Regina 2.0: Prateek Kumar Vishwakarma, on returning to the University of Regina a second time.

Ruth A. Situma, Program and Communications Manager

Regina is not a new city for Prateek. He has, in fact, returned to familiar stomping grounds — one that he visited not too long ago. A native of Uttar Pradesh in India, Prateek moved to Regina on February 14th, 2022. He owes his smooth transition to the fact that this was his second trip to Regina. In 2019, he spent a full semester at the University of Regina Campus, creating long-lasting friendships and connections— some of whom have been around to see his return back to the city. This earlier trip was funded by the Queen Elizabeth Award, which he won in 2019 while undertaking his Ph.D. work in India. Prateek is settling in and looking forward to teaching a full course in the coming summer semester. Asked about his newest discovery since arriving in Regina, he says “ My best discovery is that I can cook, and not just cook but cook well. From India, I miss the cafeteria-served food and the warm weather”. Our interview has been edited for clarity.

Tell us about your academic journey: What field are you in and how did you get there?

After completing my undergraduate degree at Banaras Hindu University in 2014, I decided to take a year-long break for self-study. During this time, I prepared for competitive exams to join a science research institute in India for higher studies in mathematics. I received an offer from the Department of Mathematics at the Indian Institute of Science (Bangalore) for the Integrated Ph.D. (M.S. and Ph.D.) program, and I joined the Institute in July 2015. During the first two years, I took various courses and completed my Integrated Ph.D. project (Master’s project) on zeros of random polynomials, under the supervision of Professor Manjunath Krishnapur. On completing this project, I had a few months to finalize my research topic and prepare for the Comprehensive Exams. I took this opportunity to meet Professor Apoorva Khare, who had joined the Department there only a week or two previously. His expertise in mathematics covers a variety of topics including algebra, representation theory, positivity, and probability. I was particularly intrigued by his relatively recent work in entrywise positivity preservers, partly because of the ease with which the problem can be explained and its connections with topics like metric geometry and combinatorics. As per his suggestion, I started reading the related pioneering and recent literature on entrywise preservers. I quickly realized that this topic has far-reaching repercussions (in other topics as well), and so I proposed to Professor Khare that I write my Ph.D. dissertation on this and he became my Ph.D. mentor.

Tabulation of the classification of tuple of entrywise maps that preserve positivity. Image credit — Prateek Kumar Vishwakarma’s Ph.D. thesis

My thesis introduced a novel and refined entrywise operation and provided the classification of positivity preservers in that environment. In addition to working for classification in this refined environment, I kept looking for collaborations and opportunities in related areas of matrix positivity. Fast forward to May 2019, as fate would have it, I saw a broadcast email from my Department Head stating that the Canadian Queen Elizabeth (QE) scholarship was available to support graduate students for a semester visit to the University of Regina. There was only one spot available. I mentioned this to my supervisor and he mentioned a connection to Professor Fallat (and about his expertise on Total Nonnegativity). Upon showing my interest and relevant experience to Professor Fallat, he immediately agreed to supervise, provided my application was approved by the funding agency. Happily, my application was successfully approved.

I was in Regina from September to December 2019. During this period, besides learning more about fundamentals in total nonnegativity (including its uniquely amazing and beautiful expression in terms of planar networks), I continued to work on entrywise positivity preservers. Several enlightening discussions with Professor Fallat helped me improve my thesis and enhance my understanding of total nonnegativity. By the end of my term, we made enough progress in total nonnegativity and preservers and we were sure to continue further research together. (In fact, we are working along that line in the present, and it is pretty exciting to see how things are unfolding!) I reported back to IISc on the first week of January, when my Ph.D. supervisor suggested that I communicate my work on entrywise positivity preservers for publication. This paper and my thesis have the same title; the article can be accessed using the links below:

Positivity preservers forbidden to operate on diagonal blocks. Trans. Amer. Math. Soc., DOI: [arXiv]

I came across the PIMS postdoctoral fellowship in 2020 and its availability at the University of Regina. Given my continued excitement about total nonnegativity, I contacted Professor Fallat with my proposal to work further on it with support from PIMS. He kindly agreed to support my application, and after almost a year-long wait (due to covid) and preparation I have been back in Regina since February 2022 and am looking forward to pursuing further research in academia, with more responsibility and experience.

What do you do to balance your research and life? What does a typical Sunday look like for you?

My formula is to keep my research and life simple and stimulating. For this, I make sure to enjoy different activities throughout the day. For instance, on a usual weekday, I tend to start with a task that really excites me. Usually, this is something that needs immediate action, which could be anything from my ongoing research, preparation for a talk, refereeing a manuscript, listening to a lecture, etc. After making some progress in any of those, I move on to the next most exciting/pressing assignment, and so on.

For the ‘life’ part, I prefer to enjoy outdoor activities like cycling, running, going for long walks in the main city and in forests (with my camera, in general). And if I get the opportunity and weather conditions are favorable, I enjoy playing any team sport that I can play. The most important activity that I do to keep my energy effervescent is investing my time into yogic and meditative processes on an everyday basis.

Preparing to practice Surya Kriya. Photo credit — Rahul Palamuru

Prateek Kumar Vishwakarma is a PIMS postdoctoral fellow at the University of Regina, working under the mentorship of Professor Shaun Fallat in the area of Matrix Analysis and Positivity. Prateek defended his Ph.D. thesis in December 2021 under the supervision of Professor Apoorva Khare at the Indian Institute of Science (IISc) Bangalore, India. He was awarded the Canadian Queen Elizabeth II Diamond Jubilee Scholarship to support his first visit to Regina as a visiting graduate student when he worked under Professor Fallat for the Fall 2019 semester. He has visited and has been invited to visit or speak at various international conferences in India, Canada, the US, Slovenia, the UK, and Ireland.

Prateek will be speaking at the PIMS Emergent Research Seminar Series, on April 6, 2022, at 9:30 AM Pacific. Details on his talk, Positivity preservers forbidden to operate on diagonal blocks can be found here.



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