polona durcikAssistant Professor of Mathematics Dr. Polona Durcik received a grant from the National Science Foundation for her research on “Multilinear Harmonic Analysis and Applications.”

Research will be done in harmonic analysis, which is an area of mathematics concerned with decomposing functions into basic waves, analyzing the pieces individually, and then synthesizing this local information into information about the original object. The methods and results from the harmonic analysis have wide applications in physics and engineering, but also within mathematics itself.  A special focus of harmonic analysis is to obtain good quantitative information on the objects in question, which can often be used to establish quantitative results in the nearby fields of ergodic theory and combinatorics. Ergodic theory investigates the behavior of dynamical systems that evolve for a long time, while one focus of combinatorics is to study the existence of structures in large but otherwise arbitrary sets. These structures can be geometric patterns such as arithmetic progressions, vertices of a square, their translates, rotates, or dilates. 

The project will focus on the study of multilinear integral transformations that naturally occur within harmonic analysis. The results and techniques will then be applied to investigate quantitative convergence properties for various types of ergodic averages along the orbits of several commuting measure-preserving transformations. Further applications will be in the combinatorics of Euclidean spaces, where the existence of linear and non-linear point configurations in large subsets of the Euclidean space will be investigated. Motivated by several major open questions, the research builds a toolset that further expands the reach of harmonic analysis into different fields of mathematics. 

Congratulations to Dr. Durcik!