MATH 428, FALL 2007, SYLLABUS
1. General Information
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Course Title: Topics in Complex Analysis
Time and Location: TR 10:50AM - 12:05PM in FLI 414 (library)
Prerequisites: A semester course in complex analysis
Instructor: Sabin Cautis
Office: Herman Brown 432, x3273
Office Hours: by appointment
E-mail: scautis@rice.edu
Website: http://www.math.rice.edu/∼sc5/math428/
Text: Raghavan Narasimhan, Several Complex Variables (should be available from the bookstore)
2. Course Description
The main idea of the course is to study various fundamental results of complex analysis in several
variables while reviewing their simpler analogues in one variable. Most of these results will be concerned
with (open) domains in Cn . We’ll briefly survey Riemann surfaces, partly as an introduction to the
world of (compact) complex manifolds and algebraic geometry and finally take a glance at the very
basic theory of cohomology.
Here are some topics we will explore:
elementary properties of holomorphic functions (Cauchy’s formula, open mapping theorem etc.)
Riemann mapping theorem
subharmonic functions and applications (Hartog’s theorem on slicewise analyticity)
automorphisms of bounded domains (Poincaré’s theorem that the polydisc and the ball are
analytically distinct)
• Riemann surfaces (holomorphic line bundles and existence of meromorphic sections)
• Basic Introduction to Cohomology
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3. Other Information
Homework: Homework will be assigned in class biweekly. Homework is not pledged. You may
get help from any source and are encouraged to seek help from other students in the class. However
it is a very good idea to attempt the problems yourself before seeking assistance. Any written work
submitted must be your own.
Tests: There will be one midterm exam. The date for the midterm will be decided at a later time.
Grades: Grades will be computed as follows: Homework: 45 %, Midterm 20 %, Final Presentation
35 %.
Final Presentation: Students will present on some topic they find interesting and want to understand in greater detail. Alternatively, they can try to present a more recent paper in complex analysis
in which case it will give them the chance to learn more about the current state of the subject and
research therein.
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MATH 428, FALL 2007, SYLLABUS
Web page: The webpage for the course is http://www.math.rice.edu/∼sc5/math428/ It includes the
syllabus, course schedule, homework assignments and other information which may come up during
the course.
Note: Any student with a documented disability seeking academic adjustments or accommodations
is requested to speak with me during the first two weeks of class. All discussions will remain as
confidential as possible. Students with disabilities will need to also contact Disability Support Services
in the Ley Student Center.