Mathematician uses topology to study abstract spaces, solve problems

CHAMPAIGN, Ill. -- Studying complex systems, such as the movement of robots on a factory floor, the motion of air over a wing, or the effectiveness of a security network, can present huge challenges. Mathematician Robert Ghrist at the University of Illinois at Urbana-Champaign is developing advanced mathematical tools to simplify such tasks.

Ghrist uses a branch of mathematics called topology to study abstract spaces that possess many dimensions and solve problems that can't be visualized normally. He will describe his technique in an invited talk at the International Congress of Mathematicians, to be held Aug. 23-30 in Madrid, Spain.

Ghrist, who also is a researcher at the university's Coordinated Science Laboratory, takes a complex physical system – such as robots moving around a factory floor – and replaces it with an abstract space that has a specific geometric representation.

"To keep track of one robot, for example, we monitor its x and y coordinates in two-dimensional space," Ghrist said. "Each additional robot requires two more pieces of information, or dimensions. So keeping track of three robots requires six dimensions. The problem is, we can't visualize things that have six dimensions."

Mathematicians nevertheless have spent the last 100 years developing tools for figuring out what abstract spaces of many dimensions look like.

"We use algebra and calculus to break these abstract spaces into pieces, figure out what the pieces look like, then put them back together and get a global picture of what the physical system is really doing," Ghrist said.

Ghrist's mathematical technique works on highly complex systems, such as roving sensor networks for security systems. Consisting of large numbers of stationary and mobile sensors, the networks must remain free of dead zones and security breaches.

Keeping track of the location and status of each sensor would be extremely difficult, Ghrist said. "Using topological tools, however, we can more easily stitch together information from the sensors to find and fill any holes in the network and guarantee that the system is safe and secure."

While it may seem counterintuitive to initially translate such tasks into problems involving geometry, algebra or calculus, Ghrist said, that doing so ultimately produces a result that goes back to the physical system.

"That's what applied mathematics has to offer," Ghrist said. "As systems become increasingly complex, topological tools will become more and more relevant."

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Funding was provided by the National Science Foundation and the Defense Advanced Research Projects Agency.


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