"Symmetry in geometry has historically been a major driver of scientific development, particularly in physics. It also has a broad and seemingly innate appeal to our sense of aesthetics, order and beauty in nature. Mathematicians have a highly evolved set of tools for working with symmetries that are in a sense ""compact"" and localized, like rotational motion. They have a much harder time with symmetries that relate points very far away from each other, like translations. It turns out that many natural ""flows"" can be interpreted in terms of non-compact symmetries. Work of the beneficiary and the supervisor addresses these non-compact flows in a range of geometric, and ultimately physical, settings.
This project exploits what may be termed ""hidden non-compact symmetries"" that are canonically attached to many geometric objects. These do not arise as symmetries of the objects themselves, but rather as symmetries of a higher dimensional object, with the original geometric object representing the higher dimensional object ""up to non-compact symmetry"". In effect, each point of the original geometric object represents a non-compact ""flow"" along the higher dimensional object. For a primitive picture, one can think of a cylinder as a higher dimensional object, which reduces to a line, using circular symmetry, and alternately reduces to a circle, using translational symmetry perpendicular to the base. The main idea is that it turns out this canonically associated higher dimensional object, equipped with the symmetry, is often much simpler than the original object. The main goal of the project is to develop and apply tools for these non-compact symmetries to a range of well-known, and in some cases very long standing, problems about the original geometric objects.
There are three broad areas of application: geometry, arithmetic (or number theory), and physics of quantum systems. The geometry questions have roots in old problems dating to the time of Newton: given n points in the plane, what is the minimal degree polynomial curve that passes through them (various constraints also can be put on how the curve looks near these points)? Various conjectures and partial results exist, but it is still wide open. For number theory, much like in Fermat's Last Theorem, one can ask how many integer solutions exist to a system of polynomial equations. Refinements of these questions in special cases, about the rate of growth of number of points solving the equations with respect to a natural ""height"" measure, can be studied with the techniques of the project. Finally, in physics, a robust quantum entanglement classification, with application to quantum information, can be studied with these techniques. Quantum information and computing has many potential applications to technology and society as a whole, as has been well-covered in the popular science media."
