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Project Information

RUI: SPECTRAL THEORY AND GEOMETRIC ANALYSIS IN SEVERAL COMPLEX VARIABLES

Agency:
NSF

National Science Foundation

Project Number:
1500952
Contact PI / Project Leader:
FU, SIQI M
Awardee Organization:
RUTGERS THE STATE UNIVERSITY OF NEW JERSEY CAMDEN

Description

Abstract Text:
Many physical and social phenomena can be modeled mathematically using partial differential operators. The Laplace operator is a differential operator that has long played an important role in mechanics, physics, and mathematics. The complex Laplace operator is a natural outgrowth of the classical Laplace operator in complex analysis of several variables, a branch of mathematics where algebra, analysis, and geometry intertwine. This research project investigates analytic and geometric properties of the complex Laplace operator, in particular, its spectrum. Spectral analysis is a major tool in scientific research, and spectral properties of the complex Laplace operator are known to be closely related to certain quantum phenomena in physics. The goal of this project is to understand how algebraic, analytic, and geometric structures of the underlying complex space interact with each other. The project combines ideas and methods from several branches of mathematics, and the techniques under development could potentially have applications in other areas of mathematics and physical sciences. This project involves undergraduate students in research activities and broadens participation of underrepresented groups in mathematics.

The complex Neumann Laplace operator is a prototype of an elliptic operator with non-coercive boundary conditions. Since the work of Kohn and Hörmander in the 1960's, there have been extensive studies on regularity theory of the complex Neumann Laplace operator that led to important discoveries in both partial differential equations and several complex variables. The main thrust of this proposal is to study spectral theory of the complex Neumann Laplace operator, with emphasis on the interplay between the spectral behavior of the operator and the underlying geometric structures. Among the problems studied in this project are stability of the spectrum as the underlying structures deform and characterization of complex manifolds whose complex Laplace operator has discrete spectrum. Also investigated are regularity theory of the Cauchy-Riemann operator on complex manifolds, reproducing kernels, invariant metrics, and their applications to problems in complex algebraic geometry. This project supports research activities of undergraduate and graduate students, facilitates the development of new courses that attract students into mathematics, and fosters interdisciplinary research.
Project Terms:
Algebra; Algebraic Geometry; Area; Behavior; Complex; Complex Variables; Development; Differential Equation; Fostering; geometric theories; Geometry; Goals; graduate student; Interdisciplinary Study; Laplacian; Mathematics; Mechanics; Methods; Modeling; physical science; Physics; Play; Property; prototype; quantum; Research; Research Activity; Research Project Grants; Role; social; Structure; Students; Techniques; theories; tool; undergraduate student; Underrepresented Minority; Work

Details

Contact PI / Project Leader Information:
Name:  FU, SIQI M
Other PI Information:
Not Applicable
Awardee Organization:
Name:  RUTGERS THE STATE UNIVERSITY OF NEW JERSEY CAMDEN
City:  CAMDEN    
Country:  UNITED STATES
Congressional District:
State Code:  NJ
District:  01
Other Information:
Fiscal Year: 2015
Award Notice Date: 22-Apr-2015
DUNS Number: 625216556
Project Start Date: 01-Jun-2015
Budget Start Date:
CFDA Code: 47.049
Project End Date: 31-May-2018
Budget End Date:
Agency: ?

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National Science Foundation
Project Funding Information for 2015:
Year Agency

Agency: The entity responsible for the administering of a research grant, project, or contract. This may represent a federal department, agency, or sub-agency (institute or center). Details on agencies in Federal RePORTER can be found in the FAQ page.

FY Total Cost
2015 NSF

National Science Foundation

$58,434

Results

i

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