Science/Engineering Number Theory Counts

2-10-96

Baltimore -- Basic research is supposed to be like caviar, you get what you pay for. But that doesn't mean there aren't bargains.

George Andrews, the Evan Pugh Professor of Mathematics and Department Chair at Penn State, thinks that basic research in his field, number theory, and especially partitions, is just such a gilt-edged bargain.

A partition is an elementary idea in number theory. Partitioning a number merely means breaking it up into a sum. For instance, 5 can be partitioned in seven ways: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, and 1+1+1+1+1. This simple concept has developed into a sophisticated and deep branch of number theory.

At the annual meeting of the American Association for the Advancement of Science, Andrews today (Feb. 10) outlined how Rodney Baxter, an Australian physicist, coupled number theory and physics to produce an exact mathematical description of the behavior of liquid helium on a sheet of graphite. This union is even more amazing when one considers that number theory is sometimes viewed as application-less, pure mathematics, while liquid helium is one of the most studied substances in the world, Andrews said.

Baxter's work initiated extensive research in statistical mechanics (the statistical study of atomic phenomena) where the theory of partitions plays a central role. Such theoretical studies are immensely helpful in understanding chemical and physical interactions, said Andrews, who collaborated with Baxter on follow-up research. As a direct consequence of their effort, mathematicians learned new and important facts about the theory of partitions, increasing the odds the theory would be used in other scientific disciplines.

Andrews recently worked with scientists in Switzerland applying number theory to programming computer software. This work centered on a study of the average running time of computer programs and how to predict when a program is likely to be hopelessly inefficient.

"I'm surprised each time that questions in other fields end up related to partitions," Andrews said. "As a number theorist, it's stimulating when partitions can be used as a tool of statistical or mathematical analysis to help another scientific field."

Andrews said pure math research is often applied to other disciplines. He points to Non-Euclidean geometry, thought to have no use until Einstein applied it to his theory of relativity.

"Basic research in mathematics is important and not terribly expensive," he said. "Often, all you need is pencil and paper or at most a small computer."

The mathematics used by Baxter in his breakthrough research was pioneered by the turn-of-the-century mathematical genius Srinivasa Ramanujan.

In 1976, Andrews discovered misplaced papers of Ramanujan among old hotel bills and letters in a box in the Trinity College Library, Cambridge University, England. Ramanujan, who died in 1920 at age 32, developed formulas that helped revolutionize the theory of numbers, particularly partitions. Andrews is considered one of the foremost authorities on Ramanujan's work, and thus it was natural that Baxter would enlist Andrews in a collaboration.

**sjt**

EDITORS: Andrews can be reached at (814) 865-7527.

Contact:
Scott Turner
(814) 865-9481 (office)
(814) 867-3834 (home)
sxt11@psu.edu

Vicki Fong
(814) 865-9481 (office)
(814) 238-1221 (home)
vyf1@psu.edu