## Markovic Elected to Great Britain's Royal Society

Vladimir Markovic, the John D. MacArthur Professor of Mathematics at Caltech, has been named a fellow of Great Britain's Royal Society. He is one of 50 new fellows and 10 foreign members elected in 2014. Markovic's election brings to seven the number of fellows and foreign members of the Royal Society currently on the Caltech faculty.

Membership in the Royal Society is bestowed each year on a small number of the world's scientists. The oldest scientific academy in existence, the Royal Society was established in 1660 under the patronage of King Charles II for the purpose of "improving natural knowledge," and it helped usher in the age of modern science. Today, the society seeks to promote science leaders who champion innovation for the benefit of humanity and the planet.

Markovic studies the shapes and structures of mathematical spaces called manifolds. A line is a one-dimensional manifold while a plane would be two-dimensional. In its citation for Markovic, the Royal Society wrote, "Markovic is a world leader in the area of quasiconformal homeomorphisms and low dimensional topology and geometry. He has solved many famous and difficult problems. With Jeremy Kahn, he proved William Thurston's key conjecture that every closed hyperbolic 3-manifold contains an almost geodesic immersed surface."

In 2004, Markovic received awards recognizing his early career achievements from the London Mathematical Society and the Leverhulme Trust. In 2012, he was awarded the Clay Research Award. Earlier this year, he was an invited speaker at the International Congress of Mathematics in Seoul, South Korea.

Born in Germany, Markovic earned a BSc and PhD from the University of Belgrade in Serbia in 1995 and 1998, respectively. Before joining the Caltech faculty as a professor in 2011, he was an assistant professor at the University of Minnesota, an associate professor at SUNY Stony Brook, and a professor at the University of Warwick. He was named Caltech's John D. MacArthur Professor of Mathematics in 2013.

Markovic is currently on leave, teaching at the University of Cambridge.

## To Supernova or Not to Supernova: A 3-D Model of Stellar Core Collapse

What happens when massive stars collapse? One potential result is a core-collapse supernova. Astronomers can make observations of such events that tell us what is happening on the surface of a star when it explodes in a supernova, but it is considerably more difficult to know what is driving the process inside the star at its hot, dense core.

Astrophysicists attempt to simulate these events based on the properties of different kinds of stars and knowledge of the fundamental interactions of mass and energy, hopefully providing astronomers with ready predictions that can be tested with observational data.

In a recent publication*,* Caltech postdoctoral scholar Philipp Mösta and Christian Ott, professor of theoretical astrophysics, present a three-dimensional model of a rapidly rotating star with a strong magnetic field undergoing the process of collapse and explosion . . . or at least trying to.

Stars with a very rapid spin and a strong magnetic field are comparatively rare: no more than one in a hundred massive stars (those at least 10 times the mass of our sun) have these features. According to Mösta and Ott's research, when these bodies undergo core collapse, small perturbations around its axis of rotation may inhibit the process that would ordinarily lead to a supernova explosion.

Previous models of the collapse of rapidly rotating magnetized stellar cores assumed perfect symmetry around the axis of rotation. In effect, these models were two-dimensional. The models yielded the expectation that as these cores collapsed, the strong magnetic field combined with the rapid spin would squeeze the stellar material out into two narrow "jets" along the axis of symmetry, as shown at left.

Assuming perfect symmetry around the axis of rotation can be excused in part as a matter of simplifying the scenario so that it could be simulated on an ordinary computer rather than the kind of supercomputer that Mösta and Ott's three-dimensional simulations require: 20,000 processors to output 500 terabytes—over 500 trillion bytes—of data that represent only some 200 milliseconds in time. But, says Ott, "Even working with paper and pencil, writing down equations and discussing them with other theoretical astrophysicists, we should have known that small perturbations can trigger an instability in the stellar core. Nothing in nature is perfect. As we learn from this model, even small asymmetries can have a dramatic effect on the process of stellar collapse and the subsequent supernova explosion."

When Mösta and Ott took on the ambitious task of simulating a magnetorotational core collapse in three dimensions, they introduced a small asymmetry into their initial conditions: a 1 percent perturbation (a kink) around the axis of symmetry. "You can think of it like the vertebrae in your spine," says Ott. "If one vertebra is slightly offset, there will be greater pressure on one side of the spinal column, and less pressure on the other side. This causes the disk and the material between the vertebrae to be squeezed toward the side with less pressure. The same thing happens when you introduce a kink in the axis of symmetry of a collapsing star with a strong magnetic field."

With an ever-so-slightly distorted magnetic field, the core is still constrained in the middle, just as it is in the axially symmetric model. But instead of producing two perfectly matched jets, the magnetic distortion—it is called a "kink instability"—produces two asymmetric, misshapen lobes, as shown at right. "Even more noteworthy," says Mösta, "is the fact that in the three-dimensional model, the explosion—the supernova—never quite gets off the ground."

This slideshow illustrates the three-dimensional simulation in a step-by-step fashion.

Setting the two simulations—two-dimensional and three-dimensional—alongside one another provide a dramatic visualization of the impact of even a small asymmetry in a rapidly-rotating, magnetized body. In a video that compares the two, 186 milliseconds of core collapse are slowed to fill two minutes of real time. The two events look very similar for about 20 milliseconds, before the kink instability in the three-dimensional model begins to deform the stellar core and constrain its progress toward supernova.

The kink instability in the three-dimensional simulation leads to a "wobbling" of the central funnel of material that is pushed out by the ultra-dense and hot stellar core, a proto-neutron star. "As the material expands, it gets wound in tubes around the spin axis of the star, like water being expelled vigorously from a garden hose left lying on the ground," says Mösta. In the three-dimensional view illustrated here, regions in which the magnetic field pressure dominates are yellow, while regions that are dominated by the normal pressure of the stellar gas are blue, red, and black.

Unlike the two-dimensional, axially symmetric simulations with their uniform jets along the axis of rotation, the three-dimensional simulations of Mösta and Ott result in two lobes of twisted and highly magnetized material that are only slowly pushed outward and do not show signs of a runaway explosion at the end of the simulation. More and longer simulations on more powerful supercomputers will be needed to determine the final fate of core collapse in a rapidly rotating magnetized star.

"We can be smarter in our simulations now," says Ott. "We are wrestling with a more interesting if less perfect universe—the one we actually live in."

The paper, "Magnetorotational Core-collapse Supernovae in Three Dimensions," appeared in the April 3, 2014, issue of *Astrophysical Journal Letters* and is authored by Mösta, Ott, Sherwood Richers, Roland Haas, Anthony Piro, Kristen Boydstun, Ernazar Abdikamalov, and Christian Reisswig, all of Caltech, and Erik Schnetter of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. This research was funded by the National Science Foundation, the Sherman Fairchild Foundation, the U.S. Department of Energy, and NASA.

## Simulating Milliseconds of Stellar Collapse: A Conversation with Christian Ott

Theoretical astrophysicists infer what sort of physical processes might cause the observed behavior of the universe; observational astrophysicists—astronomers—observe the universe to determine what is out there and how it is behaving.

Theoretical and observational astrophysics overlap more often than you might think. Astrophysicists with their varying specializations are in constant conversation with one another, weighing theory against observation and vice versa. Certainly this is true in the area of gravitational waves, first theorized by Albert Einstein nearly a hundred years ago as part of his general theory of relativity. While gravity is weak compared to other forces in the universe, gravitational waves actually squeeze and ripple space-time, creating physical effects in the universe that have not been successfully explained by any other mechanism.

There is excellent observational evidence for the existence of gravitational waves, including the behavior of the Hulse-Taylor pulsar, a binary system first discovered in 1974, and the recent finding by Caltech professor Jamie Bock and his coauthors that the cosmic microwave background has a polarization pattern specific to the gravitational waves that would have been released during the period of rapid inflation at the beginning of the universe. As of today, however, gravitational waves have not been directly detected, though not for want of trying. The Laser Interferometer Gravitational-wave Observatory (LIGO), a collaboration between Caltech and the Massachusetts Institute of Technology, is currently being refitted with a new technology called Advanced LIGO. When Advanced LIGO goes online in 2015, there is hope that it will be able to directly detect gravitational waves as they come to the earth.

Christian Ott, professor of theoretical astrophysics at Caltech, is eagerly awaiting data from Advanced LIGO. Ott formulates scenarios for what happens when stars collapse, and one result of stellar collapse is the rapid release of gravitational waves, just the kind that LIGO hopes to detect.

Much about the collapse of massive stars is well understood. But there are crucial hundreds of milliseconds in this process that determine whether a star will collapse into a black hole or into a neutron star, and these milliseconds are still a matter of highly educated and informed speculation. It is these fractions of a second that consume Ott's interest. His scenarios for stellar collapse are stories told with multiple terabytes of computer memory and petaflops of computing power—stories that are plausible, but whose truth is still unknown. One day detections of gravitational waves will help to confirm or contradict the models of stellar collapse that Ott is creating.

**How did you get interested in astrophysics?**

I've had an interest in this since I was a child growing up near Frankfurt, Germany. My father was an amateur astronomer. We had a small telescope at home, and we would look at the stars and the planets and the moon. After high school I chose to go to Heidelberg University to study physics and astronomy. As a freshman I read a book by Kip Thorne (Richard P. Feynman Professor of Theoretical Physics, Emeritus) in German translation: *Black Holes and Time Warps.* He has a way of explaining these things so that even a layperson can understand them, and I became fascinated with black holes, neutron stars, and regions of strongly curved space-time. Honestly, Caltech seemed to me to be some mythical place. I wasn't even daring to dream about a place like this, and now I'm a professor here. It seems crazy to me.

**What spurred your interest in gravitational waves?**

Heidelberg University has an exchange program with the University of Arizona, so I came to spend a year there during college. Shortly after I arrived, I was telling a graduate student about my interest in neutron stars and black holes, and he recommended that I talk to Professor Adam Burrows, now at Princeton University. I wasn't too excited about gravitational waves at that point. I remember that quite well. But Professor Burrows set me to work on calculating gravitational waves from supernovae. That was 2001, and I've been working on similar questions ever since.

**What do you find exciting about supernovae?**

What most people don't realize is that without supernova explosions, we wouldn't be here.

There are two kinds of supernova explosions. Type Ia, those that come from white dwarf stars, are responsible for about 80 percent of the iron in the universe, and core-collapse supernovae, or Type II, which come from massive stars, are responsible for the remaining 20 percent of the iron. Without supernovae, there wouldn't be iron for our blood; there wouldn't be iron in Earth's core; there wouldn't be iron to make steel. Type II supernovae are also responsible for most of the oxygen and carbon in the universe. Without this enrichment of heavy elements, there would be no life, there would be no planets . . . it would be a pretty boring place.

So blowing stuff up and chemically polluting the universe, as supernovae do, is crucially important. But for fundamental physics, it's actually more interesting to examine the collapse itself.

The physics of stars up to that time is pretty well understood: we know where the pressure comes from in the iron core of a star; we know about thermonuclear reactions. However, as a star collapses the core becomes unbelievably dense. Eventually the electrons, which are exerting pressure in the opposite direction of gravity, are themselves squeezed out in a process called electron capture. In electron capture, a proton and an electron combine to make a neutron and a neutrino, a tiny subatomic particle with no electrical charge. When all of the neutrons and protons are packed together that tightly, the nuclear force kicks in. Usually the nuclear force binds protons and neutrons together, but when you try to squeeze protons and neutrons too close to one another, the nuclear force acts in the opposite direction: it has the effect of an outward pressure against the gravitational pull of a collapsing star. We don't understand this mechanism very well, but if we didn't have the nuclear force, all stars would collapse to black holes. There would be no neutron stars or supernovae. As it is, there are three outcomes we know of when stars collapse: Stars can collapse directly into black holes with no supernova; they can experience a weak supernova and a collapse into a neutron star that then collapses into a black hole within hours or days; or there can be a strong supernova that leaves a neutron star behind, apparently forever.

**What determines whether stellar collapses result in neutron stars rather than black holes?**

You tell me.

**You don't know?**

It's what we call "an area of active research." Advanced LIGO should help us to answer this question. When you see supernovae with telescopes, you're looking at optical waves, and these come pretty late in the process of stellar collapse; it's not easily connected to what's actually happening deep inside the star. A star collapse is a highly energetic event and should create substantial gravitational waves. When we detect gravitational waves, we will get information about what is going on earlier in the process. Depending on the precise shape—the amplitudes and frequencies—of the gravitational waves we detect, we can get a finer sense of exactly what is happening in the core of a star when it collapses.

Gravitational waves would arrive on Earth up to a day before we would see the light from the supernova, depending on how far away from us the supernova occurs. The same is true of neutrinos. Although neutrinos are remarkably tiny, a supernova produces an enormous quantity of neutrinos that fly out into the universe. When a star collapses, 99 percent of the gravitational energy released goes into neutrinos; only a tiny portion of the remainder takes the form of gravitational waves. We can already detect neutrinos on Earth, and we have even detected them directly from a supernova in 1987 that occurred in the Large Magellanic Cloud, a neighbor galaxy of our Milky Way. If a stellar collapse occurs anywhere near us, we should detect tens of thousands of neutrinos.

**So if you detected these specific gravitational waves or a lot of neutrinos, you could alert the entire scientific community to point their telescopes at the sky the next day to see the supernova?**

No! I would tell everyone to turn their big telescopes away, so the instruments would not be destroyed! Imagine if Betelgeuse blows up in a supernova—it's a red supergiant star twenty times the mass of the sun. If that goes, it's going to be as bright as the full moon for an entire month. At the very least, astronomers would need to put filters on their telescopes to protect them from the intense light.

## Fu, Harrison, and Preskill Elected to the National Academy of Sciences

Three professors at Caltech have been elected to the prestigious National Academy of Sciences. The announcement was made Tuesday, April 29, in Washington D.C.

The new Caltech electees are Gregory C. Fu, Altair Professor of Chemistry; Fiona A. Harrison, Benjamin M. Rosen Professor of Physics; and John P. Preskill, Richard P. Feynman Professor of Theoretical Physics.

Fu is a synthetic organic chemist focusing on transition-metal catalysis and nucleophilic catalysis. He is currently developing enantioselective reactions and exploring the use of copper and nickel catalysts. In 2012, Fu won the Award for Creative Work in Synthetic Organic Chemistry from the American Chemical Society. He is a fellow of both the American Academy of Arts and Sciences (2007) and the Royal Society of Chemistry (2005).

Harrison specializes in observational and experimental high-energy astrophysics. She is the principal investigator for NASA's NuSTAR Explorer Mission. Harrison is recognized for her leadership in the design, development and launch of NuSTAR, as well as leading the team in the mission's scientific return. As a result of almost two decades of technology development, NuSTAR is revolutionizing our view of the high-energy X-ray sky. Harrison was elected to the American Academy of Arts and Sciences in 2014, was elected as a fellow of the American Physical Society in 2012, and won a NASA Outstanding Public Leadership Medal in 2013.

Preskill is a theoretical physicist who began his career in particle physics (in particular, the interface between particle physics and cosmology) before moving to a specialization in quantum information and quantum computing. In 2000, Preskill founded the Institute for Quantum Information with the aim of harnessing principles of quantum mechanics to aid in particularly challenging information-processing tasks. He is a fellow of the American Physical Society.

The National Academy of Sciences is a private organization of scientists and engineers dedicated to the furtherance of science and its use for the general welfare. It was established in 1863 by a congressional act of incorporation signed by Abraham Lincoln that calls on the academy to act as an official adviser to the federal government, upon request, in any matter of science or technology.

The election of Fu, Harrison, and Preskill brings the total Caltech membership to 75 faculty and three trustees.

## The Intergalactic Medium Unveiled: Caltech's Cosmic Web Imager Directly Observes "Dim Matter"

Caltech astronomers have taken unprecedented images of the intergalactic medium (IGM)—the diffuse gas that connects galaxies throughout the universe—with the Cosmic Web Imager, an instrument designed and built at Caltech. Until now, the structure of the IGM has mostly been a matter for theoretical speculation. However, with observations from the Cosmic Web Imager, deployed on the Hale 200-inch telescope at Palomar Observatory, astronomers are obtaining our first three-dimensional pictures of the IGM. The Cosmic Web Imager will make possible a new understanding of galactic and intergalactic dynamics, and it has already detected one possible spiral-galaxy-in-the-making that is three times the size of our Milky Way.

The Cosmic Web Imager was conceived and developed by Caltech professor of physics Christopher Martin. "I've been thinking about the intergalactic medium since I was a graduate student," says Martin. "Not only does it comprise most of the normal matter in the universe, it is also the medium in which galaxies form and grow."

Since the late 1980s and early 1990s, theoreticians have predicted that primordial gas from the Big Bang is not spread uniformly throughout space, but is instead distributed in channels that span galaxies and flow between them. This "cosmic web"—the IGM—is a network of smaller and larger filaments crisscrossing one another across the vastness of space and back through time to an era when galaxies were first forming and stars were being produced at a rapid rate.

Martin describes the diffuse gas of the IGM as "dim matter," to distinguish it from the bright matter of stars and galaxies, and the dark matter and energy that compose most of the universe. Though you might not think so on a bright sunny day or even a starlit night, fully 96 percent of the mass and energy in the universe is dark energy and dark matter (first inferred by Caltech's Fritz Zwicky in the 1930s), whose existence we know of only due to its effects on the remaining 4 percent that we *can* see: normal matter. Of this 4 percent that is normal matter, only one-quarter is made up of stars and galaxies, the bright objects that light our night sky. The remainder, which amounts to only about 3 percent of everything in the universe, is the IGM.

As Martin's name for the IGM suggests, "dim matter" is hard to see. Prior to the development of the Cosmic Web Imager, the IGM was observed primarily via foreground absorption of light—indicating the presence of matter—occurring between Earth and a distant object such as a quasar (the nucleus of a young galaxy).

"When you look at the gas between us and a quasar, you have only one line of sight," explains Martin. "You know that there's some gas farther away, there's some gas closer in, and there's some gas in the middle, but there's no information about how that gas is distributed across three dimensions."

Matt Matuszewski, a former graduate student at Caltech who helped to build the Cosmic Web Imager and is now an instrument scientist at Caltech, likens this line-of-sight view to observing a complex cityscape through a few narrow slits in a wall: "All you would know is that there is some concrete, windows, metal, pavement, maybe an occasional flash of color. Only by opening the slit can you see that there are buildings and skyscrapers and roads and bridges and cars and people walking the streets. Only by taking a picture can you understand how all these components fit together, and know that you are looking at a city."

Martin and his team have now seen the first glimpse of the city of dim matter. It is not full of skyscrapers and bridges, but it is both visually and scientifically exciting.

The first cosmic filaments observed by the Cosmic Web Imager are in the vicinity of two very bright objects: a quasar labeled QSO 1549+19 and a so-called Lyman alpha blob in an emerging galaxy cluster known as SSA22. These objects were chosen by Martin for initial observations because they are bright, lighting up the surrounding IGM and boosting its detectable signal.

Observations show a narrow filament, one million light-years long, flowing into the quasar, perhaps fueling the growth of the galaxy that hosts the quasar. Meanwhile, there are three filaments surrounding the Lyman alpha blob, with a measured spin that shows that the gas from these filaments is flowing into the blob and affecting its dynamics.

The Cosmic Web Imager is a spectrographic imager, taking pictures at many different wavelengths simultaneously. This is a powerful technique for investigating astronomical objects, as it makes it possible to not only see these objects but to learn about their composition, mass, and velocity. Under the conditions expected for cosmic web filaments, hydrogen is the dominant element and emits light at a specific ultraviolet wavelength called Lyman alpha. Earth's atmosphere blocks light at ultraviolet wavelengths, so one needs to be outside Earth's atmosphere, observing from a satellite or a high-altitude balloon, to observe the Lyman alpha signal.

However, if the Lyman alpha emission lies much further away from us—that is, it comes to us from an earlier time in the universe—then it arrives at a longer wavelength (a phenomenon known as redshifting). This brings the Lyman alpha signal into the visible spectrum such that it can pass through the atmosphere and be detected by ground-based telescopes like the Cosmic Web Imager.

The objects the Cosmic Web Imager has observed date to approximately 2 billion years after the Big Bang, a time of rapid star formation in galaxies. "In the case of the Lyman alpha blob," says Martin, "I think we're looking at a giant protogalactic disk. It's almost 300,000 light-years in diameter, three times the size of the Milky Way."

The Cosmic Web Imager was funded by grants from the NSF and Caltech. Having successfully deployed the instrument at the Palomar Observatory, Martin's group is now developing a more sensitive and versatile version of the Cosmic Web Imager for use at the W. M. Keck Observatory atop Mauna Kea in Hawaii. "The gaseous filaments and structures we see around the quasar and the Lyman alpha blob are unusually bright. Our goal is to eventually be able to see the average intergalactic medium everywhere. It's harder, but we'll get there," says Martin.

Plans are also under way for observations of the IGM from a telescope aboard a high-altitude balloon, FIREBALL (Faint Intergalactic Redshifted Emission Balloon); and from a satellite, ISTOS (Imaging Spectroscopic Telescope for Origins Surveys). By virtue of bypassing most, if not all, of our atmosphere, both instruments will enable observations of Lyman alpha emission—and therefore the IGM—that are closer to us; that is, that are from more recent epochs of the universe.

Two papers describing the initial data from the Cosmic Web Imager have been published in the *Astrophysical Journal*: "Intergalactic Medium Observations with the Cosmic Web Imager: I. The Circum-QSO Medium of QSO 1549+19, and Evidence for a Filamentary Gas Inflow" and "Intergalactic Medium Observations with the Cosmic Web Imager: II. Discovery of Extended, Kinematically-linked Emission around SSA22 Lyα Blob 2." The Cosmic Web Imager was built principally by three Caltech graduate students—the late Daphne Chang, Matuszewski, and Shahinur Rahman—and by Caltech principal research scientist Patrick Morrissey, who are all coauthors on the papers. Additional coauthors are Martin, Anna Moore, Charles Steidel, and Yuichi Matsuda.

## Experiences from two years of MOOCs at Caltech: A WEST Public Seminar

## Hyperbolic Homogeneous Polynomials, Oh My!

Cutting-edge mathematics today, at least to the uninitiated, often sounds as if it bears no relation to the arithmetic we all learned in grade school. What do topology and combinatorics and *n*-dimensional space have to do with addition, subtraction, multiplication, and division? Yet there remains within mathematics one vibrant field of study that makes constant reference to basic arithmetic: number theory. Number theory—the "queen of mathematics," according to the famous 19^{th} century mathematician Carl Friedrich Gauss—takes integers as its starting point. Begin counting 1, 2, 3, and you enter the domain of number theory.

Number theorists are particularly interested in prime numbers (those integers that cannot be divided by any number other than itself and 1) and Diophantine equations. Diophantine equations are polynomial equations (those with two or more variables) in which the coefficients are all integers.

It is these equations that are the inspiration for a recent proof offered by Dinakar Ramakrishnan, Caltech's Taussky-Todd-Lonergan Professor of Mathematics and executive officer for mathematics, and his coauthor, Mladen Dimitrov, formerly an Olga Taussky and John Todd Instructor in Mathematics at Caltech and now professor of mathematics at the University of Lille in France. This proof involves homogeneous equations: equations in which all the terms have the same degree. For example, the polynomial *xy *+* z*^{2} has degree 2, and *x*^{2}*yz *+* xy*^{3} has degree 4. If we take an equation like *xy *=* z ^{2}*, one solution for (

*x, y, z*) would be (1, 4, 2). Multiplying that solution by any rational number will give infinitely many rational solutions, but this is a trivial way to get solutions achieved simply by "scaling." These are not the type of answers Ramakrishnan and Dimitrov were searching for.

What Ramakrishnan and Dimitrov showed is that a specific collection of systems of homogeneous equations with six variables has only a finite number of rational solutions (up to scaling). Usually people look for integer solutions of Diophantine equations, but the first approach is to find solutions in rational numbers—those that can be expressed as a fraction of two integers.

Diophantus, after whom the Diophantine equations are named, is best known for his *Arithmetica, *which Ramakrishnan describes as "a collection of intriguing mathematical problems, some of them original to Diophantus, others an assemblage of earlier work, some of it possibly going back to the Babylonians." Diophantus lived in the city of Alexandria, in what is now Egypt, during the third century CE. What makes the *Arithmetica *unusual is that it continues to serve as the basis for some very interesting mathematics more than 1,700 years later.

Diophantus was interested primarily in positive integers. He was aware of the existence of rational numbers, since he knew integers could divide one another, but he seemed to regard negative numbers (which are also rational numbers and can be integers) as absurd and unreal. Present-day number theorists have no such discomfort with negative numbers, but they continue to be as fascinated by integers as Diophantus was. "Integers are very special," says Ramakrishnan. "They are kind of like musical notes on a clavier. If you change a note even slightly, you'll hear a dissonance. In a sense, integers can be thought of as the well-tempered states of mathematics. They are quite beautiful."

Diophantus was especially interested in integer solutions for homogeneous polynomial equations: those in which each term of the equation has the same degree (for example, *x*^{7} + *y*^{7} = *z*^{7} or *x*^{2}*y*^{3}*z* = *w*^{6}). The classic example of a homogeneous polynomial equation is the Pythagorean theorem—*x*^{2} + *y*^{2} = *z*^{2}—which defines the hypotenuse, *z*, the longest side of a right triangle, with respect to the perpendicular sides *x* and *y*. As early as 1600 BCE, the ancient Babylonians knew that there were many integer solutions to this equation (beginning with 3^{2} + 4^{2} = 5^{2}), though it was Pythagoras, a Greek mathematician living in the sixth century BCE, who gave his name to the formula, and Euclid who two centuries later proved that this equation has an infinite number of positive integer solutions, known as "Pythagorean triples" (such as 3, 4, 5; 5, 12, 13; or 39, 80, 89).

In 1637, French mathematician Pierre de Fermat famously wrote in the margin of Diophantus's *Arithmetica* that he had a "truly marvelous proof" showing that although there were an infinite number of positive integer solutions for *x*^{2} + *y*^{2} = *z*^{2}, there were no positive integer solutions at all when the variables were raised to the power of three or higher (*x*^{3} + *y*^{3} = *z*^{3}; *x*^{4} + *y*^{4} = *z*^{4} ; . . . ; *x ^{n}* +

*y*=

^{n}*z*). Fermat did not provide the actual proof; he claimed that the margin of Diophantus's book was too small to contain it. Fermat's conjecture (it was not yet a proof, though Fermat apparently believed he had one in his mind) remained unsolved until the early 1990s, when British mathematician Andrew Wiles created a complicated and unexpected proof that made use of previously unrelated mathematical principles.

^{n}In geometric terms, Fermat's conjecture and Wiles's proof, with their three variables, operate in three-dimensional space and can be described as points on a curve on the projective plane, drawn with *x*, *y*, *z* coordinates up to scaling. By moving to a greater number of variables, Ramakrishnan and Dimitrov are interested in identifying points on so-called hyperbolic surfaces. A hyperbolic surface is a negatively curved space, like a saddle—as opposed to a positively curved space like a sphere—in which the rules of Euclidean geometry no longer apply. A simple example of a hyperbolic surface is given by the simultaneous solution (where the values of the variables are held constant) of three equations: *x*_{1}^{5} + *y*^{5} = *z*^{5}; *x*_{2}^{5} + *w*^{5} = *z*^{5}; and *x*_{3}^{5} + *w*^{5} = *y*^{5}. In the 1980s, German mathematician Gerd Faltings did pioneering work on the mathematics of hyperbolic curves, work that inspired Ramakrishnan and Dimitrov.

Ramakrishnan and Dimitrov's recent finding considers rational-number solutions for several systems of homogeneous polynomial equations describing hyperbolic surfaces. One solution is to set all the variables to zero. This solution is considered trivial; but are there any nontrivial solutions?

There are at least a few nontrivial solutions that Ramakrishnan and Dimitrov use as examples. Their challenge was to determine if there are finitely many or infinitely many rational solutions. They demonstrated—in a proof-by-contradiction that took nearly two years to complete—that the hyperbolic case they consider has only a finite number of solutions.

But, as Ramakrishnan remarks, there is no rest for number theorists, because "even if we solve another bunch of equations, there are still many more that are unsolved, enough for our descendants five hundred years from now."

For Ramakrishnan, this is not a counsel of despair. He continues to find mathematics exciting, especially the concept of the mathematical proof. As he points out, "In other ancient civilizations in the Middle East or India or China, they did some very complicated math, but it was more algorithmic, more related to computer science in my opinion than to philosophy. Whereas the Greeks emphasized proofs, rigorously establishing mathematical truths. There's nothing vague about it."

Apart from the inherent joy of pushing number theory forward through another generation, Ramakrishnan points out that this field has interesting applications in both science and everyday life. "Quite often in science, you are counting. Think of balancing chemical equations such as wCH_{4} + xO_{2} —> yCO_{2} + zH_{2}O, in which methane oxidizes to produce carbon dioxide and water. This is a linear Diophantine equation."

Number theory also plays an important role in encryption. "Every time one visits a website with an https:// address," says Ramakrishnan, "it is likely that the website browser is using an encryption system that validates the certificate for the remote server to which one is trying to connect. The security keys that are exchanged point to a number-theoretic solution. Most people prefer equations with simple solutions, but in some situations, such as encryption, you actually want integer equations that are hard to solve without the key. This is where number theory comes in."

Ramakrishnan and Dimitrov's paper, "Compact arithmetic quotients of the complex 2-ball and a conjecture of Lang," is posted on the math arXiv, a Cornell University Library open e-print archive for papers in physics, mathematics, computer science, quantitative biology, and quantitative finance and statistics.

## Spring Ombudsperson Training

## Jamie Bock Wins George W. Goddard Award

James J. (Jamie) Bock, professor of physics at Caltech and senior research scientist at the Jet Propulsion Laboratory, is the 2014 recipient of the George W. Goddard Award from SPIE, the international society for optics and photonics.

SPIE selected Bock for the award in recognition of his development of sensitive bolometer arrays for studies of distant, dusty galaxies and the cosmic microwave background radiation, leading to their use on the Spectral and Photometric Imaging Receiver (SPIRE) on the Herschel Space Observatory and the High Frequency Instrument (HFI) on the Planck Surveyor spacecraft.

Bock is a principal investigator on a collaboration that successfully measured B-mode polarization signals from the period immediately following the Big Bang—a research finding that has been hailed as one of the most significant scientific developments in recent times. Bock and scientists at Caltech and JPL developed and perfected the BICEP1 and BICEP2 instruments, stationed at the South Pole, that were essential to this research project.

"I am honored to receive the 2014 George W. Goddard Award from SPIE," says Bock. "This was only possible thanks to the unique environment we enjoy at JPL and Caltech, a combination of wonderful colleagues, one-of-a-kind facilities, and support for pioneering science experiments."

Bock received his BS from Duke University in 1987 and his PhD from UC Berkeley in 1994. He served as a research scientist at JPL from 1994 to 2012, when he was named senior research scientist. He was a visiting associate at Caltech from 1994 until 2008 and a senior faculty associate from 2008 to 2012, when he was named a professor of physics.

The George W. Goddard Award is given annually in recognition of exceptional achievement in optical or photonic technology or instrumentation for earth, planetary, or astronomical science, reconnaissance, or surveillance from airborne or space platforms. The award is for the invention and development of a new process or technique, technology, instrumentation, or system.

Previous Caltech- and JPL-affiliated winners of the George W. Goddard Award include Lew Allen Jr. (1978), JPL director from 1982 to 1990; James B. Breckinridge (2003), visiting associate in aerospace; Moustafa Chahine (2010), chief scientist at JPL; Bruce Murray (1967), Caltech professor of planetary science, emeritus, and former head of JPL; and Caltech alumni (BS '65) Jerry E. Nelson (1993), currently a Thirty Meter Telescope project scientist.