Algebraic positivity preserver problems (25rit039)

The Banff International Research Station will host the “Synergy of the Formulation of Atmospheric Convection Parameterization” Research in Teams program at UBCO from August 10 - 17, 2025.

Recent mathematical advances are shedding new light on how matrices (arrays of numbers) can be transformed while preserving key notions of positivity. Understanding and classifying these transformations has profound implications in areas as diverse as cryptography, quantum computing, and high-dimensional statistics. This research brings together four experts in algebra, combinatorics, and matrix analysis, including three emerging researchers in the area -- one recent PhD from the University of British Columbia (now postdoc at Georgia Tech), and two postdocs at the University of Regina and Universit\'e Laval. Together, they aim to advance the classification of positive maps over finite fields and explore their deep connections to fundamental combinatorial structures.

The project bridges multiple mathematical disciplines, uncovering links between combinatorial matrix theory, finite field theory, and graph theory. In particular, it contributes to the study of Paley graphs—special structures with applications in coding theory, cryptography, and extremal combinatorics. These graphs also play a key role in number theory and quantum information theory, particularly in quantum error correction and nonlocal games. With positive definite matrices over finite fields introduced only recently (2022), this research is at the cutting edge of a rapidly evolving field, paving the way for new theoretical insights and practical applications across mathematics and beyond.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta’s Advanced Education and Technology.