For the first time in its 78-year history, a female mathematician has won the Fields Medal, the discipline’s most prestigious prize. Maryam Mirzakhani of Stanford University in Palo Alto, California, joined the list of 52 previous winners, along with three other recipients of the prize this year: Artur Avila of the Institute of Mathematics of Jussieu in Paris, Manjul Bhargava of Princeton University, and Martin Hairer of the University of Warwick in the United Kingdom. In all, the International Mathematical Union (IMU) gave out eight prizes in opening ceremonies at its quadrennial International Congress of Mathematicians on 13 August in Seoul.
"This is a great honor. I will be happy if it encourages young female scientists and mathematicians," Mirzakhani told Bjorn Carey of the Stanford Report. "I am sure there will be many more women winning this kind of award in coming years."
Mirzakhani has polished off numerous problems related to Riemann surfaces and their associated "moduli" spaces. Named for the 19th century mathematician Bernhard Riemann, who propelled geometry into realms of abstraction, Riemann surfaces are complex doughnuts and pretzels characterized topologically according to the number of "handles" they possess. Moduli spaces govern the ways in which one Riemann surface can be deformed to look like another. The geometry of these higher dimensional spaces in some sense controls the geometric properties of the underlying Riemann surfaces.
Mirzakhani showed exactly how the control works for a particular long-standing problem: counting the number of simple closed geodesics on a Riemann surface. These curves are hyperbolic analogs of the familiar great circle (e.g., equator) on a sphere, the key difference being that closed geodesics on a Riemann surface can come in many different lengths. Mirzakhani deduced a formula for the number of simple geodesics up to a given length from a calculation of volume in moduli space. One consequence of this result was a new proof of a theorem about moduli spaces first conjectured in a theoretical physics setting by Edward Witten of the Institute for Advanced Study in Princeton, New Jersey, himself a Fields medalist.
Avila won for work in the theory of dynamical systems. He is particularly known for working on hard problems in collaboration with other researchers. His collaborative approach, the IMU citation says, "is an inspiration for a new generation of mathematicians." Avila and colleagues have brought a unified understanding to a wide class of systems that exhibit behavior ranging from regular to chaotic. He has also applied the theory of dynamical systems to solve a long-standing problem in analysis stemming from quantum mechanics. Physicists had long suspected that the energy spectrum of an electron in a strong magnetic field is, mathematically, a fractal known as a Cantor set—a poser that came to be called the "10-martini problem" after a bounty that was offered for its solution. Avila and Svetlana Jitomirskaya at the University of California, Irvine, proved this indeed to be the case.
Bhargava has developed new methods in the theory of numbers, in part by going back to the origins of the subject in a highly influential but now little-read book by Carl Friedrich Gauss, published in 1801, with the Latin title Disquisitiones Arithmeticae. Gauss's book includes a method for combining (or "composing") two quadratic polynomials in two variables, each of the form ax2 + bxy + cy2 with integer coefficients a, b, and c, to obtain a third quadratic polynomial. Bhargava saw a way to reformulate and simplify Gauss's calculations. In the process he discovered a slew of new composition laws for polynomials of higher degree—laws that number theorists had not even suspected existed. More recently, he has turned his expertise in the geometry of numbers to questions of whether "typical" polynomial curves of higher degree possess points with rational coordinates. This includes the important class known as elliptic curves, which were central in the proof of Fermat's Last Theorem.
Hairer's work has been in the area of stochastic partial differential equations (PDEs). PDEs, which describe continuous motion in time and space, are the basis for almost all mathematical physics, as well as applications ranging from biology to economics. Stochastic PDEs are ones that incorporate an element of random noise, such as occurs, presumably, in the ups and downs of stock prices. In some cases, the stochasticity is in conflict with the normal mathematical meaning of a differential equation: It calls for solutions that are simultaneously rough and smooth. Hairer has found a new approach, called the theory of regularity structures, which gives precise mathematical meaning to a broad class of stochastic PDEs and their solutions. His starting point was an equation from mathematical physics that describes the interface between air and droplets of a liquid crystal material, in which the molecules in the droplets move at random. In broad outline, his approach assumes the noise is at a microscopic but not infinitesimal level, which restores the smoothness needed for the differential equation to have a solution. He uses this "regularized" solution as the starting point for a process that converges to a solution of the original, unregularized equation. What's crucial is that the process gives the same answer no matter how the regularization starts out.