Award Abstract # 9531806
Mathematical Sciences: Singular Integrals and Fourier Integrals

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF ROCHESTER
Initial Amendment Date: March 25, 1996
Latest Amendment Date: January 12, 1998
Award Number: 9531806
Award Instrument: Continuing Grant
Program Manager: Joe W. Jenkins
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: July 1, 1996
End Date: June 30, 2000 (Estimated)
Total Intended Award Amount: $117,550.00
Total Awarded Amount to Date: $117,550.00
Funds Obligated to Date: FY 1996 = $38,174.00
FY 1997 = $39,181.00

FY 1998 = $40,195.00
History of Investigator:
  • Allan Greenleaf (Principal Investigator)
    allan@math.rochester.edu
Recipient Sponsored Research Office: University of Rochester
910 GENESEE ST
ROCHESTER
NY  US  14611-3847
(585)275-4031
Sponsor Congressional District: 25
Primary Place of Performance: University of Rochester
910 GENESEE ST
ROCHESTER
NY  US  14611-3847
Primary Place of Performance
Congressional District:
25
Unique Entity Identifier (UEI): F27KDXZMF9Y8
Parent UEI:
NSF Program(s): ANALYSIS PROGRAM
Primary Program Source: app-0196 
app-0197 

app-0198 
Program Reference Code(s): 0000, OTHR
Program Element Code(s): 128100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

Abstract Greenleaf 9531806 Characteristic space-time estimates, previously obtained, for solutions of wave equations will be extended and applied to yield results on the approximate determination of compactly supported, time-independent potentials in L^2 of 3-space from approximate knowledge of their backscattering (or other determined sets of scattering data.) Extensions of such results to potentials with noncompact support and of N-particle type will be pursued. Estimates, in terms of Sobolev and Lebesgue norms, for Fourier integral operators associated with canonical relations exhibiting cusp singularities, both simple and of higher order, will be investigated. Such estimates have applications to regularity properties of averaging operators associated with integrals over generic families of lines and curves in n-dimensional space. Finally, classes of Fourier integral operators arising from real analogues of Goncharov's complexes of hyperplane sections of algebraic arieties of minimal degree in n-dimensional projective space will be studied, with the goal of obtaining composition calculi for them. The principal object of this project will be the study of several types of singular integral operators and Fourier integral operators. Such operators, which transform functions on one space into functions on another (possibly different) space, have become central tools in the study of linear partial differential equations which govern diverse physical phenomena, such as electromagnetic fields and sound propagation. The particular operators to be studied in this project arise in the scattering of waves by potential functions, and in tomography, the mathematical basis for a variety of medical imaging systems, such as CAT and MRI scanners. Thus, progress on the problems considered in this project will contribute to the theoretical underpinnings of procedures for reconstructing unknown quantities of physical interest from noninvasive observations. Despite the fact that they arise in d ifferent problems, the operators to be studied share several common features, which involve more complicated geometry than is present in the original versions of singular integral and Fourier integral operators. It is hoped that, eventually, improved understanding of these operators will lead to improved reconstruction techniques.

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