Featured Research

from universities, journals, and other organizations

Mathematician Uses Topology To Study Abstract Spaces, Solve Problems

Date:
August 16, 2006
Source:
University of Illinois at Urbana-Champaign
Summary:
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.

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.


Story Source:

The above story is based on materials provided by University of Illinois at Urbana-Champaign. Note: Materials may be edited for content and length.


Cite This Page:

University of Illinois at Urbana-Champaign. "Mathematician Uses Topology To Study Abstract Spaces, Solve Problems." ScienceDaily. ScienceDaily, 16 August 2006. <www.sciencedaily.com/releases/2006/08/060816004930.htm>.
University of Illinois at Urbana-Champaign. (2006, August 16). Mathematician Uses Topology To Study Abstract Spaces, Solve Problems. ScienceDaily. Retrieved July 29, 2014 from www.sciencedaily.com/releases/2006/08/060816004930.htm
University of Illinois at Urbana-Champaign. "Mathematician Uses Topology To Study Abstract Spaces, Solve Problems." ScienceDaily. www.sciencedaily.com/releases/2006/08/060816004930.htm (accessed July 29, 2014).

Share This




More Computers & Math News

Tuesday, July 29, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

Teen's Phone Ignites Under Her Pillow; How Real Is The Risk?

Teen's Phone Ignites Under Her Pillow; How Real Is The Risk?

Newsy (July 28, 2014) A Texas teen's Samsung phone apparently ignited while she slept, but what was the real problem here? Video provided by Newsy
Powered by NewsLook.com
Google's Next Frontier: The Human Body

Google's Next Frontier: The Human Body

Newsy (July 27, 2014) Google is collecting genetic and molecular information to paint a picture of the perfectly healthy human. Video provided by Newsy
Powered by NewsLook.com
Cellphone Unlocking Bill Clears U.S. House, Heads to Obama

Cellphone Unlocking Bill Clears U.S. House, Heads to Obama

Reuters - US Online Video (July 27, 2014) Congress gets rid of pesky law that made it illegal to "unlock" mobile phones without permission, giving consumers the option to use the same phone on a competitor's wireless network. Mana Rabiee reports. Video provided by Reuters
Powered by NewsLook.com
Congress OKs Unlocking Phones From Carriers

Congress OKs Unlocking Phones From Carriers

Newsy (July 26, 2014) A bill legalizing "unlocking," or untethering a phone from its default wireless carrier, has passed Congress and is expected to be signed into law. Video provided by Newsy
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:

Breaking News:
from the past week

In Other News

... from NewsDaily.com

Science News

Health News

Environment News

Technology News



Save/Print:
Share:

Free Subscriptions


Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

Get Social & Mobile


Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

Have Feedback?


Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
Mobile: iPhone Android Web
Follow: Facebook Twitter Google+
Subscribe: RSS Feeds Email Newsletters
Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins