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TEACHING: AUTUMN 2024
MA 328: INTRODUCTION TO SEVERAL COMPLEX VARIABLES
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Lecture hours
Tuesdays and Thursdays 2:00–3:30 p.m.
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Classroom
Lecture Hall 4, Department of Mathematics
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About this course
This is a first course on complex analysis in several variables. It will begin with a complete and
rigorous introduction to holomorphic functions in several variables and their basic properties.
This will pave the way to studying the following topics:
- Preliminaries: Holomorphic functions: power series
development(s), the domain of convergence of a power series, circular and
Reinhardt domains; analytic continuation: basic theory and contrasts with
the one-variable theory.
- Notions of convexity: Analytic continuation: the
definition of a domain of holomorphicity, the role of convexity,
holomorphic convexity; plurisubharmonic functions; Levi-pseudoconvexity;
characterisations of domains of holomorphy; introduction to the
∂-equation.
- The
∂-equation: Review of distribution theory,
Hörmander's solution and L2 estimates for
solutions.
- Geometry: Zeros of holomorphic functions:
Weierstrass's Preparation Theorem, analytic varieties and some of their
local and global properties; holomorphic maps; the inequivalence of the
unit ball and the unit polydisc.
Prerequisites: MA 224 (i.e., the first course in complex analysis) or any equivalent
exposure to complex analysis in one variable. Students who have not studied any one-variable
complex analysis formally but are interested in this course are encouraged to speak to
the instructor.
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Recommended books
L. Hörmander, Complex Analysis in Several Variables, 3rd
edition, North-Holland Publishing Co., 1990.
S.G. Krantz, Function Theory of Several Complex Variables (reprint
of the 1992 edition), AMS Chelsea, 2001.
K. Fritzsche and H. Grauert, From Holomorphic Functions to Complex
Manifolds, Graduate Texts in Mathematics 213,
Springer-Verlag, 2002.
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Announcements
November 12: The end-of-semester examination will be held
on November 4 at 2:00 p.m. Venue: Lecture Hall 4 (i.e., our usual venue).
November 5: The second make-up lecture (scheduled for
November 9, 3:00–4:30 p.m.) will be in Lecture Hall 4 (i.e., our
usual venue).
October 28: The second lecture in our pair of make-up lectures is scheduled for
November 9, 3:00–4:30 p.m. Venue TBA
October 17: There will be no classes during the
period October 21–25 since I will be away for a conference.
September 12: There will be no classes during the
week beginning September 16 owing to mid-semester examinations in the Department
of Mathematics.
September 3: The first of the two make-up lectures for this course is scheduled
for October 5, 3:00–3:30 p.m.
August 1: Today is the opening lecture of the course.
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Notes
I will occasionally post notes here that will either elaborate some point that
I did not go into in detail in my lectures; or present certain estimates that are
in most part routine, but are perhaps not the easiest to check.
- Note 1: How to construct a smooth
strictly plurisubharmonic exhaustion function for a domain given that it admits a continuous plurisubharmonic
exhaustion function .
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Homework assignments
Homework 3
Homework 2
Homework 1
TEACHING: LAST 5 YEARS
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