Shiv Parsad

PIC

Subject Area: Hyperbolic geometry

Background and Research Interests

I completed my M.Sc from Punjabi University, patiala in 2007 and was awardee of NBHM post graduation fellowship. I joined IISER Mohali in Aug 2010. I am working under the supervision of Dr. Krishnendu Gongopadhyay.

My research has been concentrated on problems related to the complex hyperbolic space. The main problems that I have worked on are the following. (i) Classification of isometries in complex and quaternionic hyperbolic space. (ii) Complex hyperbolic Fenchel-Nielsen coordinates.

The isometries of the n-dimensional hyperbolic space are classified as elliptic, parabolic and hyperbolic according to the dynamics of their fixed points. Classically in two dimensional real hyperbolic geometry, this trichotomy of the isometries was classified algebraically in terms of their traces. It is natural to find a similar algebraic criteria in higher dimensions , as well as in case of complex and quaternionic hyperbolic space. With Gongopadhyay I have obtained a complete algebraic classification for n-dimensional quaternionic and complex hyperbolic space.

Let Σg be a surface of genus g 2. Fenchel and Nielsen gave global coordinates for the Teichmller space Σg. Analogus to that, Parker and Platis have given global coordinates to a subset of SU(2, 1) representation variety of π1g) which contains quasi-fuchsian representations. Their coordinates determines such reperesentations up to conjugacy. I am currently interested in giving such coordinates to a subset of SU(3, 1) representation variety of π1g) consisting of complex hyperbolic quasi-Fuchsian representations.

Select Publications

  1. (with K. Gongopadhyay), Classification of quaternionic hyperbolic isometries, Conform. Geom. Dyn. 17(2013),68-76.
  2. (with K. Gongopadhyay and J. R. Parker), On the classification of unitary matrices, (preprint).