Seminar on Complex Analysis
Meeting on:
Sundays 7:0010：00 pm, Room 813
Abstract:
The goals of this seminar are of several aspects:
(1) Provide necessary complex analysis theory for the study of function spaces
(2) Serve as an introduction to harmonic analysis and distribution theory
(3) Reveal the connection between complex function theory and other areas, e.g., functional calculus, stochastic analysis, analysis on manifolds.
Topics:
(1) Analytic functions and Mobius transformations
(2) Blaschke product and Pick’s Theorem
(3) Index of a closed curve and Lifting theorems
(4) Cauchy’s Theorem and Open Mapping Theorem
(5) Singularities and Resides, Argument Principle
(6) Maximum Principle, Schwarz Lemma, Hadamard Theorem, PhragmenLindelof Theorem
(7) Spaces of analytic and meromorphic functions,:
(a) Topology and convergence
(b) Riemann Mapping Theorem
(c) Infinite product and factorizations
(d) Gamma function and Riemann zeta function.
(8) Analytic continuation, Riemann surfaces, Monodromy Theorem, and the sheaf of germs of analytic functions.
(9) Harmonic and subharmonic functions
(10) Dirichlet Problem, Potentials and regularities
(11) Banach spaces of analytic functions, Hp spaces
(12) Entire functions
(13) Tempered distributions and Entire functions of finite type
(14) Theory of function spaces and LittlewoodPaley theory
(15) L(p,q) estimates of entire functions of finite type and Nilkovski inequalities
(16) Further topics in connection with functional analysis and stochastic analysis on Levy
Processes
References:
[1] Functions of one complex variables, John B. Conway
[2] Real and complex analysis, Rudin
[3] A first course in functional analysis, John B. Conway
[4] Bounded Analytic Functions， John B. Garnett
[5] Banach spaces of analytic functions, Hoffman
[6] The analysis of linear partial operators, volume 1, Hormander
[7] Theory of function spaces, Hans Triebel
[8] Levy process and infinitely divisible distributions, KenIti Sato
