Fall Semesters
 MAT 530  Geometry/Topology I
 Basic point set topology
 Metric Spaces
 Topological spaces and continuous maps
 Comparison of topologies
 Separation axioms and limits
 Countability axioms, the Urysohn metrization theorem
 Compactness and paracompactness, the Tychonoff theorem
 Connectedness
 Product spaces
 Function spaces and their topologies, Ascoli's theorem
 Introduction to algebraic topology
 Fundamental group
 Fundamental group of S^{n}; examples of fundamental groups of surfaces
 Seifertvan Kampen theorem
 Classification of covering spaces, universal covering spaces; examples
 Homotopy; essential and inessential maps
Typical References:
 James R. Munkres, Topology: a first course, Prentice Hall, Englewood Cliffs NJ, 1975;
 William S. Massey, Algebraic topology: an introduction, 4^{th} corrected printing, SpringerVerlag, 1977.
 Basic point set topology
 MAT 532  Real Analysis I
 Measures
 Sigmaalgebras
 Measures, outer measures
 Borel measures on the real line, nonmeasurable sets
 Integration
 Measurable Functions
 Littlewood's three principles
 Integration of Nonnegative Functions
 Integration of Complex Functions
 Modes of Convergence
 Product Measures
 The ndimensional Lebesgue Integral
 Integration in Polar Coordinates
 Signed Measures and Differentiation
 The HardyLittlewood maximal function
 Signed Measures
 The LebesgueRadonNikodym Theorem
 Complex Measures
 Differentiation on Euclidean Space
 Functions of Bounded Variation
 $L^p$ spaces
 Chebyshev, CauchySchwartz, Hölder, Minkowski Inequalities, Duality
 Integral operators
 Distribution functions and Weak $L^p$
 Interpolation of $L^p$ spaces
 convolution, Young's inequality
Suggested Reading:

Folland, G.B. (1984). Real Analysis, New York, Wiley.

Royden, H.L. (1969). Real Analysis, New York, MacMillan

Rudin, W. (1986). Real and Complex Analysis, New York, McGraw Hill (3rd edition).

Stein, E.M. and Sharkarchi, R. (2005). Real Analysis, Measure Theory,Integration and Hilbert Spaces, Princeton University Press.

Wheeden, R. and Zygmund, A. (1977). Measure and Integral, An Introduction to Real Analysis, New York, M. Dekker.

Rudin, W. (1976). Principles of Mathematical Analysis, New York. McGraw Hill. (3rd edition).

Stein, E.M. and Sharkarchi, R. (2003). Fourier Analysis, Princeton University Press.

Anthony Knapp. Basic/Advanced Real Analysis. Free online at http://www.math.stonybrook.edu/~aknapp/download.html
 Measures
 MAT 534  Algebra I
Spring Semesters
 MAT 531  Geometry/Topology II
 Differentiable manifolds and maps
 Inverse and implicit function theorems
 Submanifolds, immersions and submersions
 The tangent bundle
 Vector bundles, transition functions
 Reconstruction of a vector bundle from transition functions
 Equivalence classes of curves and derivations; tangent vectors
 The tangent bundle of a manifold as a vector bundle, examples
 Vector fields, differential equations and flows
 Lie derivatives and bracket
 Differential forms
 Exterior differential, closed and exact forms
 Distributions, foliations and Frobenius integrability theorem
 Poincaré Lemma
 Integration
 Stokes' Theorem
 Integration and volume on manifolds
 De Rham cohomology
 Chain and cochain complexes
 Homotopy theorem
 The degree of a map
 The MayerVietoris theorem
Typical references:
 Michael Spivak, A Comprehensive introduction to differential geometry, 2^{nd} ed., Publish or Perish, Berkeley 1979;
 Glen Bredon, Topology and geometry, SpringerVerlag, 1993.
 Differentiable manifolds and maps
 MAT 533  Real Analysis II
 Compactness
 ArzeláAscoli, StoneWeierstrass
 Functional analysis
 Normed Vector Spaces
 Linear functionals, HahnBanach theorem
 Baire Category theorem, open mapping theorem, closed graph theorem, uniform boundedness principle
 Topological vector spaces, duality, weak and weak* convergence, Alaoglu's theorem
 Hilbert spaces
 $L^p$ spaces (completing only what was omitted in first semester)
 Ordinary differential equations
 Radon measures on locally compact Hausdorff spaces
 Elements of Fourier Analysis
 Fourier Transform on $R^n$ and the circle
 Riemann Lebesgue lemma, HausdorffYoung inequality, Plancharel, Poisson summation, $L^2(R^n)$
 Summation and convergence theorems
 Distributions
Suggested Reading:

Folland, G.B. (1984). Real Analysis, New York, Wiley.

Royden, H.L. (1969). Real Analysis, New York, MacMillan

Rudin, W. (1986). Real and Complex Analysis, New York, McGraw Hill (3rd edition).

Stein, E.M. and Sharkarchi, R. (2005). Real Analysis, Measure Theory,Integration and Hilbert Spaces, Princeton University Press.

Wheeden, R. and Zygmund, A. (1977). Measure and Integral, An Introduction to Real Analysis, New York, M. Dekker.

Rudin, W. (1976). Principles of Mathematical Analysis, New York. McGraw Hill. (3rd edition).

Stein, E.M. and Sharkarchi, R. (2003). Fourier Analysis, Princeton University Press.

Anthony Knapp. Basic/Advanced Real Analysis. Free online at http://www.math.stonybrook.edu/~aknapp/download.html
 Compactness
 MAT 535  Algebra II
 MAT 536  Complex Analysis I
 The field of complex numbers, geometric representation of complex numbers
 Analytic functions
 Definition, CauchyRiemann equations
 Elementary theory of power series, uniform convergence
 Elementary functions: rational, exponential and trigonometric functions
 The logarithm
 Analytic functions as mappings
 Conformality
 Linear fractional transformations
 Elementary conformal mappings
 Complex integration
 Line integrals and Cauchy's theorem for disk and rectangle
 Cauchy's integral formula
 Cauchy's inequalities
 Morera's theorem, Liouville's theorem and fundamental theorem of algebra
 The general form of Cauchy's theorem
 Local properties of analytic functions
 Removable singularities, Taylor's theorem
 Zeros and poles, classification of isolated singularities
 The local mapping theorem
 The maximum modulus principle, Schwarz's lemma
 The calculus of residues
 The residue theorem
 The argument principle
 Rouche's theorem
 Evaluation of definite integrals
 Power series
 Weierstrass theorem
 The Taylor and Laurent series
 Partial fractions and infinite products
 Normal families
 The Riemann mapping theorem
 Harmonic functions
 The meanvalue property
 Harnack's inequality
 The Dirichlet problem
Typical references:
 Lars V. Ahlfors, Complex analysis: an introduction to the theory of analytic functions of one complex variable, 3^{rd} ed.; McGrawHill, 1979;
 John B. Conway, Functions of one complex variable, 2^{nd} ed.; SpringerVerlag, 1978