Home page for Math 646, Several Complex Variables
Fall 2003
Instructor: Charles L. Epstein
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The Course
In this course we study the elementary properties of holomorphic functions of
several complex variables. At the end of the semester we will consider some
parts of the subject that are closely connected to contact geometry. This
subject is quite different from the theory of holomorphic functions of one
complex variables. In the immortal words of Solomon Bochner:
One complex variable is to several complex variables as S^0 is to S^n.
As a reference text we will use An Introduction to Complex Analysis in
Several Variables by Lars Hormander. Below are lecture notes for a course I
gave on this subject several years ago. A problem set will be assigned every
other week. Basic real and complex analysis are the prerequisites for this
course. Contact me if you are in doubt about your preparation.
- The class meets from 12:00 to 1:30 on Tuesdays and Thursdays.
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My office hours are 4:00-5:00 on Monday and 3:00-4:00 on Wednesday. Send e-mail if you have a question or would like to come see me at some other time.
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email: cle@math.upenn.edu.
Send e-mail or call if you need to see me at some other time.
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My office is DRL 4E7, my telephone number is: 8-8476.
Syllabus for the course
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Part 1. One complex variable, for adults
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Integration by parts in complex notation and the Cauchy-Pompieu formula
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Elementary facts about analytic functions of one variable
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The Runge approximation theorem, the holomorphic convex hull
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Solving the d-bar-equation
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The Mittag--Leffler and Weierstrass Theorems, domains of holomorphy
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Germs of holomorphic functions
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Part 2. Elementary properties of holomorphic functions in several variables
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Holomorphy for functions of several variables
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The Cauchy formula for polydiscs and its elementary consequences
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Solving the d-bar-equation in a polydisc and holomorphic extension, Hartogs'
phenomenon
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Local solution of the d-bar-equation for (p,q)-forms
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Power series and Reinhardt domains
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Domains of holomorphy and holomorphic convexity
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Pseudoconvexity and the failure of the Riemann mapping theorem: the ball
versus the polydisc
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CR-structures and the Lewy extension theorem
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The Bergmann kernel function
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Complex Geometry of the unit ball
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Part 3. Geometry and analysis on strictly pseudoconvex CR-manifolds
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Part 4. Local structure of analytic sets
Lecture Notes
Interested students might also want to look at the lecture notes I've written
for various short courses and lectures on this subject, go to lecture notes.
Announcements
I have to move my office hours on two dates:
October 27, my office hour will be 1-2PM.
November 3, my office hour will be 3-4PM.
Problem Sets