by David F. Salisbury
Take geometry, combine it with algebra and you get a new and extremely powerful mathematical field called noncommutative geometry. It’s geometry taken to a whole new level that is providing the mathematical tools that allow scientists to accurately describe many of the fundamental concepts of modern physics, including quantum mechanics, Heisenberg’s uncertainty principle and the space-time continuum.

In recent years, Vanderbilt’s Department of Mathematics has built up a stable of internationally recognized experts in this field and, as the result of a grant of two-thirds of a million dollars from the National Science Foundation (NSF), it is poised to become the national center for training the next generation of young mathematicians in noncommutative geometry.

The training grant is part of a new NSF program which is based on the recognition that scientific advances in all fields – from physics to medicine – depend increasingly on innovative mathematical approaches like noncommutative geometry. Its goals are to increase the number of U.S. citizens and permanent residents who pursue mathematics careers, to broaden their background and perspective, and to stimulate permanent, positive changes in education and training within the mathematical sciences in the U.S. Vanderbilt received one of the first six grants awarded by the program.

In much the same way that the modern view of the universe differs profoundly from the classic 19th century view, noncommutative geometry is nothing like the traditional Euclidian geometry that is still taught in school.

"We usually think of a space as a collection of points with some additional structure, like a notion of distance that allows us to tell how close two points are,” explains Dietmar Bisch, the mathematics professor who directs the noncommutative geometry research group. “However, quantum mechanics has shown that this way of thinking about space is far too limited. In the small scale structure of space-time the notion of a point no longer makes sense and profound discoveries such as Heisenberg's uncertainty principle can be deduced by introducing a new concept of space – the concept we call `noncommutative space' – where points disappear completely and are replaced by certain functions called operators.”

Although it offers new approaches to mathematical problems, the breadth and technical nature of noncommutative geometry have made it difficult for beginners to navigate its frontiers. The new training grant will help address that problem with a vigorous training program for graduate students and postdoctoral associates, Bisch says.

Co-principal investigators of the grant are Distinguished Professor of Mathematics Alain Connes – recipient of the Fields Medal, the highest award in mathematics – and Professors of Mathematics Dietmar Bisch, Bruce Hughes, Gennadi Kasparov and Guoliang Yu.

“We are very excited about the opportunities provided by this grant,” says Bisch. “Vanderbilt is already an internationally recognized center in noncommutative geometry. This grant will allow us to attract the most promising students and postdocs to work with members of our research group. It will greatly enhance the advancement of these exciting research areas.”

Posted 11/15/04