Quasiconformal Constructions in Analysis and Dynamics

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The project deals with various problems in 2 and 3 dimensional geometry by applying ideas from conformal and hyperbolic geometry to problems of computer science and statistical physics, and, conversely, utilizing ideas from discrete and computational geometry to study longstanding problems in complex analysis and dynamics. These methods, in addition to their intrinsic interest, provide new connections between discrete and continuous mathematics, as well as between theoretical and applied questions. This expands the range of tools available in all these areas, and facilitates more substantial interactions between disparate groups of mathematicians, physicists and computer scientists. The concrete, geometric, and practical aspects of these problems also make them appealing to students and can form the basis of student research projects. For example, what is the optimal way to decompose a 2 or 3 dimensional region as a union of triangles or tetrahedra? Doing this quickly (e.g., in polynomial time) with good control of the shapes of the mesh elements (e.g., no angles too large or too small) is a fundamental first step in most numerical modeling methods, and a good initial mesh leads to faster, more accurate solutions. Nevertheless, many meshing algorithms used in practice (essentially all used in three dimensions) are based on heuristics rather than rigorously justified methods. The PI's earlier work on meshing in 2 dimensions is based on exploiting ideas from conformal geometry and dynamics, and uses the PI's earlier work on optimal algorithms for computing conformal mappings. The PI plans to continue his work in 2 dimensions to find the optimal meshing algorithms and to seek analogous approaches in 3 dimensions. Conformal maps send one planar region to another in a way that preserves angles. Such maps have been studied for almost two hundred years and are fundamental to many areas including complex analysis, dynamics, fluid flow and probability. The connection with probability arises most strongly through Brownian motion, the mathematical model of continuous random motion. Conformal maps send Brownian paths to Brownian paths, and this `conformal invariance' allows probabilistic arguments to be applied in complex analysis and allows complex analysis to be used to study Brownian motion. The project considers another well known, but difficult to understand, random process related to conformal maps: diffusion limited aggregation (DLA for short). In DLA particles wander at random in space until they make contact with a fixed cluster of particles and stick to it. As time progresses the cluster grows and takes a characteristic fractal shape, at least in numerical simulations; very little is known rigorously about the geometry of the resulting random clusters. Based on numerical simulations, the project considers new conjectures on DLA that may be more tractable. The above problems seek to use classical analysis to understand problems that are discrete in nature. Another aspect of the project is to use discrete ideas to attack problems in continuous analysis. The PI previously introduced a new method, called quasiconformal folding, for constructing holomorphic maps that that emphasizes the discrete and combinatorial structure of holomorphic function. Using this method, the PI and other investigators have constructed examples that answered a number of open problems, including several in the field of holomorphic dynamics. This is the study of stable versus chaotic behavior when a function is iterated, e.g., the Julia set of a map is a fractal set where the sequence of iterates acts chaotically. Determining the possible geometries of this set is a fundamental problem, and the PI will work on specific conjectures the case of general holomorphic functions defined on the whole complex plane. The project consists of problems grouped into five main categories, but all dealing with the interaction of discrete combinatorial structures (usually trees or triangulations) with continuous problems related to complex analysis and conformal geometry (Riemann surfaces, holomorphic dynamics, hyperbolic 3-manifolds, computational geometry and statistical physics). The first part deals with conformal structures induced by trees and graphs. Grothendieck's theory of "dessins d'enfants" shows that a finite graph on a compact topological surface induces a conformal structure on that surface, e.g., makes it into a Riemann surface. Several problems in the project seek to understand this connection precisely, e.g., given a finite tree in the plane, what does the tree "look like" in the induced conformal structure? A related problem is to build a Riemann surface by identifying edges of equilateral triangles. It is known that not every compact surface occurs in this way (only countably many can occur), but the PI conjectures that every non-compact surface does occur in this way (joint work with Lasse Rempe-Gillen). The second group of problems deals with transcendental dynamics, i.e., the iteration theory of non-polynomial entire functions. Quasiconformal folding is a method of building entire functions, starting from an infinite planar tree, that gives very good control of the geometry of the function and locations of its critical values. The PI seeks to apply QC folding to various problems, such as the possible Hausdorff and packing dimension of transcendental Julia sets, and the behavior of these dimensions under perturbations of the function. The third part of the project is to expand on recent work of the PI and Claude LeBrun that constructed examples 4-manifolds where the almost-Kahler structures form a non-empty, proper open subset in the moduli space of ant-self-dual metric; such examples were not previously known to exist. The proof works by reducing to questions about harmonic measure on certain (very complicated) hyperbolic 3-manifolds. Can simpler 3-manifolds be used? Are the "exotic" examples constructed actually "common" in some sense? The fourth part deals with optimal meshing problems in 2 and 3 dimensions. The PI recently proved that any planar domain bounded by a PSLG (planar straight line graph) can be meshed by non-obtuse triangles (maximum angle 90 degrees) in polynomial time, but the optimal power remains open. The proof uses a discrete closing lemma for certain flows associated to any triangulation and perhaps ideas from surface dynamics might help close the gap. The proof uses several estimates from planar geometry, and it would be a very interesting and important to extend the results to triangulated surfaces, a stepping stone to the important problem of meshing 3-dimension regions by non-obtuse tetrahedra. A key idea in the planar case is to decompose the region into pieces where either Euclidean or hyperbolic geometry is used to generate a mesh. Can a similar strategy be used in three dimensions, perhaps using the wider variety of fundamental geometries that occur in three dimensions? Finally, the PI will consider diffusion limited aggregation (DLA), a widely studied random growth model. The basic problem is to determine the almost sure growth rate of the cluster; there is no non-trivial lower bound and no improvement of Kesten's 1987 upper bound. The PI plans break the growth rate problem into two sub-problems: estimating upper bounds for the number of vertices on the convex hull boundary of DLA, and turning these into non-trivial lower bounds for the growth rate. 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.