Top honours in Mathematics this year set new records R. Ramanujam, The Institute of Mathematical Sciences, Chennai The prize The Fields Medal is often called the "Nobel Prize of Mathematics", to indicate international recognition by the community of mathematicians, and in that sense, the highest honour. The name is in honour of Canadian mathematician John Charles Fields, who was instrumental in establishing the award, designing the medal itself, and providing the initial money. It was first awarded in 1936, and since 1950, it is awarded every four years, at the ICM, the International Congress of Mathematicians. <> There are big differences from the Nobel prize, however. For one thing, the money is much less for the Fields, being less than 1/10th of the Nobel. It is awarded only once in 4 years. Most importantly, it is awarded only to young mathematicians, not more than 40 years of age. This is because Fields wanted it to to be an encouragement for further achievement on the part of the recipients. The names of the 2014 Fields Medal recipients were announced on August 12, 2014 at the ICM in Seoul, South Korea. There is much to rejoice about in this news, we explain why below. The recipients This year's winners are: Artur Avila (Brazil and France), Manjul Bhargava (USA), Martin Hairer (England) and Maryam Mirzakhani (USA). They are all excellent young mathematicians who have made fundamental contributions to mathematics. But there are many special things in this list to note. Maryam Mirzakhani is the first woman ever to win the Fields Medal, and moreover, she is a woman of Iranian origin. In times, when girls in Islamic countries are being barred from school education, the role model Mirzakhani can provide may be immensely influential in West Asia. Mathematics is considered a "man's subject" by many, without reason, and a woman winning the top honour may silence such voices. For Indians there is some pride in Manjul Bhargava, a person of Indian origin, winning the Fields Medal. Bhargava holds an associate position at the Tata Institute of Fundamental Research in Mumbai and visits India frequently. His work may inspire many Indian students to pursue deep mathematics. Artur Avila is the first person from the South American continent to win the Fields Medal. The top honour for a Brazilian is a source of pride for people in the developing world. That all his main research was done in Brazil is important as well. Nevanlinna Prize Since 2002, the ICM also announces another award: the Nevanlinna Prize, for fundamental contributions to theoretical computer science, increasingly considered a branch of mathematics in its own right. The 2014 recipient of this prize is Subhash Khot, an American computer scientist of Indian origin, from Maharashtra. There is something noteworthy about this prize as well. Quite apart from the fact that it goes to one who grew up in India, this award honours problem posing rather than problem solving. Khot has been honoured for his Unique Games Conjecture, which, if true, would have important implications in a wide variety of subjects -- not only in computer science and optimization, but also the geometry of foams, the relationship between different ways of measuring distance, and even the stability of different election systems. Formulating a problem that has such far reaching implications shows deep insight, and it is for this insight that 36-year old Khot has been awarded the medal. Music of numbers When Manjul Bhargava was only 8 years old, he became curious about an everyday problem. He would stack oranges into pyramids before they were crushed to make juice. Could there be a general formula for the number of oranges in such a pyramid? After wrestling with this question for several months, he figured it out: if a side of a triangular pyramid has length n, the number of oranges in the pyramid is n(n+1)(n+2)/6. <> In his twenties, as a brilliant young mathematician at Princeton, Bhargava started developing an arsenal of techniques for understanding the “geometry of numbers”. In recent years, Bhargava has worked on elliptic curves, a type of equation whose highest exponent is three. These curves are one of the central objects in number theory. They were crucial to the proof of Fermat’s Last Theorem, for example, and also have applications in cryptography. <> Manjul Bhargava, 40 years old now, is also a musician, deeply interested in Indian classical music, and plays the tabla. He is also passionately interested in Sanskrit poetry, has been visiting his grandparents in Jaipur regularly. Bhargava says he was greatly influenced and inspired by his mother, a professor of mathematics herself. Hyperbolic doughnuts It is about a century and a half since hyperbolic surfaces were discovered, and have become important objects of study in mathematics and physics. These are doughnut-shaped surfaces with two or more holes that have a non-standard geometry which, roughly speaking, gives each point on the surface a saddle shape. <> When Maryam Mirzakhani started her research, some of the simplest questions about such surfaces were yet unanswered. One concerned straight lines, or “geodesics,” on a hyperbolic surface. In her Ph.D. thesis, completed in 2004, Mirzakhani developed a formula for how the number of simple geodesics of length L grows as L gets larger. Along the way, she built connections to two other major research questions, solving both. One concerned a formula for the volume of the so-called “moduli” space — the set of all possible hyperbolic structures on a given surface. The other was a surprising new proof of an old conjecture proposed by the physicist Edward Witten about topological measurements of moduli spaces related to string theory. Farb, a mathematician, says that solving each of these problems “would have been an event, and connecting them would have been an event.” Mirzakhani did all three. <> In recent years, Mirzakhani and Eskin have been studying dynamical systems and answering geometrical questions in the area. Their work has been hailed as `the beginning of a new era', bringing very powerful new techniques to the study. As a child growing up in Teheran, Iran, Maryam was more interested in fiction than mathematics, and wanted to be a writer. In fact, initially she did not do well in mathematics in high school, and it was the influence of a teacher whom she liked very much that got her interested in the subject. Mirzakhani finished elementary school just as the Iran-Iraq war was drawing to a close and opportunities were opening up for motivated students. She took a placement test that secured her a spot at the Farzanegan middle school for girls in Tehran. There she made friends with Roya Beheshti, who is now a mathematics professor in the USA as well. Together they used to work hard on challenging problems. One day the two girls marched into their school principal's office and demanded that she arrange for math problem-solving classes like the ones being taught at the comparable high school for boys. At that time, Iran’s International Mathematical Olympiad team had never fielded a girl. The principal was impressed and went about arranging problem-solving classes for them. At the age of 17, Mirzakhani and Beheshti made it to the Iranian math Olympiad team. Her score on the Olympiad test earned her a gold medal. The following year, she returned and achieved a perfect score. After completing an undergraduate degree in mathematics at Sharif University in Tehran in 1999, Mirzakhani went to graduate school at Harvard University, where her talents were recognized and appreciated by her adviser, McMullen. He says, she used to keep coming to his office with lots of questions, writing down notes in Farsi. When Mirzakhani, 37 years old now, got an e-mail saying she was getting the Fields medal, she thought it was "spam" (fraud mail). She says that her ambitious teenage self would have been overjoyed by the award, but today, she is eager to deflect attention from her achievements so that she can focus on research. Her 3-year old daughter Anahita considers her a painter since she is used to scrawling and doodling pictures all the time when she works. Understanding chaos To produce complex behaviour, it is not necessary to start with complex rules. Even simple rules when repeated again and again sometimes produce chaos: random-seeming, unpredictable behaviour in which tiny changes in the starting conditions can produce dramatically different outcomes. One of the first simple systems in which chaotic behaviour was discovered is the so-called “logistic” model of population growth, which gives a precise formulation for how a population will change from year to year. Artur Avila has written what could be considered the final chapter of this story. In nature, a small population often grows quickly because there is an abundance of resources; a larger population will grow more slowly or even decline as resources are stretched too thin. In 1838, the Belgian mathematician Pierre Verhulst captured this intuition in the logistic equation for population growth. The graph of the logistic equation is simply an upside-down parabola that rises quickly if the population is small but drops precipitously if the population is larger than the environment can sustain. As a population changes in time, it will move around on the parabola — a small population may become large the next year and a large one small. <> In the mid-1970s, mathematicians discovered many families of equations with the same basic shape as the upside-down parabola (called unimodal maps) had similar characteristics. Avila solved a fundamental problem about unimodal maps. <> But that was just the beginning. Avila began diving into other areas of dynamical systems, using his renormalization technique to solve one important problem after another. In recent years, he has also studied the evolution of quantum states in physical systems governed by “quasi-periodic Schroedinger operators,” which are crude models for quasicrystals, structures that have more order than a liquid but less than a crystal. When Avila was a child, his parents had never even heard of mathematicians. His father’s formal education growing up in the rural Amazon did not start until his teenage years. Avila was enrolled in a Catholic school at the age of 6. He focused on mathematics to the exclusion of almost everything else — he often did poorly in other subjects and was expelled after the eighth grade for refusing to take mandatory religion exams. Thankfully a master teacher took interest in Avila, and helped him transfer to a new school. Two years later, he took gold at the International Mathematical Olympiad. Through the math competitions, Avila discovered IMPA, the institute where Brazil held its Olympiad award ceremonies each year. There, he met prominent mathematicians and while still technically in high school, he began studying graduate level mathematics. Avila started his PhD when he was 19 and completed it when he was 21, and his research was entirely done in Brazil. He is 35 years old now, and spends half his time in Paris and half in Rio de Janeiro. Regularity in Randomness A class of equations arises frequently in physics. These equations are mathematical abstractions of growth, the hustle and bustle of elementary particles and other “stochastic” processes, which evolve amid environmental noise. These stochastic partial differential equations (SPDEs) have tantalized mathematicians for decades. Martin Hairer’s theory of “regularity structures” brings order to SPDEs by broadening many of the most basic concepts of mathematics: derivatives, expansions and even what it means to be a solution. The problem with SPDEs is that they involve supremely thorny mathematical objects called “distributions”, which do not easily submit to arithmetic operations like squaring etc. What Hairer has done is to bring order into this world, make them manageable. <> If you think of a mathematician as someone incapable of working with machines, Martin Hairer is your counter-example. He has an entire career outside of mathematics. A lover of rock music and computer programming, Hairer is the creator of an award-winning sound-editing program called Amadeus, a popular tool among deejays, music producers and gaming companies and a lucrative sideline for Hairer. Indeed, it was his knowledge of a signal compression technique used in audio and image processing that inspired his otherworldly new theory. Born into an Austrian family living in Switzerland, Hairer spent most of his childhood in Geneva. For his 12th birthday, in 1987, Martin’s father bought him a pocket calculator that could execute simple, 26-variable programs. He was instantly hooked. The next year, he persuaded his younger brother and sister to go in with him on a joint birthday present: a Macintosh II. He quickly became a proficient programmer, creating visualizations of fractals like the Mandelbrot set and then, at age 14, developing a program for solving ordinary differential equations — the much simpler cousins of SPDEs. At 16, Martin Hairer was interested in the physics of sound as well as Pink Floyd and The Beatles. He enjoyed recording musical notes and looking at the resulting waveforms on his computer and tried to write a program that could extract the notes from the recordings. The task was too difficult, but he ended up with a program for manipulating the recordings: version 1 of Amadeus. Pulled in different directions by mathematics, physics and computer science, Hairer only settled on mathematics in his early 20s. Outside of mathematics, 38-year old Hairer enjoys skiing, cooking and reading thrillers. His wife is from China and they apparently enjoy cooking up Western - Chinese fusion dishes. A bold conjecture One of the most interesting questions is the boundary between the tractable and the intractable. In the last 30 years of the 20th century, computer scientists had shown that hundreds of important computational challenges belong to a category called “NP-hard” problems, which most computer scientists believe cannot be solved exactly by any algorithm that runs in a reasonable amount of time. So they shifted their focus to exploring efficient algorithms that find good approximate solutions to these difficult problems. In 1992, a team of computer scientists proved a result called the PCP theorem, which enabled researchers to show that for a wide variety of computational problems, even finding good approximate solutions is NP-hard, meaning that it’s a task that, most computer scientists believe, is impossible to carry out efficiently. When Subhash Khot started his doctoral research he worked on how hard it is to find approximate solutions. He realized that one of his problems got much simpler if he made a certain assumption about how difficult a certain approximation problem is. He soon saw that several of his other problems also became easier if he made the same assumption. He eventually named this assumption the Unique Games Conjecture. The conjecture is not easy to state, so we will not do so. <> Khot's paper describing the Unique Games Conjecture appeared in 2002. Within two years, everyone was studying the implications of the Unique Games Conjecture. There was a flood of approximation hardness results — theorems of the form, “If the Unique Games Conjecture is true, then it’s NP-hard to approximate the solutions of problem X any closer than Y percent.” It remains to be seen whether computer scientists will be able to prove or disprove Khot’s Unique Games Conjecture. A proof would be a boon to computer scientists, but a disproof might be even more exciting. Researchers agree that disproving the conjecture would probably require innovative new algorithmic techniques that could unlock a host of different approximation problems. Khot himself doesn’t expect someone to prove or disprove his conjecture any time soon. Khot grew up in Ichalkaranji, a small town in Maharashtra, where he won many competitions. At 16, he was national topper of JEE, the entrance exam for the IIT's, and went to study computer science, without ever having touched a computer before. At IIT-Bombay, he realised that he wanted to focus on theoretical computer science.