Award Abstract #0134408
CAREER: Topological Methods in Applied Mathematics

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	December 3, 2001
Latest Amendment Date:	December 3, 2001
Award Number:	0134408
Award Instrument:	Standard Grant
Program Manager:	Benjamin M. Mann DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	June 1, 2002
Expires:	August 31, 2003 (Estimated)
Awarded Amount to Date:	$352690
Investigator(s):	Robert Ghrist ghrist@seas.upenn.edu (Principal Investigator)
Sponsor:	GA Tech Research Corporation - GA Institute of Technology Office of Sponsored Programs Atlanta, GA 30332 404/894-4819
NSF Program(s):	COMPUTATIONAL MATHEMATICS, TOPOLOGY, APPLIED MATHEMATICS, GEOMETRIC ANALYSIS
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR, 1187, 1076, 1045, 0000
Program Element Code(s):	1271, 1267, 1266, 1265

ABSTRACT

DMS-0134408

Robert W. Ghrist

The efficacy of topological methods in contemporary applied

mathematics is primarily attributable to the fact that topological

features of a system are inherently robust and global. This

project focuses on a technology transfer from contemporary ideas

in topology, geometry, and dynamics to bear upon application

domains which include the following: First, Robotics: tools

from configuration space theory, CAT(0)complexes, and

computational topology will be directed toward specific

problems in reconfigurable robotics, sensor-based navigation

of mobile agents, and self-assembly systems. Second, Parabolic

coupled systems: a Morse-theoretic homotopy index for braids

will be used to solve parabolic variational problems arising in pattern-formation PDE's, discrete Lagrangian mechanics, and

coupled oscillators. A Floer-theoretic extension of the braid

index will also be developed for infinite dimensional systems.

Third, Hydrodynamics: tools from contact geometry and topology

will be directed toward solving global problems of the dynamics

and stability of Eulerian fluid flows in dimensions higher

than two.

In most systems of interest in science and engineering,

multiple cooperative tasks must be globally coordinated.

A common thread is that whether the tasks involve

macro-scale robots, micro-scale devices, coupled oscillators,

or fluid particles, there is an abstract space of configurations

lurking behind the physical phenomena. Unearthing and

examining those properties of physically-motivated

configuration spaces which capture the global features, the

topology, geometry, and dynamics, holds the promise of

providing global tools which transcend the physical

instantiation of the system at hand: ostensibly different

systems possess similar topological underpinnings.

The research component of this project is the development

of contemporary topological and global-geometric techniques

for analyzing the dynamics and coordination of systems of

interest in engineering and computer science. The overall

goal is an effective technology transfer from cutting-edge

perspectives in topology to bear upon systems in application

domains which include robotics, mechanics, and fluid dynamics.

This is combined with a blend of pedagogical service across

graduate, undergraduate, and high school levels, featuring

a focused research group on topological robotics and a

high-school outreach program of expository lectures on the

relevance and joy of mathematical research.

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