Dear Commons Community,
The online version of Nature is reporting that for the first time a woman will be the recipient of the Field Medal, considered the highest honor in mathematics. Maryam Mirzakhani, a native of Iran, and a professor at Stanford University, is one of four winners for 2014. She won for her work on “the dynamics and geometry of Riemann surfaces and their moduli spaces.” As reported in Nature:
“The International Mathematical Union (IMU) has revealed the winners of the 2014 Fields medals, considered the highest honor in mathematics. The four young medallists — including Maryam Mirzakhani, the first female winner since the prizes were established in 1936 — have been selected for their contributions to topics ranging from dynamical systems to the geometry of numbers and the solution of equations of the type that describe many physical phenomena.
The IMU had planned to publicly announce the names on 13 August in Seoul at the International Congress of Mathematicians, but — owing presumably to a technical glitch — the page with the announcement was already live on the organization’s website in 12 August. In addition to the medal itself, the Fields Institute, based in Toronto, awards each winner CAN$15,000 (US$13,700) in cash. The prizes are given to researchers aged 40 years or younger every four years. Although the prizes recognize outstanding achievement in the early stages of a career, they are also seen as an indicator of mathematicians to watch in the future.
“Perhaps Maryam’s most important achievement is her work on dynamics,” says Curtis McMullen of Harvard University. Many natural problems in dynamics, such as the three-body problem of celestial mechanics (for example, interactions of the Sun, the Moon and Earth), have no exact mathematical solution. Mirzakhani found that in dynamical systems evolving in ways that twist and stretch their shape, the systems’ trajectories “are tightly constrained to follow algebraic laws”, says McMullen.
He adds that Mirzakhani’s achievements “combine superb problem-solving ability, ambitious mathematical vision and fluency in many disciplines, which is unusual in the modern era, when considerable specialization is often required to reach the frontier”.
The other three winners of the Fields Medals were:
Artur Avila of the Institute of Mathematics of Jussieu in Paris, whose “profound contributions to dynamical systems theory have changed the face of the field”. Dynamical systems evolve over time, often in complex ways: they include celestial mechanics (the orbits of bodies in the Solar System); financial markets; the weather; and populations in ecosystems. Some of these systems exhibit chaotic behavior: their state at some future time cannot be predicted from their state at an earlier time, because imperceptible differences that exist now can cause big divergences later.
Avila is a problem-solver who has worked on many aspects of this field. According to mathematician Étienne Ghys of the École Normale Supérieure in Lyon, France, “he didn’t just do one or two things, but has an incredible number of achievements”. Avila has contributed in particular to complex one-dimensional dynamics — how a mathematical function with a single variable evolves when the solution is “fed back” into the function as the next value of the variable in an iterative process. This is how some fractal mathematical objects, such as the Mandelbrot set — a set of complex numbers for which a particular iterated equation does not approach infinity — are generated. Avila has also looked at the Schrödinger equation of unusual quantum systems, although Ghys attests that his motivation is more mathematical than physical.
Manjul Bhargava of Princeton University in New Jersey, who “has developed powerful new methods in the geometry of numbers”. The geometry of numbers is a method for estimating how many points with integer coordinates (meaning their coordinates are whole numbers) are contained in a region of space of a given volume. “Bhargava has added a host of new techniques to this subject,” says Benedict Gross of Harvard University in Cambridge, Massachusetts. “His style is completely original, and combines the three main areas of pure mathematics — algebra, geometry and analysis — in a beautiful way.”
Gross says that one of Bhargava’s key results “gives an entirely new perspective on finding rational solutions [solutions that are ratios of integers] to cubic equations, a problem that goes back to the mathematics of antiquity”. Bhargava says that the award is “a source of encouragement and inspiration, hopefully also for my students, collaborators and colleagues”.
Martin Hairer of the University of Warwick, UK, who “has made outstanding contributions to the theory of stochastic partial differential equations”. Partial differential equations (PDEs), which describe the evolution of quantities that depend on several different variables, crop up throughout science — from quantum physics to the propagation of heat. PDEs are stochastic if they include some element of randomness.
By developing methods for systematically solving such equations, Hairer’s work “allows one to give a rigorous meaning to a bunch of stochastic PDEs”, says Ofer Zeitouni of the University of Minnesota in Minneapolis. The award is “an extraordinary recognition and I am very humbled by it”, says Hairer. “It still doesn’t quite feel real, even though I’ve known about it for almost half a year now.”
Congratulations to all four winners!
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