Integrating math into global applications

Past the piles of calculus homework and old theses strewn across the floor, a look into the Department of Mathematics at Carnegie Mellon provides a startling surprise: Math is in everyone else’s business.

What’s no surprise is that Carnegie Mellon values an interdisciplinary environment. Though it is the product of many generations, interdisciplinary research formally became part of the University’s “Strategic Plan” in 1998.

In the 21st century, mathematics at Carnegie Mellon is traveling not only to different departments, but out of the offices of Wean Hall and into the global centers of science and industry.

“This is happening all over the world”

According to David Kinderlehrer, professor of mathematics, the mathematics department at Carnegie Mellon and its peers at universities around the world have been doing a lot more “applied math” in recent years. For Kinderlehrer and his colleagues, “applied” necessarily means interdisciplinary.

“Most often, I find myself in the company of colleagues who are not mathematicians,” Kinderlehrer states on his website. “We are learning together what we could not do in our native disciplines: new science.”

Though journals of applied mathematics line his shelves, Kinderlehrer cannot give a concise definition for applied math — because there isn’t one.

“[Mathematicians] are trying to get math into the laboratory to answer questions in biology, in mechanics, in physics,” Kinderlehrer said, describing the implications of the move toward applied mathematics. “This is happening all over the world.”

According to Kinderlehrer, however, math should never be a service to other disciplines. The collaborative efforts of mathematicians around the world should serve to make mathematics a richer field of study. “We are working with [scientists] to create new science for them and new math for us,” Kinderlehrer said.

Kinderlehrer’s breakthroughs are in materials science, and he uses mathematical simulations to control natural conditions in various materials. He is working to discover how to manipulate the granules present in almost all materials, elements that are often only few microns across.

About two years ago, Kinderlehrer discovered a rational relationship to the composition of granules where there was previously thought to be no relationship. The discovery has far-reaching implications for materials science. With the knowledge of how materials are composed, Kinderlehrer and his partners can start to understand how to mold materials to do what they want.

“We do our tango”

One of Kinderlehrer’s colleagues, professor Irene Fonseca, works on both the mathematics of materials and computer vision and imaging.

“Mathematics is present in everything around, from [information technology] to biology to non-invasive surgery,” Fonseca said. “The challenge to mathematics is that more and more the old way, [in which] you work with a pen and a pad, is no longer.”

Fonseca stresses the need to break down barriers and work across disciplines to create a new breed of mathematicians capable of tackling many different types of problems.

“It’s a long process,” Fonseca said. “We are all aware there are things that need to be changed in the curriculum.”

Mathematical education is one of Fonseca’s biggest concerns. She is worried that modern methods of mathematical instruction — the computing environment and programming language Matlab, for instance — are changing the way students of math think and closing them down to different methods of creative problem solving.

“Are we creating technicians or thinkers?” Fonseca asked. “We cannot replace thinking creatively with all the gadgets.”

To tackle emerging issues in mathematical education, Fonseca will attend two different conferences in the coming weeks.

She will travel first to the Georgia Institute of Technology to discuss the potential for more effective kindergarten-through-college math curriculums. From there, she will travel to Lisbon, Portugal, to meet with business and industrial leaders and discuss the role mathematics is playing in the global marketplace. On a broader level, the conference will deal with the ways in which the state of the economy impacts the educational curriculum.

“To what extent are industries interested in universities? To what extent is that necessary?” Fonseca asked, indicating the need for an information pipeline between industry and academia to properly inform a study of mathematics.

“It is a dance. We do our tango, we show what we have to offer, and they present their problems,” Fonseca said.

“What kind of math do I use?”

If anyone knows the tango, it is professor Shlomo Ta’asan. He has been working closely with biologists and physicians at the University of Pittsburgh Medical Center (UPMC), trying to understand diseases by using mathematical tools. Carnegie Mellon and the University of Pittsburgh received a shared grant from the National Institute of Health in September to pay for this kind of research.

Ta’asan is trying to develop mathematical models that will describe the progress of diseases and the inner workings of the immune system.

“Once we know about it, we can stop it, slow it down, accelerate it,” Ta’asan said.

Though he is trying to answer some of the most daunting questions in immunology, Ta’asan doesn’t have any degree in biology.

“I’ve had to review my statistics,” Ta’asan said.

In addition to his work in immunology, Ta’asan also works on understanding mental illness, specifically depression. He had a hunch that the ups and downs characteristic of mild depression could be studied with a set of pre-existing simulations — the ones used in the stock market to measure fluctuations. Ta’asan took some computational finance courses and borrowed the math he learned for use in his own biological computations.

“This research has made me learn areas of math I didn’t know,” Ta’asan said. “I’m trying to use all the areas of math I ever learned. The question is, what kind of math do I use?”

Like Fonseca, Ta’asan explained that every discipline has its own language and its own code. To cope with the language of biologists, Ta’asan has tried to take a qualitative approach to math, moving away from math that uses just real numbers and toward math with only a few levels. According to Ta’asan, biologists just do not speak and think the same way as mathematicians — and vice versa.

“It’s not simple to convince them to change the way they are doing things,” Ta’asan said. “I will change anything I need to.”

According to Ta’asan, the best way to get a feel for what scientists do is to make regular trips to medical research laboratories and communicate with scientists at work.

“The purview of many fields”

Across the Mall from Ta’asan’s workspace is the office of dean John Lehoczky, who, in addition to leading the school of Humanities and Social Sciences, holds positions in both the statistics and mathematics departments.

Lehoczky’s work in computational finance has him running into many of the same difficulties that Ta’asan faces at UPMC — the same ones that Fonseca is crossing the pond to talk about in Lisbon. He has to communicate mathematical research to the giants of finance through intermediaries who do not share his understanding of mathematics.

Just like Ta’asan and Kinderlehrer, Lehoczky uses mathematical models to solve interdisciplinary problems in finance. According to Lehoczky, his collaborative work forms the basis for Carnegie Mellon’s master of science in computational finance, the first program of its kind anywhere in the world.

“We combined the statistics department, the math department, Tepper, and the Heinz school into a seamless program designed for students,” Lehoczky said, explaining that while Wall Street had barely heard of Carnegie Mellon a decade ago, computational finance masters are now in high demand.

Lehoczky attributes the program to the university’s interdisciplinary nature of academics. According to Lehoczky, it is a model of academia toward which other institutions are moving out of necessity.

“This is the era of big science projects,” Lehoczky said. “The projects are not the purview of any [one] field; they are the purview of many fields.”