2018 Hatherton Award: Ground-breaking algebra

Despite only being published in 2018, Jurij's paper on noncommutative rational functions has already been cited by leading researchers in random matrix theory and free probability theory, and was used to prove a 50-year-old conjecture in linear algebra.

Royal Society Te Apārangi presents this award to the best scientific paper by a PhD student at any New Zealand university, studying chemical, physical, mathematical or information sciences.

Jurij's ground-breaking work builds on decades of efforts by eminent researchers. In his paper, Jurij makes several discoveries that eluded these earlier researchers, and further develops fundamental new properties of rational functions.

Many optimisation problems, for example textbook classics in control theory, have matrices as variables, and the formulas naturally involve ratios of polynomials, or rational functions, in these matrices. Matrix multiplication does not commute, the order of the multiplication matters, hence the term noncommutative. These functions depend only on the system layout and do not change with the size of the matrices involved, hence such problems are dimension-free. Using free analysis to approach these issues, Jurij's paper has made major progress in this field and opened up exciting new approaches to the fundamental challenges ahead.

The highlight is a new and complete realisation theory for noncommutative rational functions; a realisation "linearises" a rational function yielding an object that is much easier to work with, in a way analogous to the approximation of a curve by a simple straight line. By deeply understanding these linearisations, Jurij was able to draw far reaching conclusions about the "curves" themselves, where curves in this case refers to much more general and abstract objects than lines on a page, the rational functions.

Of special interest are the minimal realisations, whose existence and uniqueness Jurij establishes, and which for the first time produce canonical form for noncommutative rational functions.  Significantly, the machinery developed led Jurij to the solution of the long-standing open problem of checking equality of two noncommutative rational functions.

On receiving the award, Jurij said: “I was immensely delighted when I heard about receiving the award, and also positively surprised as mathematics doesn't always show its shine due to its abstractness. This paper is at the core of my thesis, and it was a lengthy process from the initial idea that I had at the beginning of my PhD, to the final results years later. For this reason, the award is also an encouragement for me that the long hard work eventually pays off. Finally, it will always remind me of the three years in New Zealand, which have a special place in my mind, and were the most pivotal ones for my future career.”

Jurij took up a postdoctoral position at the very highly-ranked Ben Gurion University Mathematics Department in Israel before his PhD thesis was complete. In August 2018 he moved to a three-year postdoctoral position at the prestigious Texas A&M University.

Hatherton Award:

For the best scientific paper by a PhD student at any New Zealand university in chemical sciences, physical sciences, mathematical and information sciences.

Citation:

To Jurij Volčič for his paper describing fundamental new tools for measuring noncommutative rational functions in the emerging mathematical field of free analysis.

Publication details:

J.Volčič: Matrix coefficient realization theory of noncommutative rational functions, Journal of Algebra 499 (2018) 397–437.