Katarina Spasojevic joined the research group of Professor Monica Nevins to complete an honours project using an unusual number system called the p-adic numbers. For each prime p, these are an extension of the rational numbers that reveal their hidden number-theoretic structures, but they are quite unlike the real numbers! For example, p-adic distances take on only discrete values – like the energy levels in quantum systems, which they are sometimes used to model. Many problems in mathematical modeling are interpreted in matrices; taking matrices with p-adic entries gives the p-adic groups that Katarina calculated with. However, p-adic groups are infinite and complex, so we study them via how they act on the associated Bruhat-Tits building – a crazy object formed by gluing infinitely many vector spaces together along specific fold lines that reveal the inner structure of the group. Katarina’s project was to take p-adic tori – key subgroups of a p-adic group that, if we could draw in p-adic space, would look like donuts – and find out which (very few!) points of the Bruhat-Tits building they fixed in place. These calculations, which required Katarina to be at once creative and meticulous, will be leveraged into a concrete resource for researchers in p-adic representation theory. Her examples and methods lay the foundation for an algorithm to compute these fixed points in general. In turn, these are key to understanding the big problems in p-adic representation theory today.
This experience sparked Katarina’s desire to pursue graduate studies and she has since embarked on a master’s program in probability at the University of Toronto. Her interest lies in using random matrix theory to tackle problems in optimization, a subfield of mathematics that is essential for the implementation of statistical methods in data science, engineering and finance. In addition to her research, Katarina also works as a tutor for undergraduate students in engineering, mathematics and economics.
Katarina credits Prof. Nevins’ invaluable support and guidance for inspiring her passion for mathematics. She encourages undergraduate science students to seriously consider gaining research experience, which allows you to delve into a topic of interest while developing many transferable skills. Students should not be shy to approach a professor and ask about their research — it could lead to valuable experience!