# Course Descriptions

## 16:640:504 - Theory of Functions of a Complex Variable II

### Course Description:

This course is a continuation of Math 503. It will be centered around Riemann surfaces and connections with geometry, topology and algebraic geometry.

None

### Prerequisites:

Algebraic Topology

### Subtitle:

Complex Analysis II

### Course Description:

This is a continuation of Prof. Matveev’s course on complex analysis (Math 503). Complex analysis is a cornerstone of mathematics. The intuitive geometric underpinnings of complex analysis leads to Riemann surface theory.  The theory is a source in mathematics and you can learn and trace many of the original ideas in current mathematical research come from it.  The course will be an intermediate level introduction to Riemann surface theory and will focus on how it connects various disparate concepts from analysis,  geometry, topology,  and algebraic geometry.

Topics to be covered include:

1.           Basic surface topology, Riemann surfaces and holomorphic maps

2.           Calculus on surfaces, algebraic curves and algebraic functions

3.           The Dirichlet problem and harmonic functions

6.           Riemannian metrics and hyperbolic geometry

7.           Riemann-Roch, Abel and Jacobi theorems

8.           The Uniformization theorem and its applications.

### Text:

There will be no text books for the class. The main references are

[1] Forster, Lectures on Riemann Surfaces, Springer-Verlag, 1981

[2] Donaldson, Riemann surfaces, Oxford Press, 2011

[3] Farkas and Kra, Riemann Surfaces, Springer-Verlag, 1991

[4] Ahlfors, complex analysis,  McGraw-Hill Education; 3rd edition, 1979

### Prerequisites:

Math 503 Complex Analysis, Math 501 Real Analysis and Point Set Topology

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### Course Description:

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.

### Prerequisites:

503, point set topology