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20232024 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 PBWtheorem. Cartan subalgebras. Roots system, Weyl group, and Dynkin diagram. Classification of semisimple Lie algebras. Representations of semisimple algebras. Weyl character formula. HarishChandra 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, MayerVietoris 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 NavierStrokes 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, wellposedness, 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 largescale 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 CoxIngersollRoss models. Term structure models: HullWhite fitting procedure. HeathJarrowMorton pricing framework. LIBOR and swap market models, BraceGatarekMusiela approach. Affine models.

(3 units) 

Course Instructor:
Prof WU, Lixin

MATH 6450K  Random Walks on Graphs and Applications
MATH 6771  Professional Development Training in Mathematics
Description 
This onecredit 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 miniworkshops or tasks. Graded PP, P or F.

(1 units) 

Course Instructor:
Prof Jin, Tianling

20232024 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 semicontinuity, basics of curves and surfaces.

(3 units) 

Course Instructor:
Prof LI, WeiPing

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, multigrid and domain decomposition techniques.

(2 units) 

Course Instructor:
Prof CAI, Jianfeng

MATH 5353  Multiscale Modeling and Computation for Nonequilibrium Flows
Description 
Introduction of the NavierStrokes 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 ChapmanEnskog 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 nonequilibrium 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, spanningtree 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, largesample 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) 

Course Instructor:
Prof CHANG, HuaiLiang
