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