Fall Semesters

• MAT 530 - Geometry/Topology I
1. 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
2. Introduction to algebraic topology
• Fundamental group
• Fundamental group of Sn; examples of fundamental groups of surfaces
• Seifert-van 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, 4th corrected printing, Springer-Verlag, 1977.

• MAT 532 - Real Analysis I
1. Measures
• Sigma-algebras
• Measures, outer measures
• Borel measures on the real line, non-measurable sets
2. Integration
• Measurable Functions
• Littlewood's three principles
• Integration of Nonnegative Functions
• Integration of Complex Functions
• Modes of Convergence
• Product Measures
• The n-dimensional Lebesgue Integral
• Integration in Polar Coordinates
3. Signed Measures and Differentiation
• The Hardy-Littlewood maximal function
• Signed Measures
• Complex Measures
• Differentiation on Euclidean Space
• Functions of Bounded Variation
4. $L^p$ spaces
• Chebyshev, Cauchy-Schwartz, Hölder, Minkowski Inequalities, Duality
• Integral operators
• Distribution functions and Weak $L^p$
• Interpolation of $L^p$ spaces
• convolution, Young's inequality

• 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.

• MAT 534 - Algebra I

Spring Semesters

• MAT 531 - Geometry/Topology II
1. Differentiable manifolds and maps
• Inverse and implicit function theorems
• Submanifolds, immersions and submersions
2. 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
3. Differential forms
• Exterior differential, closed and exact forms
• Distributions, foliations and Frobenius integrability theorem
• Poincaré Lemma
4. Integration
• Stokes' Theorem
• Integration and volume on manifolds
• De Rham cohomology
• Chain and cochain complexes
• Homotopy theorem
• The degree of a map
• The Mayer-Vietoris theorem

Typical references:

• Michael Spivak, A Comprehensive introduction to differential geometry, 2nd ed., Publish or Perish, Berkeley 1979;
• Glen Bredon, Topology and geometry, Springer-Verlag, 1993.

• MAT 533 - Real Analysis II
1. Compactness
• Arzelá-Ascoli, Stone-Weierstrass
2. Functional analysis
• Normed Vector Spaces
• Linear functionals, Hahn-Banach 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
3. $L^p$ spaces (completing only what was omitted in first semester)
4. Ordinary differential equations
5. Radon measures on locally compact Hausdorff spaces
6. Elements of Fourier Analysis
• Fourier Transform on $R^n$  and the circle
• Riemann Lebesgue lemma,  Hausdorff-Young inequality, Plancharel, Poisson summation, $L^2(R^n)$
• Summation and convergence theorems
7. Distributions

• 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.

• MAT 535 - Algebra II

• MAT 536 - Complex Analysis I
1. The field of complex numbers, geometric representation of complex numbers
2. Analytic functions
• Definition, Cauchy-Riemann equations
• Elementary theory of power series, uniform convergence
• Elementary functions: rational, exponential and trigonometric functions
• The logarithm
3. Analytic functions as mappings
• Conformality
• Linear fractional transformations
• Elementary conformal mappings
4. 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
5. 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
6. The calculus of residues
• The residue theorem
• The argument principle
• Rouche's theorem
• Evaluation of definite integrals
7. Power series
• Weierstrass theorem
• The Taylor and Laurent series
• Partial fractions and infinite products
• Normal families
8. The Riemann mapping theorem
9. Harmonic functions
• The mean-value 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,  3rd ed.; McGraw-Hill, 1979;
• John B. Conway, Functions of one complex variable,  2nd ed.; Springer-Verlag, 1978