2007 E.W.R. Steacie Memorial Fellowship

Eckhard Meinrenken

The boundary between theoretical physics and pure mathematics, a mysterious frontier for most, is territory that Eckhard Meinrenken navigates with ease. The University of Toronto mathematician has earned a reputation in the mathematics community as a creative, ingenious thinker who successfully tackles seemingly insoluble problems.

Dr. Meinrenken's research focuses on symplectic geometry, a branch of mathematics originally developed in the early 19th century to provide a framework for optics and classical mechanics. It developed into a separate branch of mathematics during the 1960s, with applications involving various aspects of physics, as well as other areas of mathematics such as algebraic geometry and group theory.

Today, Dr. Meinrenken and others continue to expand the field, including developing ideas that are inspired by areas of theoretical physics such as string theory and conformal field theory. His highly influential work has earned him a 2007 NSERC E.W.R. Steacie Memorial Fellowship.

Among Dr. Meinrenken's achievements are proofs for several theories that had confounded mathematicians, including one that had remained unproven for more than a decade. He first attracted the attention of the wider mathematics community with his work on a theory known as the "quantization conjecture" developed by Victor Guillemin and Shlomo Sternberg. Various mathematicians, including Dr. Meinrenken, had proven that the conjecture was true in certain special cases, but he was the first to deliver a proof that applied to all its aspects.

Since then, he has continued to expand his range, quickly mastering new fields as he delves into them. Some of his work has also become part of the standard tool kit used in symplectic geometry and has been included in textbooks for graduate students.

Detailed descriptions of Dr. Meinrenken's research are populated by specialized terminology such as "twisted K-theory" and "Kashiwara-Vergne conjecture," phrases that are themselves explained with equally esoteric terms. If these concepts seem to lack an apparent connection with the stuff of everyday life, it is because they were in fact not developed in response to a pressing technical problem. Pure mathematics is as much an art as a science, and consists of a heady combination of pure reason and mathematics dedicated to solving a problem for its own sake without intentionally focusing on finding a particular application.

That's not to say that real-life applications do not exist for pure mathematics, but it can take years for them to become apparent. As with any enquiry into basic science, it is not uncommon for a researcher to develop a theory that others then spend decades or longer trying to understand, test and prove.

In the meantime, the importance of this type of research lies in the fact that it helps build a solid theoretical foundation that others can build on. Ultimately, Dr. Meinrenken's conclusions might help test physics theories that are intended to describe the fundamental nature of the universe, but proving or disproving something like string theory is not his ultimate goal.

It comes as no surprise then that Dr. Meinrenken says that his research is almost completely curiosity-driven. Mathematics has been in his blood for as long as he can remember, even though he opted to study physics at university in response to family pressure to do something with a practical focus. "My talents were in mathematics," he explains, "but I chose to study physics instead, basically to have mathematics with some application. Eventually I ended up in mathematical physics and at the end drifted into pure mathematics."

One of the things that motivates Dr. Meinrenken to keep plugging away at a particular problem is the elusive quest for an "elegant" solution. He says there are two types of results: one that involves lots of hard work but is ultimately hard to explain to people, and another in which the simplicity and beauty of the proof make it easily explainable and accessible. "I like to have very clean and clear proofs which really explain *why* they are true," he says.

In fact, the lack of an elegant proof likely indicates that the solution is not well-enough understood. In that case, Dr. Meinrenken keeps plugging away, always keeping the problem in the back of his head and occasionally coming up with surprising ideas. "Whenever a project is finished, there are always some leftover questions. I just keep thinking about them," he says. "Initially, you think it's a very small problem, a side question. Eventually it develops into something much bigger."

Like any creative person, ideas come to him without warning. "I still distinctly remember the day, sitting at my desk and it just clicked," he says, referring to coming up with the concept of "group-valued moment maps" to explain aspects of two-dimensional gauge field theory. "It's quite nice to have that kind of experience."

Given his penchant for novel solutions to difficult problems, he seems destined to have many more such moments, moments that his colleagues will await with anticipation.