Fall Semesters

The basics of smooth manifolds and smooth maps. Immersions and submersions. Sard's theorem, transversality, and applications. Elements of algebraic topology: homotopy and homotopy equivalence, fundamental group, higher homotopy groups, CW complexes. Calculations of the fundamental group by various techniques, applications. Topological classification of 1- and 2-manifolds. Covering spaces, lifting theorems, classification of coverings, deck transformations. Introduction to fiber bundles and fibrations, homotopy exact sequence of a fibration.

  • 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

    Suggested Reading:

    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

    • 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 549 - Geometry and Topology II

    Vector fields and flows, applications. Differential forms, Lie derivatives, Frobenius theorem. Integration on manifolds, Stokes' theorem. Singular and cellular homology, calculations and applications. DeRham cohomology, ring structure.  Degree of a map. Introduction to Poincare duality.

  • 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.
    • Introduction to Smooth Manifolds, by John M. Lee, 2nd Edition, Springer, 2012.
  • 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