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MATH

Mathematics

MATH 5011
Advanced Real Analysis I
[3-0-0:3]
Background
MATH 3033
Description
Basic topology, continuous function spaces, abstract measure and integration theory, Lp spaces, convexity and inequalities, Hilbert spaces, Banach spaces, Complex measure.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Recognize fundamental concepts in the measure theory.
- 2.
  Identify and describe major theorems in the measure theory.
- 3.
  Develop analytic logic and rigorous reasonings.
MATH 5030
Complex Function Theory
[3-0-0:3]
Background
MATH 3033 and MATH 4023
Description
Review 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Understand the geometric nature of residue calculus and analytic functions; lectures; homework.
- 2.
  Learn about the Möbius transformations and their geometric properties.
- 3.
  Understand the use of normal families in the proof Riemann mapping theorem; lectures; homework.
- 4.
  Acquire knowledge of Schwarz-Christoffel mappings and their applications to elliptic integrals; lectures; homework/project presentations.
- 5.
  Acquire knowledge of entire functions and their infinite product representations such as the Gamma functions; lectures; homework/project presentations.
- 6.
  Acquire knowledge of elliptic functions, modular functions and Picard’s theorems; lectures; homework/project presentations.
MATH 5111
Advanced Algebra I
[3-0-0:3]
Background
MATH 3121 and MATH 4121 (prior to 2014-15)
Description
Advanced theory of groups, linear algebra, rings, modules, and fields, including Galois theory.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Recognize the types of algebraic structures such as groups, rings, modules and fields. Build a solid foundation of algebra.
- 2.
  Identify various algebraic structures among mathematics in the framework of category theory.
- 3.
  Apply the algebraic methods and ideas learned in this course to other fields such as algebraic geometry and homological algebra.
MATH 5112
Advanced Algebra II
[3-0-0:3]
Background
MATH 5111
Description
Advanced topics in algebra: group representations, associative algebras, commutative algebra, homological algebra, algebraic number theory.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Review the basics of abstract algebra and the main concepts in representation theory.
- 2.
  Introduce the main general results about representations of associative algebras.
- 3.
  Survey the main results about representations of finite groups.
- 4.
  Study some more advanced and special topics in the representation theory of finite groups.
- 5.
  Introduce the representation theory of quivers, and classify quivers of finite type.
- 6.
  Provide an introduction to: category theory and abelian categories in particular; homological algebra and its applications to categories of representations; & representation theory of finite dimensional algebras.
MATH 5143
Introduction to Lie Algebras
[3-0-0:3]
Prerequisite(s)
MATH 2131 and MATH 3131
Description
Lie 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Understand the concepts and basic techniques of Lie algebras and their representations.
- 2.
  Descrition of the root systems of semisimple Lie algebras.
- 3.
  Learn the properties of Weyl groups.
- 4.
  Classify simply Lie algebra by irreducible root systems.
- 5.
  Study Verma modules and irreducible highest weight modules.
MATH 5145
Introduction to Lie Groups
[3-0-0:3]
Prerequisite(s)
MATH 5111 and MATH 5143
Description
This 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Understand the concepts and basic techniques of Lie groups, Lie algebras and their representations.
- 2.
  Understand the structure of Lie subgroups and Lie subalgebras, and real forms of complex Lie algebras.
- 3.
  Learn the fundamental theorems of Lie theory.
- 4.
  Study orthogonality of characters and Peter-Weyl theorems.
- 5.
  Applications to harmonic analysis.
MATH 5230
Differential Topology
[3-0-0:3]
Background
MATH 4225
Description
Manifolds, embedding and immersion, Sard's theorem, transversality, degree, vector fields, Euler number, Euler-Poincare theorem, Morse functions.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Manifold and tangent bundle.
- 2.
  Transversality theorem.
- 3.
  Intersection number mod 2.
- 4.
  Intersection number.
- 5.
  Euler number and Poincare-Hopf theorem.
- 6.
  Differential form and Stokes' theorem.
MATH 5240
Algebraic Topology
[3-0-0:3]
Description
Fundamental group, covering space, Van Kampen theorem, (relative) homology, exact sequences of homology, Mayer-Vietoris sequence, excision theorem, Betti numbers and Euler characteristic.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Fundamental groupoid and higher homotopy groups.
- 2.
  Singular homology theory.
- 3.
  CW-complex.
- 4.
  Singular cohomology theory.
- 5.
  Poincare duality.
MATH 5251
Algebraic Geometry I
[3-0-0:3]
Background
MATH 5111 or equivalent postgraduate algebra
Description
Projective 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Learn the relations between commuatative algebra and affine varieties and affine schemes.
- 2.
  Master the techchniques using algebra to study geometry.
- 3.
  Learn the concepts of schemes and basic properties of schemes.
- 4.
  Learn the sheaves of modules on schemes.
MATH 5261
Algebraic Geometry II
[3-0-0:3]
Prerequisite(s)
MATH 5251
Background
MATH 5111 or equivalent postgraduate algebra
Description
Derived 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Deeper understanding of using algebra to study geometry.
- 2.
  Learn the key geometry concepts of schemes using algebraic methods.
- 3.
  Master the viewpoint of Grothendieck's cohomology formulation.
- 4.
  Study various cohomology of coherent sheaves.
MATH 5281
Partial Differential Equations
[3-0-0:3]
Previous Course Code(s)
MATH 6050E
Background
Multi-variables calculus, linear algebra, Lebesgue integral
Description
This is an introductory postgraduate course on Partial Differential Equations (PDEs). We will start with the classical prototype linear PDEs, and introduce a variety of tools and methods. Then we will extend our beginning theories to general situation using the notion of Sobolev spaces, Holder space and weak solutions. We will prove the existence, uniqueness, regularity and other properties of weak solutions.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Recognize the types of second-order PDEs as typified by classical equations of mathematical physics, such as the wave equation, heat equation and Laplace equation.
- 2.
  Identify and describe the major theories of general PDEs.
- 3.
  Apply the tools of calculus, linear algebra and real analysis in a coherent way to PDE problems.
- 4.
  Apply the knowledge of PDEs to physical sciences and engineering, and physically interpret the solutions.
- 5.
  Use different methods, such as Fourier transform, separation variables, characteristics, similarity, power series to solve PDEs.
MATH 5285
Applied Analysis
[3-0-0:3]
Previous Course Code(s)
MATH 6050B
Background
Undergraduate course of multivariable calculus, linear algebra, and real analysis
Description
Contraction mapping theorem, Fourier series, Fourier transforms, Basics of Hilbert Space theory, Operator theory in Hilbert Spaces, Basics of Banach space theory, Convex analysis.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Explain the key ideas and concepts behind various analytical techniques.
- 2.
  Identify situations where analytical techniques are applicable.
- 3.
  Recognize limitations of the analytical techniques.
- 4.
  Develop extension of generalization for the techniques.
- 5.
  Implement the techniques to problems in various situations.
MATH 5311
Advanced Numerical Methods I
[3-0-0:3]
Description
Numerical 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 5312
Advanced Numerical Methods II
[3-0-0:3]
Prerequisite(s)
MATH 5311
Description
Direct and iterative methods. Programming techniques and softwares libraries. Sparse solvers, Fast algorithms, multi-grid and domain decomposition techniques.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Understand direct and iterative algorithms for solving linear systems, and their convergence theory.
- 2.
  Identify suitable algorithms to solve linear equations arising from practical science and engineering problems, and implement them on a computing platform.
- 3.
  Understand algorithms and theories of other numerical linear algebra problems including the least squares problem and the eigenvalue problem.
MATH 5350
Computational Fluid Dynamics for Inviscid Flows
[3-0-0:3]
Description
Derivation 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Understand the characteristics in linear and nonlinear scalar equations, the concept of entropy, and the condition for the formation of discontinuous shock. Solve the Riemann problem for the nonlinear scalar equation with concave and convex flux functions. Develop numerical scheme to solve the Burgers’ equation and traffic flow equation.
- 2.
  Identify the characteristic variables for the linear hyperbolic system. Construct the analytical solution of the Riemann problem. Develop numerical scheme to solve the linear system.
- 3.
  Obtain the characteristic variables for the nonlinear 2x2 system, such as the shallow water equations. Construct the analytical similarity solution for the Riemann problem and apply it to the wave interactions.
- 4.
  Derive the gas dynamic Euler equations. Present the Euler equations in terms of conservative, characteristic and primitive variables. Construct the Riemann solution for the Euler equations. Identify the interaction among different kinds of waves.
- 5.
  Develop numerical scheme for the Euler equations. Understand approximate Riemann solvers. Construct reliable numerical schemes beyond the inviscid Euler equations.
MATH 5351
Mathematical Methods in Science and Engineering I
[3-0-0:3]
Description
Modeling and analytical solution methods of nonlinear partial differential equations (PDEs). Topics include: derivation of conservation laws and constitutive equations, well-posedness, traveling wave solutions, method of characteristics, shocks and rarefaction solutions, weak solutions to hyperbolic equations, hyperbolic Systems, linear stability analysis, weakly nonlinear approximation, similarity methods, calculus of variations.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to describe problems in science and engineering by partial differential equation models.
- 2.
  Apply analytical solution methods to solve nonlinear partial differential equations.
- 3.
  Apply approximation methods to solve nonlinear partial differential equations.
- 4.
  Demonstrate abilities to interpret the solutions of partial differential equation models for problems in science and engineering.
MATH 5352
Mathematical Methods in Science and Engineering II
[3-0-0:3]
Prerequisite(s)
MATH 5351
Description
Asymptotic methods and perturbation theory for obtaining approximate analytical solutions to differential equations. Topics include: local analysis of solutions to differential equations, asymptotic expansion of integrals, global analysis and perturbation methods, WKB theory, multiple-scale analysis, homogenization method.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to obtain approximate analytical solutions to differential equations by asymptotic methods.
- 2.
  Identify perturbation problems in science and engineering.
- 3.
  Demonstrate abilities to understand the errors of the obtained approximate analytical solutions.
MATH 5353
Multiscale Modeling and Computation for Non-equilibrium Flows
[3-0-0:3]
Previous Course Code(s)
MATH 6385D
Background
Background knowledge in MATH 5350 is preferred
Description
Introduction of the Navier-Strokes equations and the flow modeling in the hydrodynamic scale. The derivation of the Boltzmann equation in the kinetic scale. The basic mathematical analysis of the Chapman-Enskog expansion and the numerical methods for the Boltzmann equation. The multiscale modeling from the kinetic to the hydrodynamic scales and the discretized governing equations. The study of non-equilibrium transport phenomena in gas dynamics, radiative and heat transfer, and plasma physics.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Derive the Navier-Stokes equations in the hydrodynamic scale.
- 2.
  Model the particle transport and collision and derive the Boltzmann equation in kinetic scale.
- 3.
  Extend the kinetic scale modelling to other scales and construct the corresponding governing equations.
- 4.
  Develop numerical scheme for multiscale transport equations.
- 5.
  Use multiscale method to study transport problems in rarefied gas dynamics, radiative and heat transfer, and plasma simulation.
MATH 5380
Combinatorics
[3-0-0:3]
Prerequisite(s)
MATH 2343 or MATH 3343
Background
Linear algebra; Calculus
Description
Enumerative 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Develop a kind philosophy of discrete thinking and bijective counting.
- 2.
  Develop an understanding of the core ideas and concepts of advanced combinatorics.
- 3.
  Recognize the power of abstraction and generalization, and apply logical reasoning to investigate mathematical work with independent judgement.
- 4.
  Apply rigorous, analytic, highly quantitative approach to analyze and solve problems using combinatorial technics.
- 5.
  Understand the uni¯cation of discrete and continuous thought by demonstrating part of taught theories.
MATH 5411
Advanced Probability Theory I
[3-0-0:3]
Description
Probability 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 5412
Advanced Probability Theory II
[3-0-0:3]
Description
Stable laws; infinitely divisible distributions; weak convergence on general metric spaces; Donsker’s invariance principle; basic martingale theory; optional stopping theorem; martingale convergence theorems; martingale CLT; Concentration inequalities; logarithmic Sobolev inequality; large deviation.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Recognize the basic knowledges about martingale, Brownian motion, and functional limit theorems.
- 2.
  Identify and describe the major theories of martingale CLT and functional CLT.
- 3.
  Apply the martingale approach to establish nonlinear functionals of random variables.
- 4.
  Apply functional limiting theorem to get the limiting behavior of various statistics for stochastic process.
MATH 5431
Advanced Mathematical Statistics I
[3-0-0:3]
Description
Theory 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 5432
Advanced Mathematical Statistics II
[3-0-0:3]
Description
Theory 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Recognise the problem of interest and formulate the corresponding testing hypotheses.
- 2.
  Identify the types of random mechanisms for the problem at hand, and then choose the appropriate models to describe the data.
- 3.
  Choose the most powerful tests if available and/or the most feasible tests.
- 4.
  Evaluate and compare different types of tests.
- 5.
  Decide whether it is appropriate to use parametric tests or nonparametric tests.
- 6.
  Develop problem-solving skills to handle real-life problems with uncertainty.
MATH 5450
Stochastic Processes
[3-0-0:3]
Description
Theory of Markov processes, second order stationary theory, Poisson and point processes, Brownian motion, Martingales and queueing theory.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Introduce the general concept of stochastic process and Markov chain.
- 2.
  Study the properties of Markov chain and its limiting distributions.
- 3.
  Study the periodic Markov chain and its properties.
- 4.
  Study martingale convergence theory and option sampling theory.
- 5.
  Study Brownian motion and its properties.
MATH 5460
Time Series Analysis
[3-0-0:3]
Description
Basic 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Introduce some basic concepts in time series such as auto-covariance, stationarity and ergodicity.
- 2.
  Introduce time series models, such as AR models, ARMA models, threshold models, GARCH models, and study their properties.
- 3.
  Study the theory of estimation and testing statistics of time series models.
- 4.
  Study the diagnostic tools for identifying various models in practice.
- 5.
  Introduce for possible applications such as option theory and risk management.
MATH 5470
Statistical Machine Learning
[3-0-0:3]
Previous Course Code(s)
MATH 6450A
Exclusion(s)
MFIT 5010, MSDM 5054
Description
This course covers methodology, major software tools and applications in statistical learning. By introducing principal ideas in statistical learning, the course will help students understand conceptual underpinnings of methods in data mining. The topics include regression, logistic regression, feature selection, model selection, basis expansions and regularization, model assessment and selection; additive models; graphical models, decision trees, boosting; support vector machines; clustering.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Describe the basic procedures in data analysis.
- 2.
  Explain the principles behind statistical learning tools.
- 3.
  Analyze data with real data.
- 4.
  Draw meaningful implications in decision making.
- 5.
  Write a good project report and help decision making in practice.
MATH 5471
Advanced Machine Learning with Graphs
[3-0-0:3]
Previous Course Code(s)
MATH 6450D
Co-list with
COMP 5222
Exclusion(s)
COMP 5222
Description
This course will introduce a number of advanced learning methods and modeling principles for analyzing large-scale complex data structures and graph data. Topics covered include semi-supervised learning, spectral graph theory, deep graph modeling, knowledge graph modeling, etc., as well as open research problems in this area.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate machine learning algorithm design skills for graph data analytics tasks.
- 2.
  Analyze the quality of results to domain problems.
- 3.
  Develop a program that can handle existing real problems.
MATH 5472
Computer Age Statistical Inference with Applications
[3-0-0:3]
Previous Course Code(s)
MATH 6450E
Description
This course is designed for RPg students in applied mathematics, statistics, and engineering who are interested in learning from data. It covers advanced topics in statistical learning and inference, with emphasis on the integration of statistical models and algorithms for statistical inference. This course aims to first make connections among classical topics, and then move forward to modern topics, including statistical view of deep learning. Various applications will be discussed, such as computer vision, human genetics, and text mining.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Describe the basic procedures in data analysis.
- 2.
  Explain the principles behind statistical inference.
- 3.
  Analyze real data.
- 4.
  Draw meaningful implications in decision making.
- 5.
  Write a good report.
- 6.
  Implement algorithms for statistical inference.
MATH 5473
Topological and Geometric Data Reduction and Visualization
[3-0-0:3]
Previous Course Code(s)
MATH 6380Q
Co-list with
CSIC 5011
Exclusion(s)
CSIC 5011
Description
This course is a mathematical introduction to data analysis and visualization with a perspective of topology and geometry. Topics covered include: classical linear dimensionality reduction, the principal component analysis (PCA) and its dual multidimensional scaling (MDS), as well as extensions to manifold learning, topological data analysis, and sparse models in applied math/high dimensional statistics. Extensive application examples in biology, finance, and information technology are presented along with course projects.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Explain the basics concept of dimensional reduction.
- 2.
  Use Compressed Sensing and High Dimensional Statistics techniques for data analysis.
- 3.
  Use Manifold learning approach for applications.
- 4.
  Apply the data analysis method in applications.
MATH 5520
Interest Rate Models
[3-0-0:3]
Exclusion(s)
MAFS 5040
Description
Theory 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 6050
Topics in Analysis
[2-4 credits]
Description
Advanced topics of current interest in analysis.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of analysis.
- 2.
  Apply methods of analysis to solve problems.
- 3.
  Recognize limitations of methods of analysis.
MATH 6060
Topics in Complex Function Theory
[2-4 credits]
Description
Advanced topics of current interest in complex function theory.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of complex function theory.
- 2.
  Apply methods of complex function theory to solve problems.
- 3.
  Recognize limitations of methods of complex function theory.
MATH 6150
Topics in Algebra
[2-4 credits]
Description
Advanced topics of current interest in algebra.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of algebra.
- 2.
  Apply methods of algebra to solve problems.
- 3.
  Recognize limitations of methods of algebra.
MATH 6170
Topics in Number Theory
[2-4 credits]
Description
Advanced topics of current interest in number theory.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of number theory.
- 2.
  Apply methods of number theory to solve problems.
- 3.
  Recognize limitations of methods of number theory.
MATH 6250
Topics in Geometry
[2-4 credits]
Description
Advanced topics of current interest in geometry.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of geometry.
- 2.
  Apply methods of geometry to solve problems.
- 3.
  Recognize limitations of methods of geometry.
MATH 6380
Topics in Applied Mathematics
[2-4 credits]
Description
Advanced topics of current interest in applied mathematics.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of applied mathematics.
- 2.
  Apply methods of applied mathematics to solve problems.
- 3.
  Recognize limitations of methods of applied mathematics.
MATH 6385
Topics in Fluid Mechanics
[2-4 credits]
Description
Advanced topics of current interest in fluid mechanics.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of fluid mechanics.
- 2.
  Apply methods of fluid mechanics to solve problems.
- 3.
  Recognize limitations of methods of fluid mechanics.
MATH 6388
Topics in Numerical Analysis
[2-4 credits]
Description
Advanced topics of current interest in numerical analysis.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of numerical analysis.
- 2.
  Apply methods of numerical analysis to solve problems.
- 3.
  Recognize limitations of methods of numerical analysis.
MATH 6450
Topics in Probability and Statistics
[2-4 credits]
Description
Advanced topics of current interest in probability and statistics.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of probability and statistics.
- 2.
  Apply methods of probability and statistics to solve problems.
- 3.
  Recognize limitations of methods of probability and statistics.
MATH 6510
Topics in Financial Mathematics
[2-4 credits]
Description
Advanced topics of current interest in financial mathematics.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of financial mathematics.
- 2.
  Apply methods of financial mathematics to solve problems.
- 3.
  Recognize limitations of methods of financial mathematics.
MATH 6770
Professional Development in Science (Mathematics)
[0-2-0:2]
Description
This 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Exhibit effective skills in the teaching of Mathematics.
- 2.
  Apply essential communication skills for classroom teaching.
- 3.
  Recognize the features of the departmental online teaching system.
- 4.
  Prepare mathematics research papers, teaching notes and presentation slides using software (Tex).
- 5.
  Develop basic understandings of data science and reserch data management.
- 6.
  Demonstrate an understanding of the applications of knowledge and abilities of mathematics in industry.
MATH 6771
Professional Development Training in Mathematics
[0-1-0:1]
Exclusion(s)
MATH 6770
Description
This one-credit course aims at providing research postgraduate students basic training in teaching skills, research management, career development in and outside academia, and related professional skills in Mathematics. This course lasts for one semester, and is composed of a number of mini-workshops or tasks. Graded PP, P or F.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Exhibit effective skills in the teaching of Mathematics.
- 2.
  Apply essential communication skills for classroom teaching.
- 3.
  Recognize the features of the departmental online teaching system.
- 4.
  Prepare mathematics research papers, teaching notes and presentation slides using software (Tex).
- 5.
  Develop basic understandings of data science and reserch data management.
- 6.
  Demonstrate an understanding of the applications of knowledge and abilities of mathematics in industry.
MATH 6900
Mathematics Seminar
[0-1-0:1]
Description
This 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Recognize the frontier of modern mathematical research.
- 2.
  Build potential collaborations with seminar speakers.
- 3.
  Develop skills of giving mathematical seminars.
MATH 6911-6914
Reading Course
[1-6 credit(s)]
Description
For individual students or a group of students. Specific topics under the supervision of a faculty member.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to understand the advanced concepts and techniques of the assigned specific topics.
- 2.
  Apply methods of the assigned specific topics to solve problems.
- 3.
  Recognize limitations of methods of the assigned specific topics.
MATH 6915
Scientific Computation Seminar
[0-1-0:1]
Description
This 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.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Attend 6 seminars related to the scientific computation from the home or other departments in the whole semester.
MATH 6916
Student Seminars on Computation Related Research
[0-1-0:1]
Description
Seminar presentation on computation related research by students in the Scientific Computation Concentration (SCC) within the first four regular terms of study. Students should notify their major department 2-3 weeks in advance for the presentation. Graded P or F.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate knowledge and understanding on computation related research.
MATH 6990
MPhil Thesis Research
Description
Master's thesis research supervised by a faculty member. A successful defense of the thesis leads to the grade Pass. No course credit is assigned.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to conduct research in mathematics.
- 2.
  Demonstrate knowledge in the chosen discipline of mathematics.
- 3.
  Synthesize and create new knowledge and to making contributions to the discipline of mathematics.
- 4.
  Demonstrate communication skills in presenting reporting findings in mathematics.
MATH 7990
Doctoral Thesis Research
Description
Original and independent doctoral thesis research. A successful defense of the thesis leads to the grade Pass. No course credit is assigned.
Intended Learning Outcomes
On successful completion of the course, students will be able to:
- 1.
  Demonstrate abilities to conduct independent and original research in mathematics.
- 2.
  Demonstrate mastery of knowledge in the chosen discipline of mathematics.
- 3.
  Synthesize and create new knowledge and to make original and substantial contributions to the discipline of mathematics.
- 4.
  Demonstrate a broad knowledge in mathematics and mathematical sciences.
- 5.
  Demonstrate effective communication skills in presenting and publishing findings in mathematics.

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