Professor’s boyhood dream leads to ‘cluster algebra’ discovery

 

Professor Andrei Zelevinsky.
JIM CHIAVELLI

 

By Susan Salk

 

By the time he was 7 years old, Andrei Zelevinsky had the answer to one of life’s most difficult problems.

 “My parents, both physicists, asked me what I wanted to do when I grew up. I don’t know how I knew it, or even if I knew what it was, but I told them I wanted to be a mathematician,” he said.

Seven years later, Zelevinsky represented Russia in the International Mathematics Olympiad for high school students, held in Romania, one of just a handful selected for the honor. He won a silver medal.

Since then, the victories have kept on coming for the Northeastern professor of math:

• His recent discovery and development of a new mathematical concept— cluster algebra — will be developed into a book with colleague Sergey Fomin of the University of Michigan. The American Math Society has contracted the pair to write a book based on a series of lectures given by Zelevinsky at North Carolina State University in 2006.
• In December, the Notices of the American Math Society will publish a column titled, “What is … cluster algebra?” which endeavors to explain his work to the math community. This concept, which has many connections “to different areas of mathematics and mathematical physics” expands on work done in the early 1990s by George Lusztig and Masaki Kashiwara, he explained in online publication in-cites.com.
• His work with “representation theory” continues to offer insights into symmetry, a concept understood on some level by anyone appreciating a fine painting, or other beautiful objects. “The math is another way to recognize symmetry, and to find different ways to classify what so many of us recognize when we look at a beautiful painting,” he said.
• His analysis of systems of algebraic equations has been used in theoretical work related to robotics, he said.

Considered a highly cited expert in varied areas of research, including representation theory, algebraic geometry, algebraic and polyhedral combinatorics and hypergeometric functions, Zelevinsky’s most recent work has been referenced more than 215 times, according to “Essential Science Indicators,” the basis of a recent interview published by in-cites.com. He has published more than 80 research papers and surveys and two research monographs.

Asked in the article why he felt he was referenced so often, Zelevinsky speculated it had something to do with the fact he works in several mathematical disciplines.

“First, I try to address natural questions whose significance be understood without going into too many technicalities. Second, the topics addressed cut across several mathematics disciplines, and so have a better chance attract the attention of a wider mathematical audience,” he said.

Admitting that math is not a mainstream topic that easily grabs headlines — it is by its nature difficult to put in laymen’s terms, he said — Zelevinsky nonetheless views himself as a mathematician with a “down-to-earth” approach.

At the end of the day, Zelevinsky said, his biggest hope is that he will have made some interesting contributions to his field. He has been striving to do so since he received his doctorate in math from the Moscow State University, and, since 1991, through his teaching at Northeastern.

On the verge of publishing a new book, Zelevinsky describes how he traveled to Vienna six years ago to meet with colleague and longtime collaborator Fomin on another matter. The pair intended to flesh out an outline for an entirely different book, when they began discussing some mysterious calculations done by Zelevinsky in the preceding few months.

“We were so excited about it that we forgot all about the other book idea,” he recalled, noting that the new concept of cluster algebras was born as a result of these discussions.

“In math, it’s not every day when you create an entirely new concept,” he said. “We’re not just studying historic facts from 300 years ago, but we’re making the discoveries that hopefully will be of interest to other mathematicians for years to come.”