 541DescriptionProbability spaces and random variables, distribution functions, expectations and moments, independence, convergence concepts, law of large numbers and random series.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Develop a rigorous mathematical framework to analyze randomness based on measure theory.
- 2.Derive probability inequality as a useful tool for mathematical analysis.
- 3.Demonstrate various forms of laws of large numbers and large deviation and their applications.
- 4.Establish the central limit theorem and present their applications.
- 5.Apply random walk theory to actuarial analysis.

- MATH 5412Advanced Probability Theory II[3-0-0:3]Previous Course Code(s)MATH 542DescriptionCharacteristic functions, limit theorems, law of the iterated logarithm, stopping times, conditional expectation and conditional independence, introduction to Martingales.
- MATH 5431Advanced Mathematical Statistics I[3-0-0:3]Previous Course Code(s)MATH 543DescriptionTheory of statistical inference in estimation. Topics include: sufficiency, ancillary statistics, completeness, UMVU estimators, information inequality, efficiency, asymptotic maximum likelihood theory. Other topics may include Bayes estimation and conditional inference.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Develop a rigorous mathematical framework and lay a firm foundation for statistical inference.
- 2.Recognize the most important concepts, including sufficiency, MLE, efficiency, risk, admissibility.
- 3.Employ appropriate statistical models for modeling and inference.
- 4.Recognize the different criterion for evaluation of statistical models.
- 5.Apply the theory in mathematical statistics to real-life problems.

- MATH 5432Advanced Mathematical Statistics II[3-0-0:3]Previous Course Code(s)MATH 544DescriptionTheory of statistical inference in hypothesis testing. Topics include: uniformly most powerful tests, unbiasedness, invariance, minimax principle, large-sample parametric significance tests. Concept of decision theory also covered.
- MATH 5433Statistical Learning[3-0-0:3]Previous Course Code(s)MATH 6450BBackgroundSome knowledge of linear algebra, basic probability and statistics would be very helpful.DescriptionThis course covers methodology, major software tools and applications in data mining and statistical learning. By introducing principal ideas in supervised and unsupervised learning, the course will help students understand conceptual underpinnings of methods in data mining and statistical learning.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Apply basic statistical learning methods to build predictive models.
- 2.Properly tune and select statistical learning models.
- 3.Correctly assess model fit and error.
- 4.Build an ensemble of learning algorithms.
- 5.Use statistical and machine learning software.

- MATH 5450Stochastic Processes[3-0-0:3]Previous Course Code(s)MATH 545DescriptionTheory of Markov processes, second order stationary theory, Poisson and point processes, Brownian motion, Martingales and queueing theory.
- MATH 5460Time Series Analysis[3-0-0:3]Previous Course Code(s)MATH 546DescriptionBasic idea of time series analysis in both the time and frequency domains. Topics include: autocorrelation, partial ACF, Box and Jerkins ARIMA modeling, spectrum and periodogram, order selection, diagnostic and forecasting. Real life examples will be used throughout the course.
- MATH 5510Mathematical Models of Financial Derivatives[3-0-0:3]Previous Course Code(s)MATH 571DescriptionBlack-Scholes-Merton framework, dynamic hedging, replicating portfolio. Martingale theory of option pricing, risk neutral measure. Option pricing under incomplete market. Stochastic volatility and jump-diffusion models. Path dependent options. Volatility derivatives.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Formulate and evaluate pricing models of derivatives.
- 2.Use appropriate hedging strategies for monitoring risks in derivatives.
- 3.Model dynamics of stock prices and commodity prices.
- 4.Use exotic derivatives in investment and wealth management.

- MATH 5520Interest Rate Models[3-0-0:3]Previous Course Code(s)MATH 572Exclusion(s)MAFS 5040DescriptionTheory of interest rates, yield curves, short rates, forward rates. Short rate models: Vasicek model and Cox-Ingersoll-Ross models. Term structure models: Hull-White fitting procedure. Heath-Jarrow-Morton pricing framework. LIBOR and swap market models, Brace-Gatarek-Musiela approach. Affine models.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Explain fixed-income markets and the roles of the fixed-income derivatives.
- 2.Apply advanced mathematical tools for fixed-income modeling.
- 3.Identify risk factors of fixed-income derivatives and formulate major classes of fixed-income models accordingly.
- 4.Evaluate the effectiveness of various models for different sectors of fixed-income derivatives.
- 5.Analyze exotic derivatives, identify proper pricing models and strategies for hedging.

- MATH 5530Volatility Smile Models[3-0-0:3]Previous Course Code(s)MATH 6380A, MATH 685VPrerequisite(s)MATH 5510DescriptionVolatility smile modeling actually is about option pricing in incomplete markets. We will introduce two classes of models, namely, stochastic volatility models and jump-diffusion models, and consider option pricing under models of these two classes. While stochastic volatility models are introduced on an individual basis, the jump-diffusion models are introduced under the framework of Levy models. The last part of the course is devoted to the pricing of volatility derivatives.
- MATH 6050Topics in Analysis[2-4 credits]Previous Course Code(s)MATH 605DescriptionAdvanced topics of current interest in analysis.
- MATH 6060Topics in Complex Function Theory[2-4 credits]Previous Course Code(s)MATH 606DescriptionAdvanced topics of current interest in complex function theory.
- MATH 6150Topics in Algebra[2-4 credits]Previous Course Code(s)MATH 615DescriptionAdvanced topics of current interest in algebra.
- MATH 6170Topics in Number Theory[2-4 credits]Previous Course Code(s)MATH 617DescriptionAdvanced topics of current interest in number theory.
- MATH 6250Topics in Geometry[2-4 credits]Previous Course Code(s)MATH 625DescriptionAdvanced topics of current interest in geometry.
- MATH 6380Topics in Applied Mathematics[2-4 credits]Previous Course Code(s)MATH 685DescriptionAdvanced topics of current interest in applied mathematics.
- MATH 6385Topics in Fluid Mechanics[2-4 credits]Previous Course Code(s)MATH 665DescriptionAdvanced topics of current interest in fluid mechanics.
- MATH 6388Topics in Numerical Analysis[2-4 credits]Previous Course Code(s)MATH 635DescriptionAdvanced topics of current interest in numerical analysis.
- MATH 6450Topics in Probability and Statistics[2-4 credits]Previous Course Code(s)MATH 645DescriptionAdvanced topics of current interest in probability and statistics.
- MATH 6510Topics in Financial Mathematics[2-4 credits]DescriptionAdvanced topics of current interest in financial mathematics.
- MATH 6770Professional Development in Science (Mathematics)[0-2-0:2]DescriptionThis two-credit course aims at providing research postgraduate students basic training in ethics, teaching skills, research management, career development, and related professional skills. This course lasts for one year, and is composed of two parts, each consisting of a number of mini-workshops. Part 1 of the course is coordinated by the School; and Part 2 consists of some department-specific workshops which are coordinated by the department. Graded PP, P or F.
- MATH 6900Mathematics Seminar[0-1-0:1]Previous Course Code(s)MATH 600DescriptionThis course will expose our PG students to the current mathematical research and development and provide them with opportunities to make mathematical and social contacts with the speakers and with local and international mathematical communities in general. Graded P or F.
- MATH 6911-6914Reading Course[1-6 credit(s)]Previous Course Code(s)MATH 601-604DescriptionFor individual students or a group of students. Specific topics under the supervision of a faculty member.
- MATH 6915Scientific Computation Seminar[0-1-0:1]DescriptionThis course will expose PG students to the current research and new development in scientific computation and provide them with opportunities to make social contacts with the speakers from both the local and international academic communities. Graded P or F.
- MATH 6990MPhil Thesis ResearchPrevious Course Code(s)MATH 699DescriptionMaster's thesis research supervised by a faculty member. A successful defense of the thesis leads to the grade Pass. No course credit is assigned.
- MATH 7990Doctoral Thesis ResearchPrevious Course Code(s)MATH 799DescriptionOriginal and independent doctoral thesis research. A successful defense of the thesis leads to the grade Pass. No course credit is assigned.