Award Title: Quasiconformal methods in analysis, geometry and dynamics
Federal Award ID: 1305233
A conformal map is a transformation from one 2-dimensional region to another that may distort lengths and areas, but preserves angles. Such maps have an ancient history (e.g., projecting the spherical Earth onto a flat piece of paper for navigation), but are still the focus of intense investigation. The PI's work is devoted to better understanding the basic theory of such maps (and related classes of maps that allow limited angle distortions), their dynamics and their applications.
One such application is the creation of triangulations and meshes that have good geometric properties and are efficient to make. Approximating a region by a discrete mesh is a fundamental starting point in many applications such as computer graphics, numerical analysis and data analysis. Good geometry means that the elements of the mesh should "roundish", i.e., not long and narrow. Efficient means that not too many elements are used, e.g., there is a polynomial bound on the number of elements in terms of the complexity of the region (e.g., the number of boundary edges). The PI has used ideas arising from conformal analysis to give meshing algorithms using triangles and quadrilaterals that are optimal from both the geometric and complexity viewpoints. For example, conformal maps can be used to transfer "nice" meshs from a standard domain like a disk, to a more complicated domain, so meshing is closed related to computing conformal maps numerically. The PI also developed meshing algorithms based on dynamics of flowing points along vector fields that arise by thinking of a polygonal region as a Riemann surface (using hyperbolic geometry to solve a problem in Euclidean geometry). These methods improve previously known algorithms, and in some cases, provide the first polynomial time bounds for solving a meshing problem with geometrically nice mesh elements. A specific example of such a result is the following: if the interior of a polygon is triangulated (divided into triangles by edges that join vertices of the polygon), can we subdivide the triangles so as to get a new triangulation (triangles meet along whole edges, but vertices need not be on the original polygon) where all the triangles are acute (all angles less than 90 degrees). Previous work had shown that if we start with a triangulated n-gon, then this can be accomplished with n^4 new sub-triangles. The PI reduced this n^2, which is best possible.
The second main focus of the PI's work has been on the iteration theory of entire functions. A dynamical system usually consists of a map taking some space into itself. Given a single point, we repeatedly apply the map to obtain the orbit of this point and we then try to describe the possible behaviors of the orbit (does it have a limit? does it behave chaotically?). A well studied special case is when the space is the two dimensional plane and the map is an analytic function on the plane (also called an entire function). In this case, points in the plane are either classified as being in the Fatou set of the map (if the corresponding orbit is "well behaved" in a precise sense), or in the complementary Julia set (if the orbit is chaotic). Understanding the geometry of these two sets is the basic problem. The sub-case of polynomials have received the most attention, but in recent years the more general case of non-polynomial entire functions (called transcendental functions) has received increasing attention. The PI created a new method for constructing transcendental examples that is more geometric than previous methods (which generally depended on formulas for the functions) and used this solve a number of conjectures in the field. One example was the creation of a transcendental function whose Julia set has Hausdorff dimension 1 (this is a measure of the complexity of the set). It was known since 1975 that 1 is the smallest possible value, but this was the first example to attain this dimension. Another application was the construction of new examples of transcendental functions that have "wandering domains", e.g., components of the Fatou set whose orbit under the map f never return to the same part of the plane twice. A famous 1985 theorem of Dennis Sullivan says that polynomials never have such wandering domains, so this result helps illustrate the difference between polynomial and transcendental dynamics in the plane. Other examples of "pathological" behavior in transcendental dynamics can be created in a similar way.