An
undergraduate student who earned a dual degree last Saturday has taken a complex
mathematical theory on numbers and converted it into reality -- a task that
has boggled mathematicians for decades.
By Anne Danahy
University Relations
An
undergraduate student who earned a dual degree last Saturday has taken a complex
mathematical theory on numbers and converted it into reality -- a task that
has boggled mathematicians for decades.
Rhiannon L. Weaver, whose hometown is just outside Philadelphia, was recognized May 12 for this significant contribution during a Schreyer Honors College medals ceremony. She is the first person to be honored with the Schreyer Honors College Dean's Award for Research or Creative Achievement.
Inspired by a paper on a topic known in the math world as partition congruences, Weaver wrote a C computer program to find concrete examples of the congruence of numbers that had been proven to exist only in theory by Ken Ono, the Louis A. Martarano professor of mathematics at Penn State and author of the paper. C is a computer programming language.
"Rhiannon translated the theory and found 70,000 new examples," said Ono, who is internationally recognized for his work in numbers theory. "These are things you would never find. She found them using the theory of modular forms in a completely new and clever way. These astute observations revealed concreate conditions which force congruences to exist. She then wrote a sophisticated C program to find when these conditions occur. Her doing this translates the theory into things people can check.
"Her paper will soon be one of the standard references in the subject."
A student in the Schreyer Honors College since enrolling at Penn State in 1995, Weaver also had received several scholarships including the College of Engineering Undergraduate Scholarship, the Academic Excellence Scholarship and the Schreyer Summer Research Grant in the summer of 1999 when she wrote a paper to explain her findings on congruences. Her research also has been supported by the National Science Foundation.
These findings have shed new light on a concept first introduced by Ramanujan, a mathematician from India who lived 80 years ago and whose studies of number theory make him what Weaver calls "the Mozart of math."
In spring 1999, Weaver began working with Ono on his paper on congruences in partitions. Ono proved the existence of an infinite number of congruences. Aside from being able to find two new concrete examples to corroborate this theory, the actual numbers remained elusive.
A partition function essentially counts the number of ways you can write a positive integer as a positive integer, according to Bruce Berndt, professor of mathematics at the University of Illinois, who is known as a leading expert on the work of Ramanujan. A common example is the number four, which has five partitions. The partitions are the number four by itself, three plus one, two plus two, two plus one plus one, and one plus one plus one plus one.
"Ramanujan noticed that the partition function had some remarkable divisibility properties," said Berndt. "If you take this number four and add any multiple of five, then the number of partitions of the resulting number will always be divisible by five. Ramanujun had these for five, seven and eleven. I think that the work Ken and Rhiannon have done is really the most important and exciting work that's been done on this topic since Ramanujan."
Weaver, who earned dual bachelor's degrees in mathematics and computer science, was able to fuse the two branches of the field together and write a C program that uncovered 70,000 new observable examples. She had set out to find congruences for the numbers 13, 17, 19, 23, 29 and 31. But after the computer spent two days processing the information she had given it, she realized she had generated thousands.
"What I did was write a program based on Ken's proof," said Weaver, who is modest about her accomplishment. "I think that probably anybody else that came around at the right time could have done the same thing. I guess I was lucky to be offered the opportunity, lucky to be noted at least as somebody who can program in C."
Weaver was honored with the Lockheed Martin Design Excellence Award in Software Engineering in the fall of 1999 and was invited to speak at the Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics Conference in November 1999. Her paper "New Congruences for the Partition Function" has been accepted for publication in the Ramanujan Journal.
Weaver spent her last week at Penn State before graduation finishing a project known as a library of functions in the computer programming language C++ to prove two propositions she had included in her senior thesis.
"The other thing I'm doing right now is turning this library into an interactive research tool for Ken to explain coefficients and modular forms," she said.
After that, Weaver will spend three weeks this summer with the University Choir on a tour of Eastern Europe. In the fall, she will begin pursuing a Ph.D. in statistics at Carnegie Mellon University in Pittsburgh. Wherever she goes, her discovery of numbers will surely follow.