Written Exam Syllabus and Recommended Courses

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.

References:

Dummit and Foote: Abstract Algebra

S. Lang: Algebra

N. Jacobson: Basic Algebra I, II

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:

F.W. Warner: Foundation of Differential Manifolds and Lie Groups

W.S. Massey: Algebraic Topology

J.R. Munkres: Elements of Algebraic Topology

M. Greenberg, J. Harper: Algebraic Topology

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:

L. Ahlfors: Complex Analysis

T. W. Gamelin: Complex Analysis

R. E. Greene and S. G. Krantz: Function theory of one complex variable

E. Hille: Analytic Functions

S. Lang: Complex Analysis

S. Saks and Zygmund: Analytic Functions

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:

H.L. Royden: Real Analysis

W. Rudin: Real and Complex Analysis

R.L. Wheeden and A. Zygmund: Measure and Integral

G. Folland: Real Analysis