Research

Brief Research description.

My research interests, broadly speaking, lie in the theory of functions of several complex variables and in complex geometry.  A significant part of my Ph.D. dissertation is devoted to the study of interpolation problem from the polydisc to the unit disc, based on the tools from the function theory aspects of several complex variables. Complex-geometric objects such as the invariant distances come into the picture naturally, in connection with the interpolation problem from the unit disc to the spectral unit ball. Some of the projects that I am currently engaged in are the following:

1. Schwarz lemma for holomorphic correspondences. 

Holomorphic correspondences maybe thought of as set-valued mappings where the graph of the mapping is an analytic variety in a product domain. In my Ph.D. dissertation, while studying the Pick interpolation problem from the unit disc to the spectral unit ball, I proved a Schwarz lemma-type inequality for holomorphic correspondences from the unit disc into a given planar domain. The tools that were employed there were closely related to the aforementioned interpolation problem with 2-point data set. This latter inequality involves the Hausdorff distance induced by the Caratheodory distance between the values of holomorphic correspondence in the planar domain and the Mobius distance in the unit disc. 

A natural question arises whether one could replace the Caratheodory distance by the Kobayashi distance and the planar domain with any hyperbolic Riemann surface. That would certainly provide a refinement of the above inequality even in the case of planar hyperbolic domains. Another natural question is whether it is possible to find an analogous Schwarz lemma-type inequality for holomorphic correspondences for bounded domains in higher dimensions that are nice enough. 

2. Negative curvature-type behaviour of Kobayashi distance.

Given a bounded domain in a complex Euclidean space we equip it with the Kobayashi distance. In recent times, intensive research has been done in finding out when the resulting distance space exhibits negative-curvature-type behaviour. For example: one could ask when the distance space is Gromov hyperbolic? In this direction itself, many non-trivial and deep results are now known, e.g., every bounded strongly pseudo-convex domain is Gromov hyperbolic and so is every bounded convex domain of finite type in the sense of D'Angelo.

In general, Gromov hyperbolicity of the aforementioned distance space is extremely difficult to check. Motivated by a definition of visibility spaces by Eberlein and O'neil, in the context of Riemannian manifold with non-positive scalar curvature, Bharali and Zimmer introduced a notion of visibility with respect to almost geodesics. They also showed that a large class of domains (named as the Goldilocks domain) satisfy the visibility property. They also proved several function-theoretic properties of Goldilocks domains. Later, Bharali and Maitra realized that it is the visibility property rather than the abstract and technical defining conditions of Goldilocks domains that are at the heart of some the function-theoretic properties of these domains.  Another notion of visibility in the context when the above distance space is complete is introduced in a recent manuscript of Bracci, Nikolov and Thomas. At present, working with my collaborators we are interested in understanding the connection between these two notions and also finding out more applications of these notions. 

3. Squeezing function and uniform squeezing property.

Given a bounded domain and a point in the domain, the squeezing function is a bi-holomorphic invariant that measures the radius of the largest Euclidean ball that fits in the image of the domain under a holomorphic embedding, that maps the given point to zero, inside the Euclidean unit ball centered at the origin. A domain is called holomorphic homogeneous regular or uniform squeezing if the infimum of the previously defined quantity on the bounded domain is a positive number. Such domains exhibit many nice properties, e.g., the Caratheodory metric, the Kobayashi metric and the Bergman metric on a uniform squeezing domain are all equivalent. 

My interest in the above topic is from the point of view of discovering new classes of domains that are uniform squeezing. There are many other problems, for example, how the boundary behaviour of squeezing function reflects into certain complex-analytic properties of the  bounded domain that also interest me.