수학과

Complexes, Simplicial homology, Singular homology, Homotopy invariance, Exact sequences and excision, Equivalence of simplicial and singular homology, Cellular homology, Mayer-Vietoris sequences, Homology with coefficients, Universal coefficients for homology, Axioms for homology theory, Cohomology groups, Universal coefficient theorem,  Cup product, Cohomology rings, A Kunneth formular, Orientations and homology, Poincare duality, Alexander duality, Computations and applications, Homotopy groups and homology groups.

• Allen Hatcher, Algebraic Topology, Cambridge University Press 2002.

Ring, Ideal, Factorization, Primary decomposition, Dedekind domain, Ideal class group, Extension and contraction of ideals, Modules, Injective and projective modules, Hom functors, Tensor products, Restriction and extension of scalars, Exact sequence and exactness, Localization, Quotient field, Integral dependence, Going up and down theorems, Noetherian and Artinian ring, Hilbert theorems (basis theorem, syzygy theorem, Nullstellensatz), Tensor/symmetric/alternating algebra

• M. Atiyah & I. G. Macdonald, "Introduction to Commutative Algebra", Addison-Wesley
• S. Lang, "Algebra", Graduate Texts in Mathematics, Springer
• D. Eisenbud, "Commutative Algebra with a view toward algebraic geometry", Graduate Texts in Mathematics, Springer
• O. Zariski & P. Samuel, "Commutative Algebra" vol. I, II, Graduate Texts in Mathematics, Springer

review on complex analysis, comparison of and , Hartogs’ phenomenon, domain of holomorphy, plurisubharmonic function, holomorphical convexity, Hartogs pseudoconvexity, Levi pseudoconvexity, Levi problem, unbounded operator, complex differential forms, -PDE, Hörmander L2-theory, Bochner-Martinelli formula, Bergman kernel, Cousin problem

• Krantz – Function theory of several complex variables
• Hörmander – Introduction to complex analysis in several variables
• Range – Holomorphic functions and integral representations in several complex variables

Basic properties of solutions of four significant linear PDEs: transport equations, Laplace's equations, heat equations, and wave equations including fundamental solutions, mean value formulas and energy methods, Hölder spaces, Sobolev spaces, Approximation, Extensions, Trace, Sobolev inequalities, Compactness.

• Lawrence C. Evans, Partial Differential Equations, Second edition, American Mathematical Society

Solving Nonlinear Equations; Solving Sets of Equations; Interpolation and Curve Fitting; Approximation of Functions; Numerical Differentiation and Integration; Numerical Solution of Ordinary Differential Equations; Optimization; Partial Differential Equations; Particle/Ensemble Kalman Filters; Variational Data Assimilation

• Curtis F. Gerald, Patrick O. Wheatley, Applied Numerical Analysis
• K.J.H. Law, A.M. Stuart, K.C. Zygalakis, Data Assimilation: A Mathematical Introduction

differentiable manifolds review, metrics, Riemannian manifolds, connections, geodesics, curvature tensor, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, isometric immersions, complete manifold, space forms, Rauch comparison theorem