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.

Cynthia Eller
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Simulating Milliseconds of Stellar Collapse
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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.

Cynthia Eller
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In Our Community

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.

Cynthia Eller
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The Intergalactic Medium Unveiled
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Cosmic Web Imager Directly Observes "Dim Matter"
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On the Front Lines of Sustainability

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On the Front Lines of Sustainability

The chemical processes used to make products ranging from pharmaceuticals to perfumes can have a harmful impact on the environment. However, Caltech chemist and Nobel laureate Robert Grubbs has spent several decades developing catalysts—compounds that speed up a chemical reaction—that can make the synthesis of these products more efficient and ecologically friendly, ultimately reducing their environmental footprint. Similarly, chemist Brian Stoltz is developing new strategies for the synthesis of compounds needed in the chemical, polymer, and pharmaceutical industries. His new processes rely upon oxygen and organometallic catalysts—greener alternatives to the toxic metals that are normally used to drive such reactions.

Switching from paper files to cloud-based data storage might seem like an obvious choice for sustainability, but can we further reduce the environmental impact of storing data? The theoretical work of engineer and computer scientist Adam Wierman suggests that with the right algorithms, we can. Today, data centers—the physical storage facilities Wierman calls the "SUVs of the Internet"—account for more than 1.5 percent of U.S. electricity usage. And as more data goes online, that number is expected to grow. Wierman's work helps engineers design algorithms that will reroute data, with preference to centers that use renewable energy sources like wind and solar.

Energy from the sun—although free and abundant—cannot easily be stored for use on dreary days or transported to cloudy regions. Caltech engineer and materials scientist Sossina Haile hopes to remove that barrier with a specific type of solar reactor she has developed. The reactor is lined with ceramic cerium oxide; when this lining is heated with concentrated sunlight it releases oxygen, priming it to remove oxygen from water molecules or carbon dioxide on cooling, thus creating hydrogen fuel or "syngas"—a precursor to liquid hydrocarbon fuels. This conversion of the sun's light into storable fuel could allow solar-derived power to be available day and night.

Caltech student participants in the Department of Energy's biennial Solar Decathlon competition set out to prove that keeping a house lit up, cooled down, and comfortable for living is possible—even while off the grid. The Techers teamed up with students at the Southern California Institute of Architecture to create CHIP and DALE, their entries in the 2011 and 2013 competitions, respectively. These functional and stylish homes, powered solely by the sun, were engineered with innovative components including a rainwater collection system and moving room modules that optimize heating and cooling efficiency. 

Although many of us take the nearest bathroom for granted, working toilets require resources and infrastructure that may not be available in many parts of the world. Inspired by the "Reinventing the Toilet Challenge" issued by the Bill and Melinda Gates Foundation, environmental scientist and engineer Michael Hoffmann and his team applied his research in hydrogen evolution and water treatment to reengineer the toilet. The Caltech team's design—which won the challenge in 2012—can serve hundreds of people each day, treat its own wastewater, and generate electricity, providing a sustainable and low-cost solution to sanitation and hygiene challenges in the developing world. Prototypes are being tested in India and China for use in urban and remote environments in the developing world.  

Geophysicist Mark Simons studies the mechanics of the Earth—furthering our understanding of what causes our planet to deform over time. His research often involves using satellite data to observe the movement associated with seismic and volcanic activity, but Simons is also interested in changes going on in the icy parts of Earth's surface, especially the dynamics of glaciers. By flying high above Iceland's ice caps, Simons and his colleagues can track the glaciers' melt-and-freeze response in relation to seasonal and long-term variations in temperature—and their potential response to climate change.

The production of industrial nitrogen fertilizer results in 130 million tons of ammonia annually—while also requiring high heat, high pressure, and lots of energy. However, in a process called nitrogen fixation, soil microorganisms that live near the roots of certain plants can produce a similar amount of ammonia each year. The bugs use catalysts called nitrogenases to convert nitrogen from the air into ammonia at room temperature and atmospheric pressure. By mimicking the behavior of these microorganisms, Jonas Peters and his colleagues synthesized an iron-based catalyst that allows for nitrogen fixation under much milder conditions. The catalyst could one day lead to more environmentally friendly methods of ammonia production.

Traditionally, the photovoltaic cells in solar panels have been expensive and have had limited efficiency—making them a hard sell in the consumer market. Engineer and applied physicist Harry Atwater's work suggests that there is a thinner and more efficient alternative. Atwater, who is also the director of the Resnick Sustainability Institute, uses thin layers of semiconductors to create photovoltaics that absorb sunlight as efficiently as thick solar cells but can be produced with higher efficiency than conventional cells.

The generation of chemical fuels from sunlight could completely change the way we power the planet. Researchers in the laboratory of Caltech chemist Nate Lewis are working to develop different components of a fuel-producing device that could do just that called a photoelectrochemical cell. The cell would consist of an upper layer that could absorb sunlight, carbon dioxide, and water vapor, a middle layer consisting of light absorbers and catalysts that can produce fuels, which are then released through the device's bottom layer. When such a device is created, the Joint Center for Artificial Photosynthesis, of which Lewis is the scientific director, aims to ease the transfer of these technologies to the private sector. 

Clean energy from the wind is a promising alternative to fossil fuels, but giant pinwheel-like wind turbines that are common on many wind farms can create dangerous obstacles for birds as well as being an unpleasant addition to a landscape's aesthetic. To combat this problem, Caltech engineer and fluid-mechanics expert John Dabiri is testing a new design for wind turbines, which looks a bit like a spinning eggbeater emerging from the ground. By placing these columnar vertical wind turbines in a careful arrangement—an arrangement inspired by the vortex of water created behind a swimming fish—his smaller vertical turbines create just as much energy as the "pinwheels" and on a much smaller land footprint.

In the early 1990s, Caltech bioengineer Frances Arnold pioneered "directed evolution"—a new method of engineering custom-built enzymes, or activity-boosting proteins. The technique allows mutations to develop in the enzyme's genetic code; these mutations can give the enzyme properties that don't occur in nature but are beneficial for human applications. The selectively enhanced enzymes help microbes turn plant waste and fast-growing grasses into fuels like isobutanol, which could sustainably replace more than half of U.S. oil imports, Arnold says. She's also exploring ways the technique could help factories to make pharmaceuticals and other products in much cleaner and safer ways.

The combined research efforts of Richard Flagan, John Seinfeld, Mitchio Okumura, and Paul Wennberg aim to improve our understanding of various aspects of climate change. Chemical engineer Flagan is pioneering ways to measure the number and sizes of particles in the air down to that of large molecules. Seinfeld studies where particles in the air come from, how they are produced by airborne chemical reactions, and the effect they have on the world's climate. Chemical physicist Okumura studies the chemical reactions that occur when sunlight encounters air pollution and results in smog. Wennberg, an atmospheric chemist, studies the natural and human processes that affect smog formation, the health of the ozone layer, as well as the lifetime of greenhouse gases. Wennberg and his colleagues join a legacy of Caltech researchers who have improved air quality through key discoveries about pollution.

In the past, researchers have discovered materials that can act as reaction catalysts, driving sunlight to split water into hydrogen fuel and an oxygen byproduct. However, these wonder materials are often expensive and in short supply. The research of chemist Harry Gray, who leads the National Science Foundation-funded Center for Chemical Innovation in Solar Fuels program, tests combinations of Earth-abundant metals to search for an inexpensive catalyst that boosts the water-splitting reaction with the sun. Gray also coleads an outreach project in which students in the classroom can participate in the race for solar fuels by testing thousands of materials and reporting their results to Caltech researchers.


Although Earth Week has officially come to a close, Caltech's commitment to sustainability continues. In this feature, you will meet some of the researchers at Caltech whose work is contributing to a greener planet and to the long-term improvement of our global environment.

Monday, May 5, 2014
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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 19th 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 + z2 has degree 2, and x2yz + xy3 has degree 4.  If we take an equation like xy = z2, 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, x7 + y7 = z7 or x2y3z = w6). The classic example of a homogeneous polynomial equation is the Pythagorean theorem—x2 + y2 = z2—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 32 + 42 = 52), 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 x2 + y2 = z2, there were no positive integer solutions at all when the variables were raised to the power of three or higher (x3 + y3 = z3; x4 + y4 = z4 ; . . . ; xn + yn = zn). 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.

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: x15 + y5 = z5; x25 + w5 = z5; and x35 + w5 = y5. 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 wCH4 + xO2 —> yCO2 + zH2O, 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.

Cynthia Eller
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Friday, April 11, 2014
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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.

Brian Bell
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Wednesday, April 16, 2014
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Reflecting on BICEP2

On Monday, March 17, 2014, collaborators working with the BICEP2 telescope at the South Pole presented the world with its first direct evidence of primordial gravitational waves and thus of cosmic inflation. Caltech professor of physics Jamie Bock, co–principal investigator for BICEP2 and the chief architect of the telescope's detectors, described the finding as "mind-boggling."

Cosmologists were thrilled by the news that BICEP2 had observed B-mode polarization in the cosmic microwave background at a level twice the intensity they had expected. This faint swirling polarization is thought to be a relic of the rapid inflation of the universe, faster than the speed of light, that took place in the first "trillionth of a trillionth of a trillionth of a second after the Big Bang," according to Caltech senior research associate Sean Carroll, who along with John Preskill, the Richard P. Feynman Professor of Theoretical Physics at Caltech, has been blogging about both the specifics of the BICEP2 finding and its implications.

A massively energetic event like inflation would have produced gravitational waves, ripples in the fabric of spacetime predicted by Einstein's general theory of relativity but never directly detected. As they traveled through the early universe, these gravitational waves should have left their signature on the cosmic microwave background, which is the oldest visible radiation in our universe, dating to 300,000 years after the Big Bang.

This prediction about the behavior of primordial gravitational waves and its polarizing effect on the cosmic microwave background was made in the late 1990s by, among others, Marc Kamionkowski, a professor of theoretical astrophysics at Caltech from 1999 to 2011 who is now at Johns Hopkins University. Kamionkowski describes the BICEP2 finding as "not just a home run," but "a grand slam," while Max Tegmark, a cosmologist at MIT says, "If this stays true, it will go down as one of the greatest discoveries in the history of science."

But it wasn't just cosmologists who took notice. In the days since the announcement, headlines around the world have announced the cosmological finding from BICEP2, and journalists from The New York Times to Al Jazeera have proclaimed this a "landmark in science" and an "epic discovery."

The BICEP experiments that have caught the world's attention began at Caltech in 2001 with discussions between Bock, then a research associate at JPL, and Brian Keating, a postdoctoral scholar at Caltech, about how to design a telescope that could observe the cosmic microwave background across a relatively large area of the sky. When Bock and Keating brought the idea to the late Andrew Lange, then Marvin L. Goldberger Professor of Physics at Caltech, Lange declared it a wild goose chase . . . and then happily plunged in.

As the BICEP project developed, Lange and Bock brought talented graduate students and postdoctoral scholars to join the BICEP team at Caltech and JPL. Former Caltech graduate student Randol Aikin (PhD '13), a BICEP2 collaborator now on staff at MIT's Lincoln Laboratory, says, "In addition to being superb scientists, Andrew and Jamie had an extraordinary capacity to empower students and give them room to take ownership of their work."

Among the postdoctoral scholars nurtured at Caltech are John Kovac, now a professor at the Harvard-Smithsonian Center for Astrophysics, who was the first Kilroy Fellow in Astrophysics at Caltech, and Chao-Lin Kuo, now a professor at Stanford University and an associate at the SLAC National Accelerator Laboratory; both are principal investigators on the BICEP2 project along with Bock and Clem Pryke of the University of Minnesota. (The project has a co-PI structure. At each stage of the BICEP experiments, one PI takes the lead. Lange was the leader for BICEP1; Kovac is the leader for BICEP2, and Kuo is the leader for BICEP3, already in progress.)

By the standards of other major experiments in physics, such as the Planck space telescope or the Large Hadron Collider, the BICEP2 team is quite small; there are just 47 coauthors on the paper that has disseminated the experiment's results, and only around 20 team members working closely on the core analysis. The BICEP2 team credits its success to the team members' focus, dedication, and close collaboration, and, says Keating, to the skill and determination of Bock, "one of the hardest working scientists I've ever met." Adds Hien Nguyen, a BICEP2 collaborator from JPL, says, "It's always a pleasure to sit back and see Jamie in action. There are a lot of details in the telescope that never would have been there if Jamie didn't pay attention at the beginning. He actually foresaw the intricacy of the experiment way ahead of time."

A strong public/private partnership has sustained this project throughout its 12-year history. The BICEP2 finding was made possible through grants from the National Science Foundation and the gifts of generous donors, including the W. M. Keck Foundation and the Gordon and Betty Moore Foundation. The Moore Foundation, along with Caltech and JPL internal funds provided the support to invent the unique detectors that were essential to achieving these results. A grant from the Keck Foundation funded the building of the Keck Array telescopes that have helped to provide preliminary confirmation of the BICEP2 results. The John M. Robinson estate granted additional funding to BICEP2 at a critical time, while the Jim and Nellie Kilroy Foundation provided resources to support members of the team at Caltech.

At a celebration for the Caltech/JPL BICEP2 team, Cyndi Atherton, previously of the Moore Foundation, said, "When I first took over supervision of the foundation's grant to Caltech for the BICEP2 project, my colleagues told me, 'We don't quite know what they're going to do, but there's this group of really smart people at Caltech and JPL. We're going to give them money and we're going to let them work.' I think you have made Gordon and Betty Moore and the Keck family proud to be associated with this project. I know my colleagues and I are walking taller this week, saying 'This is what science does for us.'"

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