MMA602 ALGEBRA(1)
Courses of Master Program
MMA602 ALGEBRA(1)
Grade one required (3-0)
Fields：algebraic extensions, splitting fields, normal extensions, finite fields. Galois Theory：automorphisms of field extensions, Galois extensions, fundamental theorem of Galois theory, solvability of polynomials by radicals, Galois groups of polynomials with degree up through 4, insolvability of the quintic.
Groups：permutation groups, alternating groups, simple group, finite groups, the Sylow theorems.
MMA627 ALGEBRA (2)
Grade one elective (0-3)
The structure of modules：free modules, semisimple modules, projective modules, injective modules.
The structure of rings：localization, Hilbert basis theorem, Hilbert Nullstellensatz, matrix rings, simple rings, prime rings, semiprime rings, Wedderburn-Artin theorem.
MMA601 ANALYSIS (1)
Grade one required (3-0)
Lebesgue measure、outer measure、measurable functions、Lesgue integral、Lebesgue’s dominated convergence theorem、monotone convergence theorem and uniform convergence theorem.
Prerequisite：Advanced Calculus.
MMA626 ANALYSIS (2)
Grade one elective (0-3)
Fubini’s theorem、Lebesgue’s differentiation theorem、L^p spaces、Jordan decomposition、 Hahn decomposition and Radon-Nikodym theorem.
Prerequisite：Analysis(1)
MMA621 PROBABILITY
THEORY
Grade one elective (3-3)
In this course, measure theory is used to discuss probability theory. Main contents are basic measure theory, random variables, distribution function, characteristic functions, integration, conditioning, law of large number, law of iterated logarithm, central limit theorem and other frequently used convergence theorems.
MMA624/MMA625 TOPICS
IN MATHEMATICAL BIOLOGY(1)(2)
Grade one elective (3-0)(0-3)
Introduction of various biological models and their mathematical structures. Study of selected papers.
Prerequisite：Advanced Calculus、Ordinary Differential Equations、Partial Differential Equations.
MMA622 MATRIX THEORY
Grade one elective (3-3)
This course covers classical and modern results in matrix theory, and its important applications in different areas. Topics contain eigenvalues, decomposition, canonical form, norms and positive-definiteness.
MMA630 PARTIAL
DIFFERENCE EQUATIONS
Grade one elective (3-3)
First order partial differential equations, Cauchy-Kowalevski theorem and characteristics, Classification of second order linear differential operators, Laplace、heat and wave equations.
Prerequisite：Advanced Calculus、Ordinary Differential Equations.
MMA629 GAME THEORY
Grade one elective (3-3)
Definition of A Game, Two-Person Zero-Sum Games, Infinite Games, Infinite Games, Multistage Games.
Two-Person General Sum Game, Two-Person Cooperative Games, n-Person Games, Indices of Power.
MMA637 RELIABILITY
THEORY
Grade one elective (3-3)
Reading the most recent results on Coherent System, Importance of Coherent System.
Reading the most recent results on NBU, NBUE Distribution Functions, Multi-State Coherent Systems.
MMA632 ORDINARY DIFFERENTIAL EQUATIONS
Grad one elective (3-3)
Existence and uniqueness theorems, dependence of solutions on parameters, Linear systems, autonomous systems, stability of equilibria, Lyapunov function, Poincaré-Bendixon theorem, perturbation theory, boundary value problems.
MMA730/MMA635 EQUILIBRIUM
ANALYSIS(1)、(2)
Grade one elective (3-0) (0-3)
Fixed point theorems with applications to economics and game theory. The main content includes :Sperner's Lemma,Continuity of correspondences, Fan-Browder theorem, Walrasian
equilibrium, and so on.
MMA633 GRAPH THEORY
Grade one elective (3-3)
Topics of Graph theory contain the basic definitions of graph and introductions to the properties of connectivity, Eulerian, Hamiltonian graph, planar graph, graph coloring, graph decompositions and Ramsey theory.
MMA723/MMA724 MATRIX ANALYSIS (1)(2)
Grad two elective (2-0)(0-2)
Interesting topics include combinatorial matrix theory, sign pattern matrices and numerical ranges. Related research papers will be studied.
MMA721 ADVANCED GAME
THEORY
Grade two elective (3-3)
Reading the most recent papers on Cooperative Games, Multi-Choice Cooperative Games.
Reading the most recent papers on Multi-Choice Shapley Value, Continuously-Many-Choice Cooperative Games.
MMA725 INTRODUCTION
TO STOCHASTIC PROCESSES AND APPLICATIONS
Grade two elective (3-3)
In this course, basic theories of stochastics processes, including Markov chain, Markov processes and martingales, and discussed. Stochastic integral, stochastic differential equations and their applications are also discussed.
MMA739 TOPICS IN CODING THEORY
Grade two elective (3-3)
Codes and designs, hand decoding of Golay codes, duadic codes, self-dual codes, Reed-Muller codes, and covering radius.
MMA731 DIFFERENTIAL
TOPOLOGY(1)
Grade two elective (3-0)
We discuss differential structures on manifolds. The tools include fibre bundle and characteristic classes, for instance, tangent bundle and Euler class. We also discuss maps between manifolds, their homotopy approximation, and degrees.
MMA732 DIFFERENTIAL
TOPOLOGY (2)
Grade two elective (0-3)
We discuss variational problems and Morse theory. These lead to the computation of homotopy and homolgy groups. The application of the theory is the complete classification of compact surfaces.
MMA727 FUNCTIONAL
ANALYSIS
Grade two elective (3-3)
Normed Space, Banach Space, Inner Product Space, Hilbert Space, and related Theorems such as Open Mapping Theorem, Closed Graph Theorem.
Banach Fixed Point Theorem, Spectral, Resolvent of Linear Operator, Compact Operator.
MMA631 TOPICS IN
MATHEMATICAL ECONOMICS（Original:
Mathematical Economics）
Grade two elective (3-3)
There are two kinds of contents in this program. One is microeconomic: supply, demand and equilibrium, factories and markers, consumer behavior, general equilibrium …. The other we read some papers about static equilibrium.
MMA740 COMBINATORIAL
DESIGN
Grade two elective (3-3)
Topics include Block designs, Orthogonal Latin squares, Symmetric designs, Steiner systems and Tournament designs.