After 30-year Quest, UF Math Research Leads To Breakthrough
GAINESVILLE, Fla. — Two University of Florida mathematics researchers have made a significant discovery in the theory of partitions and q-series, a branch of mathematics noted for its ties with other branches of mathematics and the sciences.
Krishnaswami Alladi, professor and chairman of UF’s department of mathematics, and Alexander Berkovich, a visiting professor of mathematics at UF, collaborated with George Andrews, a professor of mathematics at Pennsylvania State University, in the discovery. The UF researchers will present their findings in May at the Millennial Conference in Number Theory at the University of Illinois in Champaign-Urbana.
“Without any exaggeration, this is one of the most exciting things that has happened in this subject, and the consequences that will be worked out over the next several years will be quite significant,” Alladi said. “It opens up several new avenues of exploration.”
The discovery consists of the solution to a problem first posed 30 years ago by Andrews, one of the world’s foremost authorities on partitions. The solution centers around the 1967 Theorem of Göllnitz, one of the most far-reaching results in partitions. In 1995, Alladi; his former teacher, Basil Gordon of the University of California at Los Angeles; and Andrews achieved a significant generalization on this theorem. Last year, Alladi and Berkovich discovered a four-dimensional identity that subsumed the Göllnitz theorem as a special case. This year, the two researchers and Andrews completed a “proof” of the identity, solving the original problem Andrews posed.
“It has been a dream of mine since 1967 that someday I would see an extension of the Göllnitz theorem,” Andrews said. “I lost hope after 30 years, so this is an especially delicious moment.”
The theory of partitions and q-series deals with the representation of numbers as sums of other numbers.
Partitions arise in a variety of settings and have applications in many branches of mathematics, including analysis, number theory and combinatorics. They also have applications in theoretical physics and related disciplines. Andrews and other researchers have done partition-related research in connection with the Hard Hexagon model in statistical mechanics in the 1980s. More recently, researchers have used partition theory in connection with Conformal Field Theory in physics.
The theory of partitions also is significant from a historical perspective. Founded by Leonard Euler, an 18th-century mathematician considered to be the most prolific in history, the theory was greatly enhanced around the turn of the century by an Indian mathematician named Srinivasa Ramanujan. Andrews, an Evan Pugh professor at Penn State, is a leading expert on Ramanujan.
Ramanujan, born in 1887, grew up in a poor family and had little formal training in mathematics. By the time he was a young teenager, however, he was astonishing professors in India with far-reaching and unconventional theories and proofs. He was elected a Fellow of the Royal Society in England in 1918 but died shortly later at age 32.
Alladi, who in 1997 began publishing the Ramanujan Journal, devoted to all areas of mathematics influenced by the Indian genius, said the recent UF discovery is a major achievement in the area of Rogers-Ramanujan type identities, which are part of the theory of partitions.
“One of the reasons mathematics is so wonderful is that you systematically build a superstructure on previous ideas and proven results. It is like building a skyscraper,” Alladi said. “To reach and understand the Göllnitz theorem, one needs to take a path starting from Euler and build on the work of Ramanujan and Schur. And this magnificent new four-dimensional identity is erected on this solid structure.”
- Christopher Davis
- Krishnaswami Alladi