Today the Royal Swedish Academy of Sciences announced the four winners of the 2012 Crafoord Prize, an annual award that rotates between the disciplines of astronomy, mathematics, geosciences, biosciences, and arthritis research. This year's honorees came from mathematics and astronomy, fields last recognized in 2008.

The two awardees in mathematics were Jean Bourgain, a Belgian mathematician now working at the Institute for Advanced Study (IAS) in Princeton, New Jersey, and Terence Tao, an Australian-American mathematician at the University of California, Los Angeles (UCLA). Both Bourgain and Tao previously won the Fields Medal, often considered the equivalent of the Nobel Prize in mathematics (Bourgain in 1994 and Tao in 2006).

Tao is probably best known for his proof of the Green-Tao Theorem in 2004, which states that the prime numbers contain arbitrarily long equally spaced sequences (such as 5, 17, 29, 41, 53 -- a sequence of five primes spaced a distance 12 apart). This had been a major unsolved problem in number theory for decades. In addition to this very "pure" mathematics result, Tao is known for more practical work such as his invention, with Emmanuel Candes, of compressed sensing. This is an approach to image acquisition that greatly reduces the amount of raw data needed to reconstruct an approximate image—or even a perfect image—of natural objects.

Bourgain has proved several groundbreaking results on "well-posedness" of nonlinear differential equations, such as the Schrödinger equation of quantum mechanics and the Korteweg-de Vries equation of wave propagation. These theorems guarantee that the solutions to these equations evolve in a unique way, no matter how irregular the initial state of the system might be. But like Tao, Bourgain has wandered far and wide. A very recent and appealing result of his, proved in 2010, concerns Apollonian gaskets—infinite collections of circles that just touch without overlapping. In these beautiful, foamlike arrangements, the radius of each bubble is one divided by an integer. Together with Elena Fuchs, Bourgain showed that if you pick any integer n, there is a nonzero probability that some bubble in the Apollonian foam has radius equal to 1/n.

"Bourgain and Tao have similar and very unique strengths," says Peter Sarnak of IAS, a colleague of both. "Their ability to see through technical difficulties that prevent almost anyone else to move forward has allowed them to resolve long-standing problems in an amazingly diverse set of areas."

The Crafoord Prizes in astronomy for 2012 will be awarded to Reinhard Genzel of the Max Planck Institute for Extraterrestrial Physics in Garching, Germany, and Andrea Ghez of UCLA. Genzel and Ghez are like the Coke and Pepsi of research in supermassive black holes. Beginning in 1998, their groups have independently produced compelling evidence for the once controversial notion that our galaxy has at its center a supermassive black hole which is about 4 million times as massive as the sun. Ghez conducted her observations at the Keck Observatory in Hawaii, while Genzel's research was conducted at the European Southern Observatory in Chile, first at the New Technology Telescope and then at the Very Large Telescope.

A key ingredient in the work of Ghez and Genzel has been the application of speckle interferometry and then adaptive optics to compensate for atmospheric distortion. Adaptive optics have proved so successful that the atmosphere "no longer limits our resolution," Ghez says—that is, the telescopes can see just as well as they could if they were in outer space.

With these ultrasharp glasses, Ghez and Genzel have tracked the motion of individual stars around the central black hole, called Sagittarius A*, making it possible to compute the black hole's mass and volume. The spectra of the stars have also revealed some surprises, such as the "paradox of youth": Most of the neighbors of Sagittarius A* are young stars, which should not be able to form so close to a black hole. These observations and others are shedding new light on the cosmic chicken-and-egg question: Which came first, the galaxy or the black hole? It looks as if the answer is both, Ghez says: "Whatever forms the galaxy forms the black hole as well."