Research

What is Contact and Symplectic Geometry (and Topology) and why should we care about it?

Symplectic and contact geometry are branches of differential geometry that study geometric structures encoding notions of "conserved quantities" and "constraints" coming from classical mechanics, optics, thermodynamics, and modern topology. They provide powerful tools and deep theorems connecting dynamics, topology, and analysis.

In particular symplectic geometry is the study of symplectic manifolds – smooth even-dimensional manifolds equipped with a closed, non-degenerate 2-form called the symplectic form (ω). This structure preserves area/volume in a special way and provides the mathematical foundation for classical mechanics.

Contact geometry is the odd-dimensional counterpart to symplectic geometry, dealing with contact manifolds – (2n+1)-dimensional manifolds equipped with a completely non-integrable hyperplane distribution. It's the mathematical framework for thermodynamics, geometric optics, and control theory.

Publications and Preprints:

1. Deep, Prerak, and Dheeraj Kulkarni. "On A Potential Contact Analogue Of Kirby Move Of Type 1." arXiv preprint arXiv:2407.04395 (2024).
   
   

2. Deep, Prerak, and Dheeraj Kulkarni. "On Round Surgery Diagrams For 3-Manifolds." arXiv preprint arXiv:2501.09518 (2025).
   
   

3. Deep, Prerak, and Dheeraj Kulkarni. "On Contact Round Surgeries on (S3,ξst)(S3,ξst​) and Their Diagrams." arXiv preprint arXiv:2504.06074 (2025).
   
   

4. Rajeevsarathy, K., Kulkarni, D., and K. Saha. "Periodic Surface Homeomorphisms and Contact Structures." J. Korean Math. Soc. 61(1): 1-28 (2024).
   
   

5. Shah, Tanushree, Yadav, Monika, and Dheeraj Kulkarni. "On The Cost Function Associated With Legendrian Knots." Accepted in Stud. Sci. Math. Hung. (2024).
   
   

6. Kulkarni, Dheeraj, and Monika. "On a generalization of Jones polynomial and its categorification for Legendrian knots." Bulletin des Sciences Mathématiques, Volume 182 (2023).
   
   

7. Datta, Mahuya, and Dheeraj Kulkarni. "A survey of symplectic and contact geometry." Indian Journal of Pure and Appl. Math., 50(3), 665-679 (2019).
   
   

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Past Grants