The intellectual core of the proposal combines geometric measure theory, geometric function theory and differential geometry in sub-Riemannian spaces and abstract metric spaces. The proposal consists of three parts. Part I focuses on sub-Riemannian geometric measure theory, specifically dimension comparison theorems for Carnot-Caratheodory and Euclidean Hausdorff measure and dimension. Applications include sharp dimension computations for nonlinear Euclidean iterated function systems of polynomial type. Related projects concern characteristic negligibility for hypersurfaces in Carnot groups. A long-term goal is to classify constant mean curvature surfaces in jet space groups with an eye to identifying candidate extremals for their isoperimetric inequalities. Part II considers sub-Riemannian geometric function theory, specifically Heisenberg analogs of the Tukia-Vaisala quasiconformal extension theorems and a problem of Heinonen-Semmes. In Part III, the PI studies highly regular surjections to metric spaces. This line of research originates in classical point-set topology results of Peano, Lebesgue and Hahn-Mazurkiewicz and is also influenced by recent work on Morse-Sard theory and rectifiability. The PI has constructed highly regular surjections from Euclidean spaces of sufficiently high dimension onto doubling geodesic spaces. Future problems to be considered include borderline regularity, infinite-dimensional analogs and other regularity classes (Holder and Sobolev maps). A problem of Gromov on density of Lipschitz maps in Sobolev spaces with sub-Riemannian target will be studied. The proposed research encompasses a range of topics within nonsmooth geometric analysis, yet remains unified by a common framework.
Geometry studies the static structure of spaces of arbitrary complexity and dimension, while analysis studies dynamic properties and functional interrelations of such spaces. The adjective nonsmooth suggests non-Euclidean settings: fractals, stratified (sub-Riemannian) manifolds, and other abstract spaces. Sub-Riemannian geometry is the `geometry of constrained motion?: it models physical situations where motion is subject to a priori geometric constraints. It features in a remarkably broad spectrum of applications including robotic motion, digital image reconstruction, computer vision, neurobiology, and the mathematics of finance. Sub-Riemannian analysis involves an intricate blend of smooth and nonsmooth techniques as these spaces admit both smooth structure (in restricted directions) and fractal structure (in generic directions). The proposal integrates research, teaching, service and outreach on multiple levels. Graduate student training occurs via summer research programs, teaching of graduate core and topics courses, and Ph.D. supervision. Educational opportunities and outreach related to the research are proposed at the undergraduate and secondary school levels. The PI?s collaborators are located across the U.S. and Europe. Visits to and from these institutions by faculty, postdocs and students will generate new opportunities for collaboration and increase the visibility of the area. To this end, the PI will also continue to organize conferences and workshops in sub-Riemannian geometry and analysis.
The classical analytic framework of differential and integral calculus, as pioneered by Newton and Leibniz, has for centuries proved to be an invaluable tool in modeling scientific and physical processes. Nevertheless, it is widely recognized nowadays that more accurate mathematical models will employ analytic theories developed specifically for nonsmooth environments involving fractal-type structure. In many situations, techniques of smooth analysis are insufficient or inappropriate, and theories of analysis and geometry on abstract spaces of minimal inherent smoothness are necessary. The research supported by this award contributes to an emerging analytic theory valid in nonsmooth settings such as fractals and certain stratified geometric structures known as sub-Riemannian spaces. A major focus of the project concerns typical behavior, either the behavior of specific transformations acting on typical representatives in parameterized families of subsets or the behavior of typical transformations. Typical here refers to the overwhelming majority of examples, apart from a small set of exceptions. It is a general theme in mathematics that typical behavior may be quite different from, and indeed substantially better than, universal behavior. It is essential to identify the exceptional cases and to quantify their prevalence. In a series of papers, the Principal Investigator (PI) and his collaborators have formulated precise criteria of this type for a widely studied class of transformations enjoying intermediate degrees of regularity or smoothness (Sobolev mappings). The conclusions indicate the precise degree of allowed distortion for dimensions of typical subsets under fixed Sobolev mappings, or of fixed subsets under typical Sobolev mappings. The size of the corresponding exceptional sets is captured by suitable geometric measurements (Hausdorff measure) which generalize classical notions of length, area and volume. Sub-Riemannian spaces are natural models for physical processes involving motion subject to specified nonintegrable constraints. (For a concrete example, consider the problem of parallel parking: the immediate motion of the automobile at any given time is constrained, yet motion in the forbidden sideways direction is achievable via an alternating series of admissible movements.) From a mathematical perspective, optimal motion in such spaces can be interpreted as a shortest path (geodesic) connecting two locations in a suitable nonsmooth geometric structure. A second theme in the project concerns the relationship between such nonsmooth structures and their smooth (Riemannian) counterparts. The PI and collaborators have developed machinery for quantifying the discrepancy between smooth and nonsmooth structures, again using Hausdorff measure. These considerations clarify the extent to which the novel methods of modern nonsmooth analysis differ from their classical smooth counterparts. A fruitful approach to nonsmooth analysis and geometry is to realize it as a limiting case of smooth approximations. This approach is well understood for functions taking values in linear spaces, but is much more subtle for nonlinear targets. For sub-Riemannian target spaces, such results have until recently been unavailable. This award also supported research by the PI and collaborators which obtained the first results, both positive and---more importantly---negative, in the study of approximation of Sobolev mappings taking values in sub-Riemannian spaces. A complete description of all geometric structures to which the principles of modern nonsmooth analysis apply remains elusive. In collaboration with two junior postdoctoral researchers, the PI established the validity of the standard axioms of nonsmooth geometric analysis on a new class of fractal spaces, further demonstrating the power of these new methods. Research, outreach and education have been consistently and closely linked throughout the life of this award. The PI has written expository papers and a graduate research monograph on various aspects of nonsmooth analysis and their interaction with other branches of mathematics and the sciences. Training and mentoring of junior researchers has been a significant project component. Two postdoctoral researchers and six graduate students, including two students from an underrepresented population, have been mentored by the PI during the time period covered by this award. Among these students, those who have completed their employment at the PI's institution have gone on to effectively compete for subsequent teaching or research positions. The funding provided in this award has helped to advance the frontiers of knowledge in the subject of nonsmooth analysis and geometry and has contributed to the training of a new generation of mathematical researchers and educators.
