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2023-2024 Fall Semester
MATH 5111 - Advanced Algebra I

Description	Advanced theory of groups, linear algebra, rings, modules, and fields, including Galois theory..
(3 units)
	Course Instructor: Prof IP, Ivan Chi Ho

MATH 5143 - Introduction to Lie Algebras

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.
(3 units)
	Course Instructor: Prof ZHU, Yongchang

MATH 5240 - Algebraic Topology

Description	Fundamental group, covering space, Van Kampen theorem, (relative) homology, exact sequences of homology, Mayer-Vietoris sequence, excision theorem, Betti numbers and Euler characteristic.
(3 units)
	Course Instructor: Prof HO, Quoc

MATH 5251 - Algebraic Geometry I

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.
(3 units)
	Course Instructor: Prof LI, Weiping

MATH 5285 - Applied 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.
(3 units)
	Course Instructor: Prof ZHANG, Hai

MATH 5311 - Advanced Numerical Methods I

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.
(3 units)
	Course Instructor: Prof PENG, Zhichao

MATH 5350 - Computational Fluid Dynamics for Inviscid Flows

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.
(3 units)
	Course Instructor: Prof XU, Kun

MATH 5351 - Mathematical Methods in Science and Engineering I

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.
(3 units)
	Course Instructor: Prof XIANG, Yang

MATH 5411 - Advanced Probability Theory I

Description	Probability spaces and random variables, distribution functions, expectations and moments, independence, convergence concepts, law of large numbers and random series.
(3 units)
	Course Instructor: Prof BAO, Zhigang

MATH 5431 - Advanced Mathematical Statistics I

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.
(3 units)
	Course Instructor: Prof GUO, Xinzhou

MATH 5471 - Advanced Machine Learning with Graphs

Description	This course will introduce a number of important statistical methods and modeling principles for analyzing large-scale data sets, with a focus on complex data structures such as text and graph data. Topics covered include sequential models, structure prediction models, deep learning attention models, reinforcement learning models, etc., as well as open research problems in this area.
(3 units)
	Course Instructor: Prof SONG, Yangqiu

MATH 5472 - Computer Age Statistical Inference with Applications

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.
(3 units)
	Course Instructor: Prof YANG, Can

MATH 5520 - Interest Rate Models

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.
(3 units)
	Course Instructor: Prof WU, Lixin

MATH 6450K - Random Walks on Graphs and Applications

Description	Advanced topics of current interest in probability and statistics.
(3 units)
	Course Instructor: Prof NITZSCHNER, Maximilian Alexander

MATH 6771 - Professional Development Training in Mathematics

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.
(1 units)
	Course Instructor: Prof Jin, Tianling

2023-2024 Spring semester
MATH 5112 - Advanced Algebra II

Description	Advanced topics in algebra: group representations, associative algebras, commutative algebra, homological algebra, algebraic number theory.
(3 units)
	Course Instructor: Prof MARBERG, Eric Paul

MATH 5261 - Algebraic Geometry II

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.
(3 units)
	Course Instructor: Prof LI, Wei-Ping

MATH 5281 - Partial Differential Equations

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.
(3 units)
	Course Instructor: Prof JIN, Tianling

MATH 5312 - Advanced Numerical Methods II

Description	Direct and iterative methods. Programming techniques and softwares libraries. Sparse solvers, Fast algorithms, multi-grid and domain decomposition techniques.
(2 units)
	Course Instructor: Prof CAI, Jianfeng

MATH 5353 - Multiscale Modeling and Computation for Non-equilibrium Flows

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.
(2 units)
	Course Instructor: Prof XU, Kun

MATH 5380 - Combinatorics

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.
(3 units)
	Course Instructor: Prof CHEN, Beifang

MATH 5432 - Advanced Mathematical Statistics II

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.
(3 units)
	Course Instructor: Prof NITZSCHNER, Maximilian Alexander

MATH 5450 - Stochastic Processes

Description	Theory of Markov processes, second order stationary theory, Poisson and point processes, Brownian motion, Martingales and queueing theory.
(3 units)
	Course Instructor: Prof BAO, Zhigang

MATH 5470 - Statistical Machine Learning

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.
(3 units)
	Course Instructor: Prof YAO, Yuan

MATH 5473 - Topological and Geometric Data Reduction and Visualization

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.
(3 units)
	Course Instructor: Prof YAO, Yuan

MATH 6250L - Higher Loops in Topological Strings

Description	Advanced topics of current interest in geometry.
(3 units)	Advanced topics of current interest in geometry.
	Course Instructor: Prof CHANG, Huai-Liang