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Seeing the Symmetry: Using Lie Theory to Solve Equations

May 26, 2009

The excerpt below is taken from "Number Sense and Sensibility: Math in a Modern World" by Leslie J. Thompson and originally appeared in the spring issue of the Arts and Sciences Magazine. The full text is available here.

Seeing the Symmetries

The application of math in areas like cryptology and clinical trials is relatively easy for the average layperson to understand. Equally common real-world applications include more abstract and multilayered mathematics such as Lie theory.

Named after Sophus Lie, a prominent Norwegian mathematician in the late 19th Century, Lie (pronounced "lee") theory relates to the symmetry of certain objects and can be used to solve differential equations--for example, how sound propagates, how heat is transferred, or how particles behave when they interact.

A similar process to analyzing symmetries can be used to break a difficult problem into smaller problems, which is really what much of mathematics is about. For example, Dr. John Davis, an associate professor of Mathematics in the College of Arts and Sciences, is researching the interplay between discrete and continuous dynamical systems.

A dynamical system "is any kind of mathematical model that describes something that changes or is dynamic," he explains. "A lot of times, we will study models that depend continuously on time. In physics, time is a continuous variable--it's not a discrete entity. But people also study things that vary discretely in time. For example, when you buy a stock, there's a discrete moment when you bought it and when you sold it, as opposed to [a continuous variable] like velocity."

Currently, Davis is focused on using hybrid dynamical systems--or time scales theory--to maximize the efficiency of how electronic information travels through a network. In a car, for example, sensors are continuously monitoring things like the oil levels, engine heat, whether a door is open, or whether an airbag needs to be deployed.

"As cars become more complex, there are more sensors put into them. But a computer can only handle so much traffic at any given time," says Davis, adding that this is similar to the amount of water that can flow through a pipe. "If you want to have 1000 sensors or 10,000 or 100,000, unless you make the pipe bigger, how do you expect them to talk to each other?"

Since increasing the size of the network pipeline is not always feasible--especially if you're talking about the sensors in an F-16 instead of a Ford Taurus--Davis is working with a group of graduate students and colleagues from the School of Engineering and Computer Science to use mathematical equations as a means of deriving a more efficient way for the sensors to communicate on different time scales.

"Some traffic needs to come along very fast--like an airbag [sensor]--whereas other traffic, like a taillight going out, doesn't need to come along with the same urgency," he explains. "We can say you've got so many sensors and the pipe is only so big, we'll let the most important things come through at one rate and the less important things come through at another," thereby making the communication more efficient rather than expanding the size of the pipe, says Davis. Theoretically, this will allow at least 20 percent more information to be transmitted--which in industry terms could means billions of dollars saved on building new equipment.

Recognizing Math in the Real World

Many people admit they feel intimidated by math, because unlike reading and writing, it is not a skill they use every day. Ironically, although we may not be performing complex calculations, each of us interacts with mathematics in almost everything we do. Whether you're flying to Hawaii on vacation or entering your password on a Web site to purchase the tickets online, you are putting into practice many of the mathematical theories that Baylor students and professors explore in the classroom each semester.

So the next time the gas light comes on in your car, you can thank a mathematician for keeping you from getting stranded by the side of the road.