This is the second semester course of Complex Analysis. We will start with a complete proof of the Riemann mapping theorem making use of the Ascoli-Arzela theorem. Then we switch to Harmonic Function Theory (including the Poisson integral representation, various mean-value theorems and Schwarz reflection principle), as well as its application to the construction of elliptic functions. After these, we study the Riemann-Zeta function and give a very quick and short proof of the Prime Number Theorem based on the of Newman. ( Another optional topic to be included in this classical work theory part would be the monodromy theory and multiple-valued holomorphic functions.) After these classical topics, depending on students' interest, we will present a detailed discussion to one of the following two topics: Basic Riemann surfaces theory, Perron's method on Riemann surfaces and the proof of the Riemann Rich theorem and the uniformization theorem for simply connected Riemann surfaces---through the method of Koebe, Poincare and Hilbert. Basic Riemann surfaces theory the proof of the classical Riemann-Roch theorem through Hodge theory and Serre Duality This is a foundational course for anyone who likes to know more about pure mathematics and applied mathematics involving some analysis. It fits perfectly to undergraduates who have taken a course equivalent to Math 503, also first year and second year graduate students. The instructor has prepared lecture notes (to be distributed during the class) which cover all the materials to be present in the class.
503, point set topology