Gautam Bharali

               Department of Mathematics

                  Indian Institute of Science

                  Bangalore 560012


Home Education Research Publications Abridged CV Miscellanea Teaching



  • Lecture hours

    Mondays, Wednesdays, and Fridays 2:00–3:00 p.m.

  • Classroom

    Lecture Hall 4, Department of Mathematics

  • About this course

    This topics course is being run as an experiment in approaching the properties of holomorphic maps in several complex variables (SCV) in a self-contained manner: i.e., without requiring any prior exposure to SCV.

    The course will begin with a complete and rigorous introduction to holomorphic functions in several variables and their basic properties. This will pave the way to motivating and studying certain objects that are perhaps entirely indigenous to SCV, such as invariant metrics and plurisubharmonic functions. This will allow us to establish the inequivalence of the (Euclidean) ball and the polydisc in higher dimensions, to discuss appropriate analogues of the one-variable Riemann Mapping Theorem in higher dimensions, and results of a similar nature.

    Next, we shall study the properties of the Kobayashi metric (which is one of the invariant metrics mentioned above) and the Kobayashi distance. This will be used to study the behaviour of automorphisms of bounded domains and refinements of some of the results hinted at above—to the extent that time permits.


    • A first course in complex analysis at the level of MA 224: i.e., our first course in complex analysis.
    • Students who are unsure of the contents of MA 224 (e.g., students who completed their M.Sc. elsewhere) and are interested in this course are encouraged to speak/write to the instructor.

    Grading policy: Click here for the grading policy for this course.

  • Recommended books

    L. Hormander, Complex Analysis in Several Variables, 3rd edition, North-Holland Publishing Co., Amsterdam, 1990.

    M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter Expositions in Mathematics, no. 9, Walter de Gruyter, Berlin, 1993.

  • Announcements

    Nov 12: The end-of-semester exam is scheduled for November 29 at 2:00 p.m. Location: TBA.

    Nov 11: The second make-up lecture will be held at the usual location: i.e., Lecture Hall 4.

    Oct 27: The second of the two make-up lectures will be on November 12 at 2:00 p.m. Location: TBA.

    Oct 1: The first of the two make-up lectures will be held on Saturday, October 22, at 3:00 p.m. Venue: TBA.

    Aug 25: As announced earlier in class, I shall be away from August 27 attending a conference. There will thus be no lectures on August 26 and September 2 and 5 (August 31 is a holiday at the Institute).

    Aug 2: There will be no lecture on August 3.

    July 30: The first lecture of the course is on August 1.

  • Notes

    This is the space for occasional notes—usually references to optional reading—that expand upon some point that I did not go into in detail in my lectures.

    • A domain on which the Lempert function does not satisfy the triangle inequality: Example 2.3 of Chapter III in:
      Several Complex Variables III (G.M. Khenkin, ed.), Encyclopedia of Mathematical Sciences 9, Springer-Verlag, 1989.

    • The Lempert function equals the Kobayashi pseudodistance on a convex domain: Theorem 4.8.1 in Hyperbolic Complex Spaces (Shoshichi Kobayashi), Grundlehren der mathematischen Wissenschaften 318, Springer-Verlag, 1998 [this is a presentation in English of the original result in the article:
      Laszlo Lempert, La métrique de Kobayashi et la répresentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427–474.]

  • Homework assignments

    Homework 3

    Homework 2

    Homework 1


  • ANALYSIS–II: MEASURE AND INTEGRATION (MA222)   [Spring 2017, Spring 2020 ]

  • UNDERGRADUATE ANALYSIS & LINEAR ALGEBRA (UM101)   [Autumn 2017, "Autumn" 2020 ]

  • ANALYSIS–I (MA221)   [Autumn 2018]

  • INTRODUCTION TO BASIC ANALYSIS (UM204)   [Spring 2019, Spring 2022 ]


  • INTRODUCTION TO COMPLEX DYNAMICS (MA380)  [ Autumn 2016, Spring 2021 ]

Page last updated on November 12, 2022