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
      • The Lebesgue-Radon-Nikodym Theorem
      • 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

    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

  • 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

    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

  • 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