Grant: Discovery Grants Program - Individual

The proposed research explores the interplay between two different areas of mathematics, namely linear algebra and graph theory. One particular type of problem being explored is how mathematics on arrays of numbers is affected by qualitative information, such as the pattern of positive, negative and zero entries in the array. Sometimes this information alone is sufficient to observe significant restrictions on results, independent of the magnitude of the individual numbers in the array. These sign patterns can be modeled using graph theory and this theory can provide insights into the restrictions for given patterns. Part of this research entails determining what restrictions can be discovered with these models, and likewise, how these models can help demonstrate lack of restrictions. The study of sign patterns was sparked by Nobel Laureate P. Samuelson in economics. Various problems in economics involve modeling situations when it is impossible to get exact numbers, but the more qualitative information is available. Such problems occur also in ecology, population biology, modeling epidemics, chemistry and sociology. Pattern analysis can also be helpful in inverse problems. These problems can arise in engineering problems, such as in structural analysis. An inverse problem generally involves trying to physically reconstruct a system with some prescribed behaviours. Usually there are constraints on the system. Pattern analysis could be useful in determining if, given the constraints, the inverse problem is feasible. I will focus on the mathematics of patterns in order to develop theory that may be useful in future development of modeling applications. As such this project is pure mathematics. I will be studying various abstract patterns to determine ways in which the graph theoretical information provides indicators as to restrictions inherent in the way the pattern can be used. I will be developing the methods used to determine what can be known about restrictions based on patterns.