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, P., Kulkarni, D. (2025). [ArXiv PDF] On Round Surgery Diagrams For 3-Manifolds. (Submitted) arXiv:2501.09518. 

2. Deep, P., Kulkarni, D. (2025). [ArXiv PDF] On Contact Round Surgeries on (S³, ξst) and Their Diagrams. Accepted in Topology and its Applications. arXiv:2504.06074. 

3. D. Kulkarni, T. Shah and M. Yadav. [Article] On the cost function associated with Legendrian knots, Studia Sci. Math. Hungar. 62 (2025), no. 1, 21–46. 

4. Deep, P., Kulkarni D. (2024).  [Arxiv PDF] On A Potential Contact Analogue Of Kirby Of Type 1. arXiv:2407.04395.

5. D. Kulkarni, K. Rajeevsarathy and K. Saha. [Article] Periodic surface homeomorphisms and contact structures, J. Korean Math. Soc. 61 (2024), no. 1, 1–28. 

6. S. Alape, A. Bhattacharya and D. Kulkarni. [Article] On certain rigidity results of compact regular (κ, μ)-manifolds, Results Math. 79 (2024), no. 8.

7. R. D. Holkar, M. A. Hossain and D. Kulkarni. [Article] The fundamental groupoid: topology, Haar system and actions, Münster J. Math. 17 (2024), no. 1, 203–240. 

8. D. Kulkarni and M. Yadav. [Article] On a generalization of Jones polynomial and its categorification for Legendrian knots, Bull. Sci. Math. 182 (2023).

9. M. Datta and D. Kulkarni. [Article] A survey of symplectic and contact topology, Indian J. Pure Appl. Math. 50 (2019), no. 3, 665–679. 

10. J. Conway, A. Kaloti and D. Kulkarni. [Arxiv PDF] Tight planar contact manifolds with vanishing Heegaard Floer contact invariants, Topology Appl. 212 (2016), 19–28. 

11. S. Gadgil and D. Kulkarni. [Article] Relative symplectic caps, 4-genus and fibered knots, Proc. Indian Acad. Sci. Math. Sci. 126 (2016), no. 2, 261–275.