Current affiliation
CRM-Laval Postdoctoral Fellow
Université Laval, Québec City, Canada
Previous affiliation
Research Associate-III
Indian Institute of Science, Bangalore, India
Emails
prateekv[at]alum.iisc.ac.in
prateek-kumar.vishwakarma.1[at]ulaval.ca
[IWOTA 2026] [AMS Sectional Meeting 2025]
[CMS Summer Meeting 2025] [WCLAM 2023]
[University of Nevada, Reno - Matrix Theory Seminar]
[05C50 Online Seminar]
Feedback is welcome.
My research centres on questions about matrices, especially positivity, algebraic and geometric structure, and reconstruction from partial information. I am drawn to concrete matrix problems that reveal unexpected connections across geometry, graph theory, algebraic combinatorics, and probability. I also enjoy learning about unfamiliar problems and exploring whether a matrix perspective can offer fresh insight or suggest a new approach.
Here are a few recent projects that illustrate the kinds of questions I enjoy exploring.
Hyperbolic distance matrix completion
When only some pairwise distances are known, can we complete them consistently in hyperbolic space? This question brings together geometry and the graph describing the available measurements. Working with Mihai Putinar, we construct a canonical completion for chordal graphs and show how the graph's structure governs distance distortion.
Cholesky decomposition for symmetric matrices, Riemannian geometry, and random matrices
The Cholesky decomposition is a familiar tool for positive definite matrices. What happens when we look beyond positive definiteness? Working with Apoorva Khare, we explore a broader setting for this classical factorization, revealing connections with Riemannian geometry and random matrices.
Positivity preservers over finite fields and its Part II
How can we study positivity over a field that has no compatible ordering? Working with Dominique Guillot, Himanshu Gupta, and Kyle Yip, we investigate a finite-field analogue of matrix positivity and classify entrywise transformations that preserve it, uncovering connections with Paley graphs along the way. This leads us to ask which aspects of positivity depend on the real numbers and which reflect a more general algebraic structure.
Finite-orbit obstructions for multipliers on the polydisk
Can repeated application of multiplication operators compensate for having only finitely many starting functions? Working with Ilya Krishtal and Javad Mashreghi, we show that, on the Hardy space of the polydisk, using fewer multipliers than variables always leaves infinitely many independent directions unreached. The Jury product -- a convolution operation on matrices -- provides a concrete perspective on this obstruction: finite Taylor sections reveal a dimension deficit that grows with their size, connecting matrix algebra with operator theory and dynamical frames.
You can find more about my work in my [curriculum vitae] and on [Google Scholar].
I am always glad to exchange ideas and learn about new problems, including those outside my immediate research areas. If you see a possible connection with your work -- or wonder whether a matrix perspective might help with a question you are exploring -- please feel free to get in touch. I would be happy to hear what you are thinking about.