| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 20, 2016 |
| Latest Amendment Date: | May 21, 2018 |
| Award Number: | 1600753 |
| Award Instrument: | Continuing 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: | July 1, 2016 |
| End Date: | June 30, 2020 (Estimated) |
| Total Intended Award Amount: | $300,000.00 |
| Total Awarded Amount to Date: | $300,000.00 |
| Funds Obligated to Date: |
FY 2018 = $100,000.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1500 HORNING RD KENT OH US 44242-0001 (330)672-2070 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Kent OH US 44242-0001 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
01001819DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The problems addressed in this project ask about properties of shapes in an ambient space (like a human heart in the body) that can be inferred from information about their shadows or slices (as in medical imaging). The advantage of the problems is that the solutions to many of them are "intuitively clear" not only to graduate students, but also to undergraduate students, and in some cases, even to high-school pupils! On the other hand, the answers are (very often) counterintuitive, requiring the use of the most advanced and sophisticated tools belonging to the different branches of modern mathematics. Moreover, many of the problems originated not in "pure math," but in medical imaging and tomography. For this reason the solutions might find very interesting biomedical applications.
The current project is a continuation of the long-time collaboration between the two principal investigators. They will continue to use and develop the methods of harmonic analysis to solve problems arising in convex and discrete geometry. These problems include ones about Brunn-Minkowski-type inequalities for general measures, as well as questions related to different versions of slicing inequalities, including a slicing inequality for general measures and a discrete version of the slicing inequality. The principal investigators also plan to continue their work on the unique determination of convex bodies given information on the size (or some other properties) of projections and sections. This will involve a mixture of topological, probabilistic, and Fourier-analytic methods, and part of the work will be concentrated around a classical problem of Bonnensen that the principal investigators and their collaborators have previously solved in even dimensions. Many of the techniques developed in that work are new, and there are high hopes that the methods could help to solve a number of classical (but still open) questions in geometric tomography.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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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 major goal of the project was to use and develop the methods of Harmonic and Classical Analysis to solve the problems from Convex and Discrete Geometry as well as questions connected to Geometric Tomography. These problems include the ones about the Brunn-Minkowski type inequalities for general measures, the questions related to slicing inequalities and to the unique determination of convex bodies from the information about projections and sections as well as classical inequalities on Mixed Volume. The project also included questions on duality and volume and questions related to discrete versions of classical notions of Convex Geometry. The advantage of many problems addressed in the project is that most of them are ?intuitively clear? not only to graduate students, but to undergraduate students as well as the high-school pupils. These problems ask about properties of shapes in the ambient space (like a human heart) based on the information on their shadows or slices (as in medical imaging). On the other hand, the answers are (very often) counterintuitive, requiring the use of the most advanced tools belonging to the different branches of modern mathematics. Moreover, many of the problems are originated not in ?pure math?, but in medical imaging and tomography, and the solutions might find very interesting applications.
The PI and Co-PI explored the connections between such questions and modern tools of Harmonic Analysis, Functional Analysis, Topology Algebraic and Differential Geometry. The project outcome included a new study of an approximation of convex bodies by polytopes, and a generalization of Grunbaum inequality. A number of results on the behavior of convex body under duality, including the study of volume product for Lipschitz-free Banach spaces and a new proof of 3-dimensional Mahler conjecture; discovery of new properties of floating bodies; new Bezout type inequality for mixed volume which characterizes simplices. Finally it included an affirmative answer to the local version of the fifth and eights Busemann-Petty problems.
Educational part is an essential component of the work on the project, the PI and co-PI had an opportunity to work with four Doctoral level graduate students, three of which have graduated during proposal time and continued their research work in academia. The PI and co-PI also had a pleasure to collaborate with five postdoctoral fellows, one of them have continued his postdoctoral appointment at Carnegie Mellon University and four others found Assistant or Associated Professor positions.
The results of the project were disseminated in a number of publications in the leading mathematical journals and in a number of invited lectures at national and international universities and conferences, done by PI, co-PI as well as by their graduate students and Postdoctoral fellows.
Last Modified: 12/09/2020
Modified by: Artem Zvavitch
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