The Mathematics of the Universe

“M stands for magic, mystery or matrix, according to taste,” says Witten, one of the founders of string theory. “Some cynics have occasionally suggested that M may also stand for murky, because our understanding of the theory is so primitive.”

Dr. Lisa Jeffrey is helping to clear up this cosmic murkiness. With chalk and intense collegial conversation, the University of Toronto mathematician is giving mathematical substance to a mind-blowing vision of our Universe.

“What I have done is largely to give mathematical proof of results found by theoretical physicists,” says Dr. Jeffrey, a professor of mathematics and one of six recipients of a 2004 NSERC Steacie Fellowship.

Dr. Jeffrey is to leading-edge physical and mathematical theory what a great player is to a Stanley Cup team. She's relentlessly working the boards, blackboards in her case, pounding away at mathematical proofs day after day to get her team into the finals.

It's the kind of work highly acknowledged by peers, and less understood by those outside the game. With good reason. Very little of Dr. Jeffrey's brand of mathematics is accessible to the outsider – a title that applies to anyone outside the rarefied field of symplectic geometry. (Even the term symplectic is arcane. It's a homologue of “complex,” coined because the term complex was already overused in geometry.)

Geometry is generally concerned with the study of shapes. Dr. Jeffrey's description of her work is filled with examples of locating points on the two dimensional surface of an imaginary donut, orange or car tire. It's the mathematics that is at the heart of studying phase space, the combination of location, motion and time that is used to describe the behaviour of everything big and small in the Universe (other than gravity, an exception left to Einstein's theory of general relativity).

But these vernacular examples have themselves become the basis for mathematically exploring more mysterious higher dimensional shapes.

For example, what's the shape of the Universe? Her mathematics doesn't directly address this question, but rather pokes at it through core questions. For example, is it possible for a particular form to be twisted so that intersecting lines drawn on its surface no longer cross?

“This is the point of departure for a piece of work I did in recent years, in which I created an orthodox mathematical proof of something Witten had discovered by physical methods that was closely related to string theory,” says Jeffrey, who worked under the supervision of Witten in 1992, as a postdoctoral fellow at Princeton's Institute for Advanced Study. “We proved formulas that determined whether lines on certain surfaces intersected or not. It's important because in studying shapes which are apparently different, there's always the core question of whether one could be deformed into the other. If they don't have the same properties, in terms of the intersection of the lines, then you can't deform one to the other.”

This heralded proof was developed during five years of work with Dr. Jeffrey's symplectic geometry colleague Dr. Frances Kirwan at Oxford University. The work has taken Dr. Jeffrey to the forefront of a field that is itself on the edge of a fascinating intersection of theoretical physics and mathematics.

As part of her NSERC Steacie Fellowship, she'll be co-leading a year-long, international program in the Geometry of String Theory at the University of Toronto's Fields Institute.

“I hope that this program will supply some more wonderful problems to work on,” says Dr. Jeffrey. “One problem that I'm already working on has to do with non-orientable surfaces, such as the Klein bottle, ones in which the two ends, if folded to touch one another, don't match up exactly in two dimensions. Witten also proposed a formula for these shapes, but so far no satisfactory mathematical proof exists.”

While Dr. Jeffrey continues to help clear the mists from Witten's cosmic vision, she chuckles when describing his high public profile, including being picked as one of Time Magazine's 25 most influential Americans in 1996. “That will never be me,” says Jeffrey.

But with an academic career that already includes schooling at Princeton, Cambridge and Oxford, a Sloan Foundation Fellowship, an Ontario Premier's Research Excellence Award, and now an NSERC Steacie Fellowship, perhaps this powerful geometer can't imagine the potential scope of her own trajectory.

Contact:

Dr. Lisa Jeffrey
Tel.: (416) 978-5142
E-mail: jeffrey@math.utoronto.ca
Web site: http://www.fields.utoronto.ca