Coloring with geometry, with PIMS PDF Andrii Arman at the University of Manitoba.
By Lisa Sammoh, Communications and Events Assistant.
Andrii Arman obtained his Ph.D. from the University of Manitoba under the supervision of David Gunderson, where he worked on the problem of determining the maximal number of cycles in graphs with certain restrictions. He obtained his Bachelor’s degree from Kyiv National University (Ukraine) and Master’s degree from the University of Manitoba.
After obtaining his Ph.D., he was a research fellow at Monash University (Australia) working with Jane Gao and Nick Wormald on uniform generation of contingency tables with given marginals. Following that, he was a Visiting Assistant Professor at Emory University (USA). Currently, he is a PIMS postdoctoral fellow at the University of Manitoba working with Andriy Prymak on covering problems in convex and discrete geometry.
Tell us about your academic journey.
I obtained my Bachelor’s degree from Kyiv National University (Ukraine) under the supervision of Andriy Bondarenko. During my third semester, I met Andriy Prymak who suggested that I consider the University of Manitoba for my Master’s degree. I followed his advice and enrolled in the Master’s program at the University of Manitoba under the supervision of Nina Zobroska. After completing my Master’s degree, I began a Ph.D. program at the University of Manitoba under the supervision of David Gunderson where I worked on the problem of determining the maximal number of cycles in graphs with certain restrictions.
After obtaining my Ph.D., I worked as a research fellow at Monash University (Australia) with Jane Gao and Nick Wormald on uniform generation of contingency tables with given marginals. I then became a Visiting Assistant Professor at Emory University (USA) where I worked with Vojtěch Rödl on finding colorful matchings in edge-colored graphs.
During my studies, I developed an interest in applying combinatorial methods in geometry, which led to a paper on a problem in Euclidean Ramsey theory that I co-wrote with Sergei Tsaturian, and also took a course in geometry taught by Andriy Prymak. At Emory University, I started a project with Andriy Prymak on determining the chromatic number of the space, which culminated in a joint paper with Andriy Bondarenko, Andriy Prymak, and Danylo Radchenko.
Currently, I am a PIMS postdoctoral fellow at the University of Manitoba, working with Andriy Prymak on covering problems in convex and discrete geometry. I also teach a numerical analysis course, which I hope will provide new insights and growth as a teacher and mathematician.
What field are you in and what is your current research on?
My research focuses on graph theory and combinatorics, specifically utilizing random methods for combinatorial problems.
Together with Dr. Prymak, our current work is centered on discrete and convex geometry. Our first project involved determining the chromatic number of n-dimensional space — the minimal number of colors needed to color the points of the space so that the points that are distance one apart receive different colors — for which we derived new upper bounds. Our results generalized the well-known coloring of a plane in 7 colors to colorings of 4-,6-,8-, and 24-dimensional spaces in 72, 73, 74 , 712 colors, respectively (see Figure b).
More recently, we achieved an exponential lower bound on the illumination number of bodies with constant width. Our proof relied on a combination of geometric and probabilistic approaches, involving a small cone-like configuration and numerous random rotations to construct the desired body. We demonstrated that such a “random” body of constant width has a high probability of possessing a large illumination number.
What does research and life balance mean to you?
I am married and have a three-year-old child, so spending time with my family is my main way of recharging. Cross-country skiing is one of the ways I relax after work, which I started during my Ph.D. studies. It makes winter in Winnipeg much more enjoyable. I also play soccer in low-level leagues, as well as badminton, biking and running. During summer, I like camping and hiking. Origami is also quite meditative for me, helping me to wind down and relax.
What has been your best discovery since arriving at UManitoba? What are you looking forward to learning/ seeing about the place and your postdoctoral position at the university?
In terms of discovery, the University of Manitoba’s Mathematics department has grown considerably since my last time here, with more postdoctoral fellows (mostly PIMS PDFs) and collaboration opportunities.
As a biker, I appreciate the bike trail from downtown to the university, which provides a healthier and eco-friendlier commute in the warmer months. I am also excited for the CanaDAM 2023 conference in Winnipeg this summer, and hope to see more math activities, conferences, and workshops in the city.
Andrii will be speaking at the PIMS Emergent Research Seminar Series, on May 10, 2023, at 9:30 AM Pacific. Details on his talk and poster, On illumination number of bodies of constant width, can be found here.
