### Berkeleyan

Mathematician Vaughan Jones to discuss 'a new kind of algebra' in next Faculty Research Lecture

28 February 2007

As encountered in school, algebra is a one-dimensional affair in which multiplication, addition, and other operations must be applied one after another, says Berkeley mathematician Vaughan Jones. In the second 2007 Faculty Research Lecture - titled "Flatland, a Great Place to Do Algebra" - Jones will discuss a new kind of algebra based on two-dimensional configurations, hoping to "communicate its flavor and give some idea of where it should be useful."

"Although two-dimensional, this algebra is intimately related to three- dimensional gadgets like knots," he says. "We will also have a quick look at places in mathematics and physics, such as quantum computing, where this planar algebra appears to be necessary."

The lecture takes place at 5 p.m. on Wednesday, March 7, in the Bancroft Hotel, 2680 Bancroft Way. Jones promises copious anecdotes and illustrations.

Describing Jones' work, Professor Emeritus Calvin Moore, a colleague in the math department, says the following: "A theme running through his work is the study of von Neumann algebras, a type of structure - infinite dimensional or continuous versions of finite-dimensional matrices - that John von Neumann discovered in the 1930s. These new objects were of interest not only for purely mathematical reasons, but von Neumann and others also saw great potential that these objects might serve as mathematical models in quantum mechanics, quantum field theory, and statistical mechanics."

Moore goes on to relate how, in 1983, Jones discovered "some fascinating, and totally unexpected, relations concerning the dimensions of certain subalgebras of these von Neumann algebras. This work has unleashed new lines of investigation that remain vibrant today. But then he combined these results with observations from knot theory, a totally unrelated area of mathematics, that tries to understand the topological structure of knots - illustrated by the different ways a piece of string can be knotted. Jones discovered a new algebraic invariant of knots, now called the Jones polynomial, a vast improvement on the Alexander polynomial that had been discovered in 1920s. Jones' work was a huge advance that rejuvenated the field. The Jones polynomial is particularly well suited to study how DNA molecules knot and coil."

Jones was born in New Zealand, did his thesis in Geneva, and has been professor of mathematics at Berkeley since 1985. He has won wide recognition, most notably a Fields Medal in 1990, the equivalent in mathematics of a Nobel Prize. ("This equivalence," notes Moore, "is recognized by no less an authority than the UC Berkeley Parking Office, which accords our Fields Medalists the same parking privileges as it does for Nobel laureates.")

He has also been awarded a number of honorary degrees and been elected to the Royal Society of London, the American Academy of Arts and Sciences, the Norwegian Royal Society, the London Mathematical Society, and the U.S. National Academy of Sciences. In 2002, Queen Elizabeth II awarded Jones the title of Distinguished Companion of the New Zealand Order of Merit, noting that his mathematical work is being used to understand the complex knotted configuration of DNA.

"Although not a knighthood," quips Moore, "it is the functional equivalent of one, except that we do not have to address him as Sir Vaughan."

Each year the campus Academic Senate celebrates research excellence by honoring several faculty members whose work they hold in high esteem. For details on the series, see www.urel.berkeley.edu/faculty.