Why round worlds and Ricci flows are good news

Eye on the Carriers By Johna Till Johnson, Network World, 08/21/06

In 2005, noted New York Times columnist Thomas Friedman published a bestseller titled The World is Flat. What he was trying to articulate is the way communications technology (especially real-time collaboration and mobility) has leveled the playing field and minimized the impact of geography and distance.

Unfortunately, his flat world metaphor is inaccurate. The geometrical construct in which every point is equally far away is actually a sphere, not a plane. (In a flat world, points can be infinitely far away from one another - not so in a sphere.)

Friedman's main point is correct, even if his metaphor isn't: Technology, particularly communications technology, is having profound effects on the global economy by bringing faraway places close together. Moreover, it's gradually eliminating the barriers posed by distance and lack of infrastructure, creating an increasingly uniform platform on which to conduct business.

If you're fond of mathematical metaphors, another way to state this is that communications technology serves as the Ricci flow of the Poincaré conjecture. The conjecture, which was created in 1904 by the mathematician Heni Poincaré, essentially states that all three-dimensional objects without holes are topologically equivalent to spheres. The Ricci flow is a mathematical process by which some oddly shaped topological entities can be transformed into spheres - thus "smoothing out" all the bumps and depressions. Until recently, though, nobody knew whether Ricci flows worked all of the time or just in a few special cases.

A reclusive Russian mathematician named Grigory Perelman recently demonstrated that Ricci flows ultimately will transform all three-dimensional objects without holes into spheres within a finite amount of time - thus proving the Poincaré conjecture. The New York Times had a great story on this last week for those of you with access to the newspaper's archives.

The key point is that in the long run, communications technology makes distance immaterial - just the way in the long run, the Ricci flow transforms weird and bumpy objects into spheres. This is a huge deal in mathematics. Proving the Poincaré conjecture has been one of the top 10 goals of mathematicians for more than 100 years. And it's an even bigger deal in economics, because it means that the global economy, over time, will become even more interconnected, just as Friedman posits.

Is this a good thing? It depends. Every economic seismic shift benefits some folks at the expense of others. But in the long run, it is good thing. However you define it, progress is always a collective effort, and it's not a zero-sum game. Any technology that lets us leverage the gifts and talents of a broader range of people ultimately will help accelerate the development of ideas and advance the state of knowledge.

Who knows? Thanks to the new round world, we may not need to wait another century to solve the next great challenge in mathematics.