Strategic Communications and Marketing News Bureau

Mathematician uses topology to study abstract spaces, solve problems

Mathematician Robert 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.

Mathematician Robert 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.

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.”

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

Editor’s note: To reach Robert Ghrist, call 217-244-5857; e-mail: rghrist@uiuc.edu.

Read Next

Life sciences Portrait of the research team posing together.

Minecraft players can now explore whole cells and their contents

CHAMPAIGN, Ill. — Scientists have translated nanoscale experimental and computational data into precise 3D representations of bacteria, yeast and human epithelial, breast and breast cancer cells in Minecraft, a video game that allows players to explore, build and manipulate structures in three dimensions. The innovation will allow researchers and students of all ages to navigate […]

Arts Photo of seven dancers onstage wearing blue tops and orange or yellow flowing skirts. The backdrop is a Persian design.

February Dance includes works experimenting with live music, technology and a ‘sneaker ballet’

CHAMPAIGN, Ill. — The dance department at the University of Illinois Urbana-Champaign will present February Dance 2025: Fast Forward this week at Krannert Center for the Performing Arts. February Dance will be one of the first performances in the newly renovated Colwell Playhouse Theatre since its reopening. The performances are Jan. 30-Feb. 1. Dance professor […]

Honors portraits of four Illinois researchers

Four Illinois researchers receive Presidential Early Career Award

CHAMPAIGN, Ill. — Four researchers at the University of Illinois Urbana-Champaign were named recipients of the Presidential Early Career Award for Scientists and Engineers, the highest honor bestowed by the U.S. government on young professionals at the outset of their independent research careers. The winners this year are health and kinesiology professor Marni Boppart, physics professor Barry Bradlyn, chemical and biomolecular engineering professor Ying […]

Strategic Communications and Marketing News Bureau

507 E. Green St
MC-426
Champaign, IL 61820

Email: stratcom@illinois.edu

Phone (217) 333-5010