수학과

Group Theory (Basic concepts, Isomorphism Theorems, Group action, p-Group, Sylow Theorems, Solvable group, Nilpotent group, Free group, Group presentation), Ring Theory (Basic concepts, Principal ideal domain, Unique factorization domain, Field of quotients, Maximal ideal, Prime ideal, Polynomial ring, Factorization), Module Theory (Basic concepts, Exact sequence, Free module, Projective module, Injective module, Tensor product), Field and Galois Theory (Field extension, Automorphism group, Galois theory and applications, Finite field).

• T. W. Hungerford, Algebra, GTM 73, Springer-Verlag New York (1980)
• W. K. Nicholson, Introduction to Abstract Algebra
• D. S. Dummit & R. M. Foote, Abstract Algebra
• N. Jacobson, Basic Algebra (I), (II)

Complex differentiability, Cauchy-Riemann equation, Complex line integral, Cauchy integral theorem, Cauchy integral formula, Liouville theorem, Singularities, Laurent series, Residue theorem, Open mapping theorem, Maximum modulus principle, Argument principle, Schwarz lemma, Harmonic functions, Poisson integral formula, Schwarz reflection theorem, Normal family, Montel theorem, Riemann mapping theorem.

topological manifolds review, smooth manifolds, smooth maps, tangent vectors, submersions immersions and embeddings, submanifolds, Sard's Theorem, Lie groups, vector fields, integral curves and flows, 1-forms, vector bundles, cotangent bundle, tensors, differential forms, orientation, integration on manifold, Stokes’ theorem

• John Lee의 Introduction to Smooth Manifolds
• Spivak의 A Comprehensive Introduction to Differential Geometry, Vol. 1
• Warner의 Foundations of Differentiable Manifolds and Lie Groups

Topology, Continuous functions, Subspaces, Product spaces, Quotient spaces, Hausdorffness, Compactness, Connectedness, Homotopy, Fundamental groups, Covering spaces, Homotopy equivalences, CW complexes, Operations on spaces, Seifert– van Kampen theorem, Equivalences of covering spaces, Classification of covering spaces

• J. Munkres, Topology, Second edition, Prentice hall, 2002,
• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

Metric Spaces, Normed Linear Spaces, Norms of Matrices, Energy Norm for Uniqueness of PDE, Metric Space Topology, Continuity, Theory of Distribution, Test Function Spaces, Analytic Methods

Normed space, Linear operator, Linear functional, Boundedness, Compactness, completeness, Inner product space, Banach spaces, Hilbert spaces, Hahn-Banach theorem, Uniform boundedness theorem, Open mapping theorem, Closed graph theorem

• E. Kreyszig, Introductory Functional Analysis with Applications
• J.B. Conway, A Course in Functional Analysis