John Garnett - US grants

This project is mathematical research in certain areas of function theory. Functions in the plane satisfying rather simple partial differential equations, for instance analytic and harmonic functions, turn out to have properties which make them both useful and fascinating. A fundamental property exhibited by such functions is that their values on a simple closed curve completely determine their values at points inside. One can study harmonic and analytic functions so to speak concretely, or one can take a more abstract approach by forming Banach spaces or algebras of appropriate kinds of functions and applying the methods of functional analysis. The investigators specialize in attacking concrete problems with abstract methods. More specifically, the topics to be investigated include uniform algebras, algebras of bounded analytic functions, the corona problem in the plane, estimates for harmonic measure, and the spectral theory of certain types of Schroedinger operators.

Professor Garnett's research project is in function theory. The topics to be investigated include the algebra of bounded analytic functions on the unit disk, the corona problem in the plane, analytic capacity, and quasiconformal mappings. The main focus of Professor Garnett's research is the theory of functions of one complex variable which are differentiable in the sense of elementary calculus. These functions are defined on a domain in the plane but not necessarily on its boundary. Some of the most interesting problems in analysis concern the behavior of these functions as the complex variable approaches the boundary. Professor Garnett's research will focus in part on such questions.

In this project the principal investigators will study the geometric properties of harmonic measure; in particular, they will focus on the relations between harmonic measure on domains in the plane and one-dimensional Hausdorff measure. The principal investigators will also look into problems involving harmonic measure and analytic capacity, the relationship between rectifiable sets and harmonic measure, and the estimation of eigenvalues of the Laplace operator on subdomains of the sphere in higher dimensions. The principal investigators are interested in studying the relationship between an analytical quantity known as the harmonic measure of a set and the geometry of this set. Harmonic measure is an important and useful concept that arises in probability theory and complex function theory. In this project the principal investigators will seek to characterize properties of the harmonic measure of a set in terms of geometric properties of the set such as its Hausdorff dimension.

This award provides support for postdoctoral research on problems arising in the general area of geometric function theory. The work focuses on the geometric properties of quasiextremal distance domains. These are domains characterized by the property that points and closed sets can be joined by curves within the domain whose lengths do not be arbitrarily long. They are singled out because they have the characteristics believed to be fundamental for the study of quasiconformal mappings. Work on this project will concentrate on showing how the modulus of quasiextremal distance domains carries over to the dilatation constant for quasiconformal mapping defined on them. A second line of research will consider the length of level sets of univalent mappings and the question of the largest power to which such a mapping's derivative has a finite integral. Complex function theory encompasses the study of differentiable functions of a complex variable and related classes of functions such as harmonic functions and quasiconformal mappings. The subject is highly geometric; many of the problems concern the properties of various sets under transform by functions from one of the above classes. Applications of the theory to potential theory and fluid dynamics is now standard in engineering circles

Garnett 9401269 The research supported by this award focuses on problems arising in the mathematical theory of functions of one complex variable and one-dimensional Schrodinger operators. The work continues investigations in four problem areas. In the first, work will be done investigating the density of Blaschke products whose roots are restricted to interpolating sequences for the algebra of bounded analytic functions. It is expected that all Blaschke products are uniform limits of the restricted ones and that the closed convex hull of these Blaschke products is the unit ball for the algebra. A second line of investigation concerns analytic capacity to help understand the nature of sets having positive Hausdorff dimension but which have zero capacity. Work will also be done on determining for which powers the derivative of a conformal mapping is integrable along straight lines in simply connected domains. Related investigations will consider the same question when the integrals are taken with respect to area measure. Efforts to understand the gap sequences (bands) in one-dimensional periodic Schrodinger operators will be taken up for operators with complex potentials. Recent work has shown that the spectrum can be realized in terms of Riemann surfaces. The purpose of this work is to obtain a more intrinsic description of the spectrum. Complex function theory is the study of the analytic and geometric properties of differentiable functions of a complex variable. The subject has grown in time to include ancillary lines of investigation such as quasiconformal mapping, quasiregular mapping and potential theory. Work in the field is motivated by problems in geometry, differential equations, fluid dynamics and control theory. Recent work also involves studies of computational conformal mapping through new applications of circle packing. ***

ABSTRACT Tao will study geometrical analysis conjectures and results, and relate them to oscillatory integral statements such as the restriction conjecture. A typical example of the former is the Kakeya conjecture, that a set which contains a line segment in every direction in R^n must have dimension n. A typical example of the latter is the generalized Strichartz estimate for various PDE (the wave equation, the Schrodinger equation, etc.). It has been known since the 1970s that there is an intimate relationship between the two types of statements, but no systematic approach exists, and the few concrete connections that are known are not very satisfactory. We hope to collect, simplify, unify, and extend previous results in this direction. Tao will also attack some of these conjectures (notably the Kakeya conjecture) directly, using some new and promising techniques; for example, Tao will exploit the affine invariance of the Kakeya conjecture. One of the aims of this work is to deepen our understanding of oscillatory integrals, which are a type of mathematical expression which occur in many places in physics (optics, quantum mechanics, acoustics, and any other field of physics dealing with waves), as well as having theoretical importance in other fields of mathematics. Understanding these integrals, and in particular knowing how large they can get, may ultimately lead to new designs for physical applications (e.g. tennis rackets that maximize the area of the "sweet spot", or curved reflectors that have a large number of focus points for a wide range of frequencies), or at least place theoretical limits on such designs. There are also numerical applications when modeling certain physical systems (e.g. the seismic behavior of the Earth); if one knows that a certain oscillatory integral will never become very "large" in a certain technical sense, then this will provide a theoretical guarantee to the accuracy of the computer sim ulation of the physical system. Tao will study these oscillatory integrals with the aid of geometry; a connection between these two fields of mathematics is known (being somewhat similar to the relationship between geometrical optics and the wave theory of light), but is not understood completely. If this connection is developed thoroughly enough, we may be able to reduce difficult questions in oscillatory integrals to simpler problems in geometry, or at least use geometrical techniques to obtain partial progress on the oscillatory integral problems.

DMS 0401720

J B Garnett

UCLA



Garnett and his students will work on several problems in classical one dimensional complex analysis. The first problem is to approximate any Blaschke product uniformly on the open disc by Blaschke products whose zeros are
sufficiently spread apart and thin that the corresponding Riesz mass is bounded in all holomorphic coordinate systems (i. e. is a Carleson measure). The approximation should be effected using explicit constructions. The second problem is to give a direct proof of the equivalence of two weight conditions, the Muckenhoupt $A_2$ condition and the Helson-Szeg\"o condition, that are necessary and sufficient for the Hilbert transform to be bounded on $L^2$(weight).
The third problem is to construct non-constant bounded analytic functions on the complement of a positive length subset of a Lipschitz graph. The fourth problem
is a corona problem for infinitely connected plane domains whose boundaries lie on certain regular Cantor sets. It too requires some new explicit constructions. The fifth problem is to prove the $n$-dimensional Lipschitz
harmonic capacity is a bilipschitz invariant.

The methods to be used on these problems will be constructive so that they can be give explicit computer aided constructions of analytic functions and conformal mappings. Analytic functions and conformal mappings have
broad applications in fluid dynamics, acoustics, and electrical engineering, and in these applications constructions are more useful than general existence theorems.

The investigators will study new computational techniques with
applications to inverse problems and image analysis. They will seek new
methods that combine variational arguments with ideas from computational
harmonic analysis and partial differential equations in order to overcome
limitations of existing methods. The research will have three objectives:
propose new models for cartoon and texture separation in images by working
with spaces of distributions; propose completely new techniques for
multiscale hierarchical decomposition of images; propose efficient
algorithms for solving inverse problems.

From the proposed research and educational program, computational
mathematics, image processing, as well as more general areas of science and
engineering will benefit. Applications include image analysis, medical
imaging, satellite imaging, material science, and terrain data analysis,
surveillance and inverse problems.

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The UCLA Analysis Research and Training Groups (RTG) program consists of a variety of initiatives with the common aim of increasing the number of US citizens and permanent residents who study mathematics and then go on to careers in mathematics, including university teaching and research. These initiatives include: recruitment and training of graduate students and postdoctoral scholars in the area of mathematical analysis, taken in its broadest meaning; better preparation of these students and postdocs for their future careers through a series of participating seminars, research workshops, and intensive summer school programs; increasing interest in mathematics among undergraduate students by creating new junior-senior level undergraduate seminar courses in analysis and involving undergraduate students in an annual Mathematics Summer Camp; and showcasing mathematics to school children and giving graduate and undergraduate students experience working with school children through the continued operation of the Los Angeles Math Circle.

The initiatives supported under the RTG grant will strengthen the department's research and training program in analysis as well as our training and outreach programs at all levels -- from middle school through postdoctoral scholars. The outcomes of these programs -- attracting, training, and supporting students to create a mathematically skilled workforce -- represent the national and state-wide effect of this grant.

The investigators, their students and collaborators study mathematical formulations and efficient computational techniques for applications arising in image analysis, materials science, and atmospheric and climate modeling. This multidisciplinary research combines areas of computational mathematics, inverse problems, image analysis, interfaces and free boundaries, and atmospheric sciences. They study image restoration using cartoon and texture decompositions, restoration of images in the presence of a stochastic point spread function, implicit open curve evolution and applications to free boundary problems in materials sciences, and variational data fusion of atmospheric records acquired by multiple instruments. The research team develops novel variational approaches, iterative and numerical analysis techniques for solving these inverse problems.

The proposed activity provides the link between efficient mathematical formulations, imaging approaches, and applications in climate and materials sciences, where similar approaches have not yet been attempted. In particular, capability for merging data acquired by multiple sensors is a key part to our understanding of the Earth's climate system, and therefore, is of importance when making projections about climate change and climate impacts. Current atmospheric data fusion methods are largely ad hoc and establishing a firm mathematical foundation and computational methods for combining important records enhances their scientific credibility and further a wide range of scientific analyses. The investigators promote multidisciplinary teaching, training and learning. Mathematics students are exposed to a broad range of topics and techniques: (i) in applied and computational mathematics, image processing and analysis; and (ii) topics outside mathematics, including materials and atmospheric sciences.

The ``Conference: Spectral Theory and Partial Differential Equations'' will be held June 17-21, 2013 at the University of California-Los Angeles. The roots of microlocal analysis,one of the main tools employed in the conference subject area, go back to the beginning of the twentieth century, but its systematic development began about fifty years ago with the theories of Pseudodifferential Operators and Fourier Integral Operators. Since then these theories have revolutionized the subject of partial differential equations. For example, they have played decisive roles in finding the Atiayh-Singer formula for the index of an elliptic operator and in solving the Weyl conjecture about the distribution of eigenvalues for elliptic operators on manifolds, and they provided the precise tools needed for studying the propagation of singularities for solutions of hyperbolic equations on manifolds. Though microlocal analysis has been an important part of linear PDE theory, it has also been successfully applied to non-linear equations as well and there are numerous connections of microlocal analysis to spectral theory, scattering theory, and inverse problems.

This mathematics research conference will bring together leading experts in partial differential equations and it will consider a wide variety of research topics in differential equations. The conference will increase communication among mathematicians from different specialties and it will accelerate research progress in several fields. The organizers plan to publish the talks presented on the conference in a dedicated volume of the series Contemporary Mathematics published by the American Mathematical Society. Special efforts will be made to bring graduate students and postdoctoral fellows to the conference and thereby to stimulate the research of future mathematicians.