PECASE: Topological Methods in Applied Mathematics

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Proposal Title: PECASE: Topological Methods in Applied Mathematics Institution: University of Illinois at Urbana-Champaign 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. This project was originally funded as a CAREER award, and was converted to a Presidential Early Career Award for Engineers and Scientists (PECASE) award in May 2004.