Gautam Bharali

               Department of Mathematics

                  Indian Institute of Science

                  Bangalore 560012


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  • Lecture hours

    Tuesdays and Thursdays 2:00–3:30 p.m.

  • Classroom

    Lecture Hall 5, Department of Mathematics

  • 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.

  • 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.

  • Announcements

    November 28: The end-of-semester examination will be held in Lecture Hall 1.

    November 28: The graded copies of the third homework assignment can be collected between 5:30 and 6:30 p.m. on Friday, November 29.

    November 5: The end-of-semester examination will be held on December 4 at 2:00 p.m. Venue: TBA.

    November 5: The third lecture in our series of make-up lectures will be on November 9, and will be held at the usual venue. Time: 2:00–3:30 p.m.

    October 4: There will be no classes during the period October 7–20 since I will be away on a research trip.

    October 3: The second lecture in our series of make-up lectures will be on October 5, and will be held at the usual venue. Time: 2:00–3:30 p.m.

    September 20: There will be no classes during the week beginning September 23 owing to mid-semester examinations in the Department of Mathematics.

    September 6: I shall be travelling in October. I plan to hold classes on certain Saturdays to make up for the lectures lost. The first of these make-up lectures will be on September 14, and will be held at the usual venue. Time: 2:00–3:30 p.m.

    September 6: There will be a lecture on September 10 even though there is a holiday at the Institute on this day.

    August 27: The photo session for members of the mathematics department for the IISc smart card could not be held on August 22. Members of the math department will be photographed on the afternoon on August 29. Thus, the lecture of August 29 will end at 3:00 p.m.

    August 22: Since it is the turn of members of the math department to be photographed for the IISc smart card this afternoon, the lecture today will end at 3:00 p.m.

    August 6: Today is the opening lecture of the course.

  • 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 .

    • Note 2: Solution of the Levi Problem
      Having studied the existence and regularity results for the -problem, students can now read the solution of the Levi Problem on their own. Here is the reference—as promised in class—to an easy-to-read exposition of the solution to this problem: Section 5.1 (i.e., in Chapter 5) of the book by S.G. Krantz mentioned above.
  • Homework assignments

    Homework 3

    Homework 2

    Homework 1


  • UNDERGRADUATE ANALYSIS & LINEAR ALGEBRA (UM101)   [Autumn 2015, Autumn 2017]


  • INTRODUCTION TO BASIC ANALYSIS (UM204)   [ Spring 2019 ]

  • ANALYSIS–I (MA221)   [Autumn 2018]

  • ANALYSIS–II: MEASURE AND INTEGRATION (MA222)  [Spring 2012, Spring 2017 ]

  • COMPLEX ANALYSIS (MA224)  [ Spring 2016 ]

  • TOPICS IN COMPLEX ANALYSIS (MA324)  [Spring 2014]

  • INTRODUCTION TO SEVERAL COMPLEX VARIABLES (MA328-329)  [experimentally as a "topics course" (MA329) in Autumn 2014]


Page last updated on December 7, 2019