# Making sense of sensors

December 05, 2006Providence, RI -- A forest ranger helicopter flies over a forest, scattering sensors that can relay temperature data to the ranger station. To ensure minimal environmental impact with maximum robustness, the sensors are very simple: they are basically tiny, sturdy thermometers. After the sensors are scattered, they might be moved further by winds, rains, rivers, or even animals. Is there a way to take the local information sent by the sensor network and turn it into global information about the existence and location of fires in the forest" In particular, without knowing the exact locations of the sensors, can one nevertheless glean information about the coverage area of the sensor network"

As sensor technology has exploded, such fundamental questions have come to the forefront in many areas. In particular, national security measures increasingly depend on sensor technology to detect, for example, radiological or biological hazards, hidden mines and munitions, or specific individuals in a crowd. Mathematics, especially the area of topology, provides a way of addressing such questions.

The January 2007 issue of the Notices of the AMS will carry the article "Homological Sensor Networks" by Vin de Silva and Robert Ghrist. The article describes new results by the authors, which demonstrate how homology theory provides fundamental insights useful in analyzing sensor networks.

Suppose you have a network of sensors, each with a unique ID, scattered around a two-dimensional domain D---for example, D could be a region of forest, an open field, or a portion of the ocean floor. The sensors have a "broadcast radius", within which they can detect the identity of any other sensor, and a "cover radius", within which the sensors perform their sensing tasks. You can think of each sensor as surrounded by a disk whose radius is the coverage radius. The union of these disks is the "sensor cover". A basic question is, Does the sensor cover contain D"

Topology, which is the study of shapes, is well suited to attacking this question. In particular, homology theory provides a way of detecting whether shapes contain holes. De Silva and Ghrist were able to use homology theory to pinpoint some simple topological conditions that, if met by the sensor network, guarantee that the sensor cover contains the whole domain D without holes. What is striking about this result is that it provides information about the sensor cover without requiring knowledge of the exact locations of the sensors. Only the broadcast and cover radii are needed.

De Silva and Ghrist also adapted the above result to networks where the sensors are going on- and off-line periodically, so that holes open up and close in the sensor cover. Can an "evader" move through the sensor network, taking advantages of holes that open up in order to slip through undetected" The authors present topological conditions on the sensor network that guarantee that the evader will be caught, regardless of the evader's speed or cunning.

"It seems counterintuitive that one can provide rigorous answers for a network with neither localization capabilities nor distance measurements," the authors remark. "A topologist is not surprised that such coarse data can be integrated into a global picture. Some engineers are." De Silva and Ghrist call for mathematicians and engineers to collaborate on the design of effective sensor networks.

Ghrist is building such collaborations as a lead investigator for a research project called SToMP, short for "Sensor Topology & Minimal Planning." Funded by the Defense Advanced Research Projects Agency (DARPA), the $7.98 million project will run over four years. SToMP will support research at Ghrist's home institution, the University of Illinois at Urbana-Champaign, as well as at Bell Labs/Lucent, Arizona State University, Rochester University, Carnegie-Mellon University, Melbourne University, the University of Pennsylvania, and the University of Chicago.

-end-

The article will be posted to the web on December 5, 2006, at the following URL:http://www.ams.org/notices/200701/fea-ghrist.pdf

For Further Information, Contact:

Professor Vin De Silva

Pomona College, Claremont, California

Email: vin.desilva@pomona.edu

Telephone: +1-909-607-8656

Founded in 1888 to further mathematical research and scholarship, the more than 30,000-member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.

American Mathematical Society

201 Charles Street

Providence, RI 02904

401-455-4000

Image by Robert Ghrist.

American Mathematical Society

## Related Mathematics Articles from Brightsurf:

A new method for boosting the learning of mathematics

How can mathematics learning in primary school be facilitated? UNIGE has developed an intervention to promote the learning of math in school.

Could mathematics help to better treat cancer?

Impaired information processing may prevent cells from perceiving their environment correctly; they then start acting in an uncontrolled way and this can lead to the development of cancer.

People can see beauty in complex mathematics, study shows

Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

Improving geothermal HVAC systems with mathematics

Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.

How the power of mathematics can help assess lung function

Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.

Mathematics pushes innovation in 4-D printing

New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.

More democracy through mathematics

For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

US educators awarded for exemplary teaching in mathematics

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Read More: Mathematics News and Mathematics Current Events

How can mathematics learning in primary school be facilitated? UNIGE has developed an intervention to promote the learning of math in school.

Could mathematics help to better treat cancer?

Impaired information processing may prevent cells from perceiving their environment correctly; they then start acting in an uncontrolled way and this can lead to the development of cancer.

People can see beauty in complex mathematics, study shows

Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

Improving geothermal HVAC systems with mathematics

Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.

How the power of mathematics can help assess lung function

Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.

Mathematics pushes innovation in 4-D printing

New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.

More democracy through mathematics

For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

US educators awarded for exemplary teaching in mathematics

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Read More: Mathematics News and Mathematics Current Events

Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.