| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
|
| Initial Amendment Date: | August 2, 2016 |
| Latest Amendment Date: | August 2, 2016 |
| Award Number: | 1650810 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Marian Bocea
mbocea@nsf.gov (703)292-2595 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | May 16, 2016 |
| End Date: | June 30, 2019 (Estimated) |
| Total Intended Award Amount: | $111,243.00 |
| Total Awarded Amount to Date: | $111,243.00 |
| Funds Obligated to Date: |
|
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
1001 EMMET ST N CHARLOTTESVILLE VA US 22903-4833 (434)924-4270 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
Charlottesville VA US 22904-4195 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
|
| Program Reference Code(s): |
|
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Harmonic analysis studies how signals (functions) break up into a superposition of basic harmonics--signals with a well-specified duration, intensity and frequency--and how operations (filtering) applied to these components affect the reconstructed signal. Variants of this time-frequency decomposition process are performed in countless real-world applications, such as audio or image compression and filtering, image pattern recognition, data assimilation and denoising. One of the broad objectives of this project is the investigation of the theoretical feasibility threshold of the time-frequency techniques in terms of the relative size and smoothness of the input. An analogous procedure is adopted in tomographic imaging, where a solid body is reconstructed by means of sampling its density along penetrating waves, mathematically described as lines in three-dimensional space. This project will study mathematical toy models of sampling along lines or curves, whose theoretical understanding may play a significant role in the derivation of improved analytical image reconstruction methods. An integral component of the project is the training of graduate and undergraduate students within the active research group in harmonic analysis at Brown University, with the particular intent of attracting young and promising researchers to the field.
The central objects of study of this project are modulation-invariant singular integrals and their behavior at or near the boundary of their known boundedness range. The model question, involving Carleson's maximal partial Fourier sum operator, is the characterization of the sharp integrability order sufficient for the almost-everywhere pointwise convergence of the Fourier series of a periodic function. The second, deeply related question concerns the extension of the Lacey-Thiele Holder-type estimates for the bilinear Hilbert transform to the boundary of the known range. Together with his collaborators, the principal investigator has recently obtained the current best results for both problems, relying in particular on a newly developed Calderon-Zygmund decomposition adapted to the modulation-invariant setting. It is expected that further developments of this technique will lead to additional improvements towards the solution of these two central questions, as well as of other significant open problems. A standout question is the extension of the known uniform estimates for the bilinear Hilbert transform to the full expected range of exponents, completing the original program of Calderon for the boundedness of the first commutator. Another central direction of the proposed investigation is the study of singular integral operators with rotational symmetries, a prime example of which is the Hilbert transform along a smooth vector field in the plane, by means of multiparameter time-frequency analysis techniques. Further improvements of the aforementioned techniques are also expected to impact on several questions concerning summability of multiple Fourier series.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The main goal of the project was to improve the state-of-the-art knowledge on the local and endpoint behavior of singular integral operators with modulation invariance properties. These operators dictate the fine behavior of Fourier and series and more general time-frequency decompositions of signals, as well as that of mathematical models of the Radon and X-ray transforms.
Among the discoveries made within the framework of the award, we mention
a) the introduction of the sparse domination technique to the realm of modulation invariant operators and rough singular integrals. Broadly speaking, sparse domination allows to control very oscillatory, nonlocal operators by a wise choice of positive, localized averaging operator. In particular, this discovery led to a completely novel weighted theory for modulation invariant multilinear operators such as the bilinear Hilbert transform. The latter was an open problem since the seminal 1995, 1997 boundedness results of Lacey and Thiele.
Findings directly or indirectly related to this discovery have been published (with coauthors) in eight journal articles, including journals such as Analysis and PDE, J. London Math. Soc., J. d'Analyse Math.
b) progress on the Stein and Zygmund conjectures on differentiation along vector fields. These questions relate, in broad terms, to the possibility of reconstructing a two-dimensional, or higher-dimensional, function as a limit of its averages along line segments pointing along a smooth, or otherwise suitable, choice of directions. In an article published on Journal of Functional Analysis, the PI and coauthors prove that Stein's conjecture holds for Lipschitz choices of directions, under a conditional frequency-localized estimate which is verified in several cases of interest. In a series of two articles published on Israel Journal of Math and International Math. Research Notices, the PI and coauthor prove the first sharp estimate for directionals singular integrals in ambient dimension three, under structural assumptions on the set of directions.
Among further aspects of the broader impacts of the research done within the framework of this project, we mention that
a) an American Institute of Mathematics weekly workshop on Sparse Domination of Singular Integral operator has been co-organized by the PI. This workshop was the culminant point of a networking effort related to sparse domination where the PI, and the resources of the present project, played a significant role. In particular, several early career researchers such as Amalia Culiuc (former PhD Student at Brown, now tenure track faculty in Amherst College), Yumeng Ou (former PhD Student at Brown, now tenure track faculty in CUNY), Gennady Uraltsev (currently postdoc in Cornell), Laura Cladek (former Phd Student in U Wisconsin, now postdoc and UCLA) and others were partially supported within this award and contributed with publications and original research to the development of the subject.
b) the PI has been active supporting underrepresented groups in Mathematics, by taking an advisory role in the U Virginia chapter of the Association for Women in Mathematics, and organizing a research and career-development oriented workshop (Women's Intellectual Research Network Symposium at University of Virginia) targeting faculty, postdoctoral fellows and graduate students from underrepresented groups in the mid-atlantic region.
Last Modified: 09/28/2019
Modified by: Francesco Diplinio
Please report errors in award information by writing to: awardsearch@nsf.gov.
