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Written Exam Syllabi and Recommended Minimal Course ContentGeneral Algebra (Math 8201-02)Groups Cyclic subgroups, normal subgroups, groups acting on sets, permutation groups. Sylow Theorems, Jordan-Holder theorem, simple groups, solvable groups, extensions, direct sums and free abelian groups, finitely-generated abelian groups. Rings, Algebras Homomorphisms, prime ideals, maximal ideals, principal ideal domains, unique factorization domains, polynomial algebras, Euclidean algorithm, Gauss' lemma, Eisenstein's criterion, derivatives and multiplicity of roots, symmetric polynomials, discriminants. Modules Homomorphisms, direct sums, direct products, free modules, exact sequences, chain conditions, noetherian modules, Jordan-Holder theorem, Hilbert basis theorem. Modules over principal ideal domains Elementary divisor theory, characteristic and minimal polynomials, Jordan normal form. Fields Finite and algebraic extensions, algebraic closure, splitting fields, normal extensions, separable extensions, finite fields, perfect fields, primitive elements. Galois Theory Galois extensions, roots of unity, norm and trace, cyclic extensions, solvable and radical extensions. The Finite Dimensional Spectral Theorem Hermitian, symmetric, unitary, orthogonal, and normal operators. Introduction to homological algebra Exact sequences, free modules, projective and injective modules.
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Manifolds / Topology (Math 8301-02)Covering Spaces Fundamental group, van Kampen Theorem, covering groups, universal covering spaces, compact surfaces, applications Algebraic Topology Singular homology, relations with fundamental group, Mayer-Vietoris sequences and excision. Applications: fixed-points theorems, invariance of domain, degree of mappings Smooth Manifolds Submanifolds, diffeomorphisms, Inverse Function Theorem, Implicit Function Theorem, Sard's Theorem, vector fields, differential forms, Lie brackets, Frobenius' Theorem and Maximal Integral Manifolds, (Local existence and uniqueness theorems for O.D.E.'s) Orientations, Stokes' Theorem, statement of DeRham's Theorem, degree of a mapping References:
Complex Analysis (Math 8701-02)Complex analysis from point of view of advanced calculus Complex derivatives, Green's theorem and Cauchy's theorem and the Integral Theorem, geometric distortion of affine mappings, conformal affine mappings Geometry of Complex Numbers Stereographic projection Möbius (fractional linear) transformations Classification, cross ratio, symmetry, introduction to the hyperbolic plane, other conformal mappings by elementary functions Local properties of analytic functions Classification of isolated singularities, open mapping theorem, Taylor's Theorem with remainder, statement of Picard's Big Theorem Global properties of analytic functions Cauchy's Theorem and the Integral Theorem revisited, Residue Calculus, Morera's Theorem, Liouville's Theorem, maximum principle, Schwarz Lemma, argument and reflection principles, Rouché's Theorem Harmonic functions Harmonic and conjugate harmonic functions and differentials, Poisson integral formula, Mean Value Theorem, Harnack's inequality Taylor and Laurent series Mittag-Leffler and Weierstrass product representations, introduction to the Gamma and Riemann-Zeta functions, Stirling's Formula Normal familes Statement of Montel's Theorem on omitting three values The Riemann Mapping Theorem Statement of boundary value theorems, the Schwarz-Christoffel Formula, rectangle mappings, the Dirichlet problem, Green's function Rank one and rank two lattices The modular group, introduction to Weierstrass elliptic functions References:
Real Analysis (Math 8601-02)Preliminaries Continuity, semi-continuity, inverse and implicit function theorems, functions of bounded variation and Riemann-Stieltjes integrals, spaces of continuous functions, uniform convergence, equicontinuity, Ascoli-Arzela theorem, Stone-Weierstrass theorem, Baire Category theorem Lebesgue measure and integrals Lebesgue outer measure, measurable sets, measurable functions, Egorov's theorem, Lusin's theorem, Lebesgue Integral, convergence theorems, Fubini's theorem, Tonelli's theorem Differentiation Maximal functions, Lebesgue differentiation theorem, Vitali's covering lemma, absolutely continuous functions, monotone functions, convex functions Abstract measure and integration (introduction) Convergence theorems, Hahn decomposition, Radon-Nikodym theorem, Caratheodory-Hahn extension theorem, Borel measures Harmonic Analysis, introduction to Functional Analysis Approximation of the identity, convolutions, Lp-spaces, orthonormal sets and Fourier series, Hilbert Spaces, inner products and linear functionals, Plancherel Theorem References:
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