Mathematician Shing-Tung Yau Profiled in Discover Magazine

 May 21, 2010 

The June, 2010, issue of Discover Magazine features a lengthy interview with Field's Medalist Shing-Tung Yau, which is titled "The Math Behind the Physics Behind the Universe." 

In the article Yau indicates that when looking at the world through the lens of geometry and topology, nonlinear equations become fundamental because in nature curves abound. The Einstein equation "described the curvature of the universe, and it was nonlinear," Yau points out. Yau learned nonlinear equations from a master, although he admits he didn't know it at the time. That master was Charles Morrey. 

Concentrating on complex manifolds, Yau found that a manifold is just a space, with each point immediately around you looking like Euclidean space—the familiar kind of space that we see around us.

"Imagine the earth is covered with a checkerboard or a grid, like latitude and longitude," he said. "This is the kind of coordinate system that Descartes introduced to geometry in the 17th century. At each point on the grid the space appears flat and finite, but it’s actually curved, a sphere. Instead of being measured with real numbers, though, we measure complex manifolds with complex numbers." 

Space, Yau continues, is not necessarily something you see in day-to-day life. "You can define geometry locally, but globally you cannot visualize the big picture, you can only imagine it and represent it through coordinates. We draw lines of latitude and longitude in a coordinate system for the continents. But that system doesn’t work well at the north or south poles, where all the lines converge. In order to get a more complete picture in those regions, we need another, more localized coordinate system for more detail. In the end, we need several such coordinate systems patched together to get a detailed picture of the entire globe." 

Yau also describes the various notions of curvature that a space might have, and discusses his role in the controversy concerning Grigory Perelman's proof of the Poincare Conjecture. In short, Yau guided a graduate student in filling in some of the “missing details” of Perelman's proof. 

Source: Discover Magazine (June 2010)