The principle investigator will study properties of the Loewner differential equation. More specifically, via conformal maps the Loewner equation provides a correspondence between increasing families of 2-dimensional sets and continuous 1-dimensional functions (called driving functions). The nature of this correspondence is not well understood and is the source of several questions. How, for example, do properties of the driving functions affect the 2-dimensional growth? And how is 2-dimensional geometry converted into 1-dimensional data? One of the ways in which this work will address these questions is by studying how deformations of the driving function affect the corresponding geometry. A further goal of this research is to build on the exciting recent probabilistic work involving the Loewner equation and to explore deterministic properties that can be recovered from the current probabilistic understanding.
The Loewner differential equation is a long-standing tool in complex analysis. In the past ten years, a new use of the Loewner equation has attracted attention not only inside complex analysis, but also in probability and theoretical physics: the Loewner equation is an integral part of the random processes called Schramm-Loewner Evolution (SLE). Introduced by O. Schramm in 2000, these processes have been a tool in the solution of several open problems of interest to both mathematicians and theoretical physicists. This research will lead to a deeper comprehension of the Loewner equation and will further the work in this collaborative field. In addition, this project will contribute to human resource development in the training and mentoring of undergraduate students, with a goal of encouraging a broader participation of under-represented groups.
