Award Abstract # 0748283
CAREER: Discrete and Generalized Riemannian Geometry and Curvature Flows

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF ARIZONA
Initial Amendment Date: February 5, 2008
Latest Amendment Date: February 4, 2015
Award Number: 0748283
Award Instrument: Continuing Grant
Program Manager: Joanna Kania-Bartoszynska
jkaniaba@nsf.gov
 (703)292-4881
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: May 1, 2008
End Date: April 30, 2016 (Estimated)
Total Intended Award Amount: $401,686.00
Total Awarded Amount to Date: $401,686.00
Funds Obligated to Date: FY 2008 = $103,258.00
FY 2009 = $96,794.00

FY 2010 = $72,385.00

FY 2011 = $63,327.00

FY 2012 = $65,922.00
History of Investigator:
  • David Glickenstein (Principal Investigator)
    glickenstein@math.arizona.edu
Recipient Sponsored Research Office: University of Arizona
845 N PARK AVE RM 538
TUCSON
AZ  US  85721
(520)626-6000
Sponsor Congressional District: 07
Primary Place of Performance: University of Arizona
845 N PARK AVE RM 538
TUCSON
AZ  US  85721
Primary Place of Performance
Congressional District:
07
Unique Entity Identifier (UEI): ED44Y3W6P7B9
Parent UEI:
NSF Program(s): GEOMETRIC ANALYSIS
Primary Program Source: 01000809DB NSF RESEARCH & RELATED ACTIVIT
01000910DB NSF RESEARCH & RELATED ACTIVIT

01001011DB NSF RESEARCH & RELATED ACTIVIT

01001112DB NSF RESEARCH & RELATED ACTIVIT

01001213DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 0000, 1045, 1187, OTHR
Program Element Code(s): 126500
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

In this project the PI proposes to study discrete geometry, curvature flows, and collapsing solutions to smooth geometric flows. The original motivation for this work is the landmark work on Ricci flow that began with R. Hamilton and includes the solution of the Poincare conjecture by G. Perelman. The PI proposes to work on both combinatorial curvature flows on piecewise linear manifolds and discrete approximations of Ricci flow and other smooth flows. In particular, the PI plans to study discrete flows numerically, to develop visualization techniques for abstract manifolds, to further develop the theory of discrete geometries in the spirit of differential geometry, and to prove convergence of these geometries and related geometric operators to the continuum. The PI also plans to study geometric flows on generalizations of Riemannian manifolds, such as Riemannian groupoids, in order to better understand those flows at singularities. The experimental part of this proposal will be run by a laboratory of undergraduates supervised by graduate students.

The recent solution of the Poincare conjecture by G. Perelman both stunned and invigorated the mathematics community. The PI proposes to study similar techniques involving geometric flows in two settings: (1) Discrete Geometries, which may be applied both to other types of geometric questions and to mathematical modelling in a variety of settings, including physics and computer graphics, and (2) Generalized Geometries, which may clarify the implications of Perelman's results and how it may be applied to both mathematical and physical applications.
The hope is not only to solve geometric problems, but develop techniques applicable to other areas of science and engineering, both theoretically and computationally. In the process, the PI plans to rely on the laboratory science model to form a group of graduate and undergraduate students developing research tools and presentation tools. The PI hopes to use these tools to communicate the excitement of modern geometry to researchers, teachers, students, and the general public.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Daniel Champion, David Glickenstein, Andrea Young "Regge's Einstein-Hilbert Functional on the Double Tetrahedron" Differential Geometry and its Applications , v.29 , 2011 10.1016/j.difgeo.2010.10.001
Daniel Champion, David Glickenstein, Andrea Young "Regge's Einstein-Hilbert Functional on the Double Tetrahedron" Differential Geometry and its Applications , v.29 , 2011 , p.109 10.1016/j.difgeo.2010.10.001
David Glickenstein "Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds" Journal of Differential Geometry , v.87 , 2011 , p.201
David Glickenstein, Tracy Payne "Ricci flow on three-dimensional, unimodular metric Lie algebras" Communications in Analysis and Geometry , v.18 , 2010 , p.927

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