MATH 6130: Topics in Analysis (Fall 2012)

Instructor: Yulij Ilyashenko

The idea of the course is to present the advanced topics in analysis with an essential use of the basic complex analysis, both in one and many variables. Note that basic topics of multidimensional complex analysis are totally missed at the existing curriculum. Some main topics of the course, like Fourier transform, existence and smoothness theorem for the differential equations, special functions (including Gamma and Beta-functions) will be presented in the natural (complex) context. The L_2 space will be introduced as a completion, without any use of the Lebesgue measure. The course will include a simple and invariant (coordinates independent) presentation of the theory of differential forms and vector analysis, as well as generalized functions and their applications (time permitting). A more detailed list of topics follows.

  1. Holomorphic functions in one and many variables
  2. Parameter depending integrals, their smoothness and holomorphy.
  3. Analyticity and analytic dependence on initial conditions for solutions of analytic differential equations.
  4. Completion theorem for metric spaces. The L2 space as a completion of the space of continuous functions with the compact support with respect to the L2 norm.
  5. Fourier series and Fourier transform. Relations between smoothness (analyticity) and rate of decay of a function and its Fourier coefficients (Fourier transform).
  6. Special functions
  7. Vector analysis in R2 and R3. Green, Gauss and Stokes formulas. Maxwell equations. Differential forms. General Stokes formula.
  8. Asymptotic series. Laplace and stationary phase methods.
  9. Generalized functions, their properties and Fourier transform
  10. Fundamental solutions of PDEs