Computing Infinity
by Steven N. Koppes

Jon Carlson has spent the past five years trying to compute infinity.

"Of course, you can’t compute anything that’s really infinite," said Carlson, a UGA Research Professor of Mathematics. "But if you can compute the most important chunk of the infinite, that will allow you to understand all the rest."

As part of his research program, Carlson applies computer algebra to a highly abstract specialty called cohomology. His work is purely theoretical, but it is directed at fundamental mathematical issues in phenomena as complicated as the global economy.

"From a strictly mathematical standpoint, cohomology measures the dependence of the whole system on the subsystem," Carlson said.

The global economy, like the mathematical system that Carlson studies, is a large system containing many variables that influence one another. The trick is to identify the most influential variables without becoming overwhelmed by the ones of little consequence.

Interest rates and other key economic indicators, for example, might be independent of the type of cars that people buy in Athens, Ga. But car-buying habits in Athens might be firmly linked to leading economic indicators.

Only within the past 10 years have computers grown powerful enough to tackle the most complex cohomology problems, which still can take nine or 10 days of continuous calculations to solve. The Sun Ultra 2200 computer that he uses contains a gigabyte of random-access memory, enough to hold the contents of a thousand books. "On some calculations, I use every bit of it," Carlson said.

He also depends on MAGMA, a sophisticated computer algebra system devised by John Cannon at the University of Sydney, Australia, to avoid becoming engulfed in a volcanic torrent of numbers. Carlson has written several thousand lines of computer code that eventually will become a part of the MAGMA package. Nevertheless, adapting MAGMA to his purpose has proven challenging.

For example, some of the calculations are extremely sensitive to considerations that might seem inconsequential. Carlson ran one problem on his computer for two weeks without coming near a conclusion. He finally killed the program and started over after modifying only the ordering on the variables. The conclusion ran in three seconds.

"Computer algebra is a very different kind of research because it’s not just that you want to figure out whether something is true, or whether something will work," Carlson said. "You’ve also got to figure out how to make it work in a reasonable amount of time, with a reasonable amount of memory storage."

Carlson’s cohomology research is supported by the National Science Foundation and the UGA Research Foundation, Inc. His recent grant from the Alexander von Humboldt Foundation will enable him to spend a year working at the University of Stuttgart and other universities in Germany. The German government awards Humboldt grants to eminent foreign scientists in recognition of their research achievements.

For more information, e-mail Jon Carlson at jfc@sloth.math.uga.edu or access www.math.uga.edu/~jfc/