Gautam Bharali

               Department of Mathematics

                 Indian Institute of Science

                 Bangalore 560012

 

Home Education Research Publications Abridged CV Miscellanea Teaching


BRIEF RESEARCH DESCRIPTION

My area of research is Several Complex Variables (SCV). Mathematical problems and techniques in this area make contact with a broad range of disciplines, from analysis to geometry to partial differential equations. My own interests in SCV, and the techniques that I am most familiar with, are closer to analysis. In recent times, I have also worked on problems of a geometric nature (although the solutions thereof feature a considerable amount of analysis) with several wonderful collaborators. Take a look at my list of publications to see whom I have been collaborating with.

A broad classification of the types of problems I am working on at the moment is as follows:

  • Dynamics of holomorphic correspondences: A holomorphic correspondence is a special type of relation between a pair of complex manifolds of the same dimension (of which holomorphic maps are a special case). A compact hyperbolic Riemann surface admits only finitely many holomorphic self-maps. Thus, the iterative dynamics of any such map is uninteresting. However, the class of holomorphic correspondences on a hyperbolic Riemann surface is very rich. Thus, the dynamical system arising from iterating a holomorphic correspondence (that is not a self-map) exhibits very interesting and complex behaviour. The study of this type of complexity is a rather new field, with lots of scope for exploration.

  • The Kobayashi geometry of domains: This aspect of my research involves the geometric properties that domains in Cn acquire when viewed as metric spaces equipped with the Kobayashi distance. My core interest is to obtain a complete theory for the boundary regularity of complex geodesics in convex domains. This has—in joint work with collaborators—led to investigations of when and how the above-mentioned domains exhibit weak forms of negative curvature.

  • Rigidity of holomorphic mappings: "Rigidity" here refers to the phenomenon wherein every member of a specified family of holomorphic maps turns out to obey severe structural constraints (and is thus easy to describe, often explicitly). This type of rigidity usually arises due to metrical or topological properties of the target space in question (and on the way these properties interact with holomorphicity).




RESEARCH SUPERVISION

CURRENT Ph.D. STUDENTS

  • Annapurna Banik

  • Mayuresh Londhe

  • Anwoy Maitra

FORMER Ph.D. STUDENTS (in reverse-chronological order of graduation)

FORMER M.S. STUDENTS (in reverse-chronological order of graduation)