ABSTRACT

In this project, the PI aims to develop techniques for the deeper understanding of fractals; that is, objects whose shape is not smooth and potentially have cusps and wrinkles, or objects with possibly self-similar repeating patterns. Such objects appear in nature as coastlines, mountainous landscapes, river networks, lightning bolts, snowflakes, growth models of plants and crystals, and soap films. The questions the PI plans to study have applications whenever storage of three-dimensional information (landscapes, faces, human brain surface) in a two-dimensional image is desired without loss of information. While in the case of smooth objects (the opposite of fractals) the corresponding mathematical theory is well understood, this is not the case for fractal objects, which require the development of new techniques. Another focus of this project is on rigidity problems, asking whether it is possible to deform a fractal object that is made out of a flexible material into another fractal object, with controlled distortion. Also, fractal sets appear sometimes as boundaries of otherwise smooth objects; another rigidity problem concerns whether these fractals are removable, in the sense that their presence can be ignored for transformation purposes. Rigidity problems on fractal sets have applications in mathematical problems that require "gluing" together two functions, or two dynamical systems, or two surfaces, and could result in the better understanding of dynamical systems in physics. This project will also incorporate the training and professional development of graduate students.

The main focus of the project is on two interrelated types of problems on fractals: uniformization and rigidity problems. The uniformization problem asks for geometric conditions on a fractal metric space so that it can be transformed to a smooth space with a well-behaved transformation that preserves the geometry, such as quasiconformal or quasisymmetric maps. Major progress has been made recently towards the quasiconformal uniformization problem with the involvement of the PI. The current project expects to develop an analytic theory for two-dimensional surfaces of locally finite area under no other assumption; the classical approaches in the field of analysis on metric spaces require instead several additional and restrictive geometric assumptions. Specifically, the PI will study the quasiconformal classification of non-smooth surfaces, the embedding of fractal surfaces in Euclidean space, the uniformization of 2-dimensional spheres of infinite area, and potential theory on fractal surfaces. Regarding rigidity problems, the PI will work on the problem of conformal removability, which asks whether a given compact subset of Euclidean space is negligible from the domain of a conformal map. The PI in recent works has displayed several new examples of removable and non-removable planar sets and has found a striking connection between the problems of uniformization and removability. Moreover, the PI has identified a new general class of sets that he conjectures to provide a characterization of removable sets. The PI will study this conjecture, as well as several related removability and rigidity problems in complex dynamics, geometric group theory, and circle domains.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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