Math Institute Serves As Bridge for Pure and Applied Mathematics

For a time, many people believed that the world of pure math, which involves solving problems in the abstract, was distinct from that of applied math, which uses math to solve real world problems, such as in engineering, and relies more heavily on the use of computers.

The perception of the pure mathematician was of someone with a pencil and paper, proving a theorem, while people who used math for practical applications through computations simply traveled a different path.

But not anymore. In fact, the reality is that the two never have been all that far apart. The mathematician working to prove a theorem, for example, was no stranger to software when it came to tackling a problem unsolvable by hand; thus, computers, to some extent, always have been a part of both disciplines.

Today they are bridging the gap, and at a faster pace.

"The pure and applied areas are closer together than they were 50 years ago, in part because of the greater use of computational tools," says Jill Pipher, professor of mathematics at Brown University. "Pure mathematicians now appreciate the tools that have driven the application of mathematics to other disciplines and to industrial problems. We are trying to accelerate that process."

Pipher directs Brown University's Institute for Computational and Experimental Research in Mathematics (ICERM), a National Science Foundation- (NSF) funded institute whose goal is to support and broaden the relationship between mathematics and computation. Specifically, it seeks to expand the use of computational and experimental methods in math, and support theoretical advances related to computation.

NSF is funding the institute, which began in 2010, with $15.5 million for its first five years.

There are eight NSF-supported mathematics institutes, each with its own niche. What makes ICERM unique is its emphasis on the intersection of math with computers, an approach that "stems from the philosophy that pure and applied mathematics are really converging," Pipher says. "There really has been a sea change."

The institute brings together the best mathematical minds from all over the world for a series of wide-ranging projects, all of which have a strong computational component. These include a single, semester-long program that concentrates on one topic, and where researchers "have a whole collection of very exciting open-ended problems that they would like to solve," she says.

ICERM also sponsors stand-alone workshops, public lectures and a summer undergraduate research program. The institute runs an "idea lab," bringing 20 early career researchers to the institute for a week-long series of tutorials and brainstorming, most recently, for example, on topics such as "tipping points in climate modeling" and "how to compute on encrypted data stored in the cloud."

The institute's first semester-long program, in the autumn of 2011, focused on computation and kinetic theory, the latter a field which involves the thermodynamic behavior of matter, especially the relationships among pressure, volume and temperature in gases.

Kinetic theory plays a key role in many areas of mathematical physics, from nanoscales to continuum mechanics, and is an important tool in the mathematical description of applications in physical and social sciences, from its origin in dilute gases to applications in semiconductors, polymers, cells, plasma, galaxies (mathematical issues arising from questions about the stability of the solar system over time), and traffic networking (models of packet flow in large networks, like the Internet), among other things.

In June, the institute plans to host an Air Force-funded follow-up conference that focuses on computational modeling challenges in kinetic theory that arise in aerospace applications.

"The problems that arise in kinetic theory and come up in engineering and mathematic physics can be approached in two ways: By people trying to understand the overriding truths by proving theorems, and by people creating simulations and understanding what is happening in these computational models," she says.

"Each of the researchers had a collection of problems they wanted to solve," she adds. "Our computationally oriented research programs have a special focus on the training and mentoring of the next generation of mathematical scientists: ICERM provides funding for a postdoc program and for a cohort of graduate students in residence for the semester."

The summer undergraduate research program exposes young college students to research by engaging them in different projects. The institute hosted 14 students last summer.

"We had at least four published or about-to-be-published papers and another two projects that involved coding, and an honors thesis--at least six major research outcomes out of that program," she says.

(Software created at this research program by one of the students visualized a dynamical process called "outer billiards with contraction.")

Finally, the institute is participating in a pilot project sponsored by NSF's Science Across Virtual Institutes (SAVI) program, whose goal is to promote and strengthen interaction among scientists, engineers and educators internationally. The belief behind SAVI is that STEM fields (science, technology, engineering and mathematics) can produce advances at a faster pace when scientists and engineers work together across global borders.

ICERM is part of a "virtual" institute for mathematical and statistical sciences (VIMSS) connecting it and the Statistical and Applied Mathematical Sciences Institute (SAMSI) at Duke University with five research institutes in India.

The United States and India, with each institute having its own strengths, plan to collaborate on several global research challenges, including sustainability, cybersecurity, health and the extraction of useful information from massive and complex data, among other things. India's Department of Science and Technology is partially supporting the Indian institutes, which are located in Mumbai, Bangalore, Kolkata and Chennai.

Researchers from both countries will participate in exchange visits and satellite conferences.

"VIMSS started out with a workshop on cryptography, and we have three Indian graduate students coming to ICERM's fall program on low-dimensional geometry and dynamics," Pipher says. "The hope is that this will catalyze more research contacts between the best math and research institutes in India and in the United States, and create lasting and productive collaborations."