When was the last time you made a withdrawal at an ATM? Or had a prescription filled? Perhaps you recently ordered a gift online, or downloaded pictures from your digital camera to your computer.
Most of us take everyday activities like these for granted, giving little thought to the inventions behind them. But all of these scenarios share a common factor: They would not be possible without mathematics.
This is the realm of the professors and students in the Mathematics and Statistical Sciences departments whose mission it is to research complex problems and to devise solutions that become tomorrow's technology breakthroughs. From pharmaceuticals to aeronautics, from electrical engineering to cryptology, Baylor's mathematics and statistics students and instructors are making an impact on the world around us.
Sending Secret Messages
One of the most commonly recognized areas of mathematics in modern society is cryptology, which is the science of creating and reading secret messages. Though historically developed for use by the government and the military, cryptology is now used daily by ordinary citizens in today's technology-driven society.
"If you're sending e-mail, that could be encrypted," notes Dr. David Arnold, the Ralph and Jean Storm Professor of Mathematics at Baylor. "Online purchases, ATM machine transactions, cell phones--you see examples of encryption in all those areas."
Theoretically, cryptology appears simple to understand as a means of creating a secret code to safeguard information. However, because of digital communication, the modern process of cryptology is quite complex. Arnold helps his students understand the mathematics that underlie the solutions to common problems, like sending a wire transfer between banks, safeguarding voters' privacy in electronic voting machines, or preventing high-definition DVDs from being pirated. Using these skills, his students have gone on to work for the FBI, the CIA and the National Security Agency.
Devising Trials With Fewer Errors
A number of Baylor's statistics students are also very involved in the world of pharmaceuticals, thanks in large part to a new partnership with drug manufacturer Eli Lilly that provides students the opportunity to work on challenging statistical research problems.
"We help define the model for clinical trials and try to provide the ability to determine how many patients are necessary to draw a valid conclusion about a certain drug," explains Dr. John Seaman, professor of Statistics. He is referring specifically to Bayesian inference, his graduate students' major area of research, which provides a means of accounting for uncertainty in things that cannot be seen in their entirety, such as a large population.
Think of it this way: If you're developing a new medication to treat high blood pressure, you can't test it on everybody in the U.S. who suffers from hypertension in the clinical trials. Instead, you work with a sample--a small group that is representative of the larger population. Seaman and his students not only develop the models to put together these groups, but also work on adaptive designs, which attempt to reduce the number of patients involved in a clinical trial while still providing accurate data.
"Our students have gone to Eli Lilly, the Mayo Clinic, the FDA--these places all engage in or monitor clinical trials," he notes proudly.
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, an area of research for Dr. Mark Sepanski, associate professor of Mathematics.
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 simple illustration of Lie theory is the transmission of a JPEG file online. The original digital image is comprised of more data than feasibly can be sent over the Internet in a reasonable span of time.
"Using the symmetry of a circle, you can convert that image into a set of numbers," explains Sepanski. The computer then breaks the data into smaller pieces and sends only one small piece over the Internet, essentially discarding the rest of the information. The program on the receiving end can "reconstruct the picture in its original form, and the eye can't tell the difference," Sepanski says. "As a result, you save an enormous amount of space and time."
Of course, this is a very elementary application using the baby elements of Lie theory. Theoretical physics uses the theory in its entirety, says Sepanski, noting, "If you want to describe how particles interact and where they will be at a particular time, then the symmetries of the universe come into play."
Putting the Pieces Together
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.