Grant: Alliance Grants

Fractals are mathematical objects with intricate, complex structure at many scales. The surface of a lung, the drainage basin of a river, the boundary of a tumour, and the fluctuations of the stock market can be modelled using the mathematics of fractals. In the 17th century, Newton and Leibniz developed calculus to help understand the fundamental physical principles that describe our world. While calculus has been wildly successful for understanding "smooth" phenomena such as the trajectory of a ball or the orbits of the planets, it is unable to cope with the kinds of "rough" phenomena described by fractals. To date, these types of rough phenomena are much less well understood, and there is a clear need for new tools to understand the mathematics of fractals. The proposed collaboration seeks to develop a new mathematical theory that describes how fractals interact with certain fundamental mathematical operations, called projections and intersections. Questions of this type are called "incidence problems in fractal geometry." Over the past 20 years, researchers have found surprising connections between incidence problems and harmonic analysis, a branch of mathematics that investigates objects through their frequency decompositions, similar to how a complicated musical chord can be created by superimposing individual notes. In recent years, there has been great progress in tackling incidence problems using new and highly innovative tools, some of which were developed by the applicants and the international collaborator. However, while each of these tools has proved effective individually, researchers don't know how to combine them. The central goal of the proposed collaboration is to achieve this unification of the theory.