Research

For a summary of my research, here is my research statement (from 2019). At some point, I'll write a statement for a broader audience, but that's a work in progress.

 Preprints

  • G. Khan, An Illustrated Introduction to the Ricci Flow. [Available here]

  • G. Khan and J. Zhang, A Hall of Statistical Mirrors. [Submitted, Available here]

  • G. Khan and F. Zheng, Kahler-Ricci Flow Preserves Negative Anti-Bisectional Curvature. [Submitted, Available here]

  • G. Khan, J. Zhang, and F. Zheng, The Geometry of Positively Curved Kahler Metrics on Tube Domains. [Available here]

  • G. Khan, On the Total Curvature of Confined Equilateral Quadrilaterals. [Submitted, available here] 

  • G. Khan, Eigenvalue estimates without Bakry-Emery-Ricci bounds. [Submitted, Available here]

  • G. Khan, On the Hermitian Geometry of k-Gauduchon Orthogonal Complex Structures. [Available here]

Published Works

  • G. Khan and J. Zhang Recent developments on the MTW tensor. Proceedings of Geometric Science of Information 2021 
  • J. Zhang and G. Khan Connections with torsion in (para-)complexified structures. Contributed chapter to Progress in Information Geometry: Theory and Applications.
  • G. Khan, M. Khan, J. Saha, P. Zhao, A Conjectural Inequality for Visible Points in Lattice Parallelograms. Accepted for publication at the Proceedings of Combinatorial and Additive Number Theory IV [Available here]
  • J. Zhang and G. Khan, Statistical Mirror Symmetry. Differential Geometry and its Applications 
  • G. Khan and J. Zhang, The Kahler Geometry of Certain Optimal Transport Problems, Pure and Applied Analysis. [Available here]
    (An extended abstract of these results appeared in the proceedings of Geometric Science of Information 2019  under the title Hessian Curvature and Optimal Transport)
  • G. Khan, Hall's Conjecture on Extremal Sets for Random Triangles, Accepted at the Journal of Geometric Analysis.  [Available here]
  • G. Khan, B. Yang, and F. Zheng, The Set of All Orthogonal Complex Structures on the Flat 6-Tori, Adv. Math. [Available here]
  • J. Zhang and G. Khan, From Hessian to Weitzenbock: Manifolds with Torsion-Carrying Connections, Accepted at Information Geometry. (An extended abstract of these results appeared in the proceedings of Geometric Science of Information 2019 under the title New Geometry of Parametric Statistical Models)
  • G. Khan, Eigenvalues of the Complex Laplacian on compact non-Kahler manifolds,  AGAG. [Available here]
  • T. Hart, G. Khan, and M. Khan, Revisiting Toom’s Proof of Bulgarian Solitaire, Annales des Sciences Mathématiques du Québec. [Available here]

 

Not For Publication

  • G. Khan, On the Behavior and Singularities of Spacial Curves Under Curve-Shortening Flow, Boston University. Distinction Thesis.
  • G. Khan, A Condition Ensuring Spatial Curves Develop Type-II Singularities Under Curve Shortening Flow, preprint. (September 2012) [available here]

Posters

  • G. Khan "Optimal Transport and Complex Geometry" Optimal Control, Optimal Transport, and Data Sciences, November 2020 
  • G. Khan "Optimal Transport and Complex Geometry" Conference on Nonlinear Partial Differential Equations and Applications, July 2019 [available here]