UMass Boston Professor on Mathematics “Dream Team”
A century-old mathematical problem that may help solve the mysteries of the universe and lead to new breakthroughs in science, engineering and finance has been computed by an international group of mathematicians, including UMass Boston Professor Alfred Noel.
“(It is) a mathematical breakthrough that may change the way mathematicians and scientists approach research,” Noel said.
Noel is one of 18 mathematicians, mostly US-based, who worked for four years to unravel E8, a calculation so large that, if it was printed out, would cover the island of Manhattan. A mathematician-programmer, Noel’s role within the group was to develop mathematical techniques that could be programmed on a computer, most importantly an algorithm that computed “standard representations.”
In basic terms, E8 is a mathematical tool to study symmetry. A 248-dimenional structure, E8 is the most complicated of the so-called “Lie groups.” named for the 19th century Norwegian mathematician who first studied them. Cylinders, cones and balls – which remain symmetrical under one degree of rotation – are examples of Lie groups.
“A thorough understanding of such symmetries will help mathematicians tackle numerous unsolved problems,” said Noel.
Among those problems is the unified theory – better known as the theory of everything – which physicists have sought for nearly two centuries, and could also aid research by Noel’s colleagues in the mathematics and computer science at UMass Boston.
Noel, who joined the research team in February of last year, currently splits his time between teaching Calculus and Probability and Statistics courses at UMass and conducting research at MIT, where he is a visiting scholar.
Working with a driven group of researchers, including one of his mentors, Massachusetts Institute of Technology mathematics professor David Vogan, is nothing new for Noel, who said he is constantly challenged by UMass math professor Steven Jackson in their representation theory research group. His role on the dream team, he said, is “business as usual.”
The discovery is part of a larger project funded by the National Science Foundation. Called the Atlas of Lie Groups and Representations, it aims to solve one of math’s greatest problems, “the determination of the unitary dual,” and provide computing software.
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This picture represents the root system of type E8, projected from 8 dimensions down to 2. Picture by John Stembridge, based on a drawing by Peter McMullen. |
“The E8 computation, although exceptional, is only the first step in a vast and complex program which will last for several years,” Noel said. “This is one of the most important problems in mathematics or even in all the so-called hard sciences. This is the Holy Grail of representation theory.”
Before joining the Department of Mathematics at UMass in 1998, Noel was a research engineer at Peritus Software Services in Billerica and a lecturer at local colleges and universities.
Noel’s research on representation theory of Lie groups and math education has been published in dozens of mathematics journals, including Experimental Mathematics and the Journal of Algebra. In 2005 and 2004 he was awarded outstanding achievement awards in research (jointly with Professor Steven Jackson) from UMass.