academics & research

Nature's patterns revealed by mathematics



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Grant Schumacher, an undergraduate majoring in math and physics, explains a complicated math concept using equations during a meeting with his research group on Wednesday. Marlie Bruns and Marlie Bruns Buy Photos

Working away at equations on a chalkboard might seem as interesting as watching paint dry, but, from crystals to deoxrybonucleic acid, math is infinitely more than meets the eye.

An insight into math research from undergraduates to professors reveals natural patterns and the reality of the work to the general public.

Since January, an undergraduate research group at IU has studied fractals, a type of mathematical pattern that appears the same at all scales, to understand the phenomenon better.

“It’s an interesting communication gap because everyone takes math,” said mathematics professor David Fisher on educating the public about mathematics. “In some sense, we’ve squandered an opportunity because people do know a little bit about this.”

Communicating math to the general public, though difficult, would provide an opportunity for everyone to learn, Fisher said.

The edges of a snowflake or the path of a lightning bolt, both fractals in nature, may appear jagged at any level.

From DNA to Saturn’s rings, fractals permeate nature and the fancies of mathematicians.

“Fractals are usually bounded in size,” said Tristan Tager, graduate student in mathematics. “We’re making fractals that are unbounded in size.”

Tager leads the group on making these unbounded fractals that extend infinitely within themselves.

Extending another professor’s theorems about fractals, the group studies operations used to create fractals, said Michael Peters, an undergraduate in mathematics and physics.

“We spend more of our time figuring out how to say what we wanna say,” said Grant Schumacher, another undergraduate in mathematics and physics.

The students try and try again until they succeed. They’re working to put these ideas into words to publish a paper on fractals.

“When you’re deeply formalizing the parts of the paper you wanna make, that’s when you go to the chalkboard and other people attack it,” Tager said. “And that’s where you’re wrong nine out of ten times.”

Getting the gist of math research out to the public is still difficult and hard to do.

Math is often so difficult to communicate that researchers spend hours working in person and sometimes devote entire workshops to understanding papers, Fisher said.

“Science reporting on mathematics happens only when there’s a tremendously big event,” Fisher said.

Fisher cited recent mathematical breakthroughs on problems such as the Poincaré conjecture and Fermat’s last theorem.

Russian mathematician Grigori Perelman solved the Poincaré conjecture, a general problem about reducing loops on spheres to single points, a decade ago.

British mathematician Andrew Wiles proved Fermat’s last theorem in 1994, more than three centuries after it was proposed, and led to advances in algebra and geometry.

Fisher and his colleagues recently published a paper that proved some cases of Zimmer’s conjecture, first made by Bob Zimmer, Fisher’s thesis adviser and president of the University of Chicago.

“Zimmer’s conjecture is about things that are sort of like crystal patterns, but these crystal patterns are very high-dimensional crystal patterns,” Fisher said.

In this conjecture, something between a guess and a bet, mathematicians resort to abstract theory to imagine these objects in two dimensions.

Plowing through equations and theory at the University of Chicago for three or four hours at a time, Fisher’s group worked through conversation.

“We all had pieces of the puzzle,” Fisher said. “It was just surprising how fast it’s all come together.”

Struggling to put their elusive ideas into words, the mathematicians’ work would eventually become a 40-page paper that took three years to write.

“It’s one of those things that’s odd about mathematics,” Fisher said. “We knew we could prove something, but I didn’t know what statement it was going to be.”

By the end of their meetings, Fisher said he knew they were going to write a paper together.

Fisher acknowledged there is still much more work to be done.

“We proved many more cases than anyone ever has before, but there are substantial cases that are still left open,” Fisher said.

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