JCAP Stabilizes Common Semiconductors For Solar Fuels Generation

Caltech researchers devise a method to protect the materials in a solar-fuel generator

Researchers around the world are trying to develop solar-driven generators that can split water, yielding hydrogen gas that could be used as clean fuel. Such a device requires efficient light-absorbing materials that attract and hold sunlight to drive the chemical reactions involved in water splitting. Semiconductors like silicon and gallium arsenide are excellent light absorbers—as is clear from their widespread use in solar panels. However, these materials rust when submerged in the type of water solutions found in such systems.

Now Caltech researchers at the Joint Center for Artificial Photosynthesis (JCAP) have devised a method for protecting these common semiconductors from corrosion even as the materials continue to absorb light efficiently. The finding paves the way for the use of these materials in solar-fuel generators.

"For the better part of a half century, these materials have been considered off the table for this kind of use," says Nate Lewis, the George L. Argyros Professor and professor of chemistry at Caltech, and the principal investigator on the paper. "But we didn't give up on developing schemes by which we could protect them, and now these technologically important semiconductors are back on the table."

The research, led by Shu Hu, a postdoctoral scholar in chemistry at Caltech, appears in the May 30 issue of the journal Science.

In the type of integrated solar-fuel generator that JCAP is striving to produce, two half-reactions must take place—one involving the oxidation of water to produce oxygen gas; the other involving the reduction of water, yielding hydrogen gas. Each half-reaction requires both a light-absorbing material to serve as the photoelectrode and a catalyst to drive the chemistry. In addition, the two reactions must be physically separated by a barrier to avoid producing an explosive mixture of their products.

Historically, it has been particularly difficult to come up with a light-absorbing material that will robustly carry out the oxidation half-reaction. Researchers have tried, without much success, a variety of materials and numerous techniques for coating the common light-absorbing semiconductors. The problem has been that if the protective layer is too thin, the aqueous solution penetrates through and corrodes the semiconductor. If, on the other hand, the layer is too thick, it prevents corrosion but also blocks the semiconductor from absorbing light and keeps electrons from passing through to reach the catalyst that drives the reaction.

At Caltech, the researchers used a process called atomic layer deposition to form a layer of titanium dioxide (TiO2)—a material found in white paint and many toothpastes and sunscreens—on single crystals of silicon, gallium arsenide, or gallium phosphide. The key was that they used a form of TiO2 known as "leaky TiO2"—because it leaks electricity. First made in the 1990s as a material that might be useful for building computer chips, leaky oxides were rejected as undesirable because of their charge-leaking behavior. However, leaky TiO2 seems to be just what was needed for this solar-fuel generator application. Deposited as a film, ranging in thickness between 4 and 143 nanometers, the TiO2 remained optically transparent on the semiconductor crystals—allowing them to absorb light—and protected them from corrosion but allowed electrons to pass through with minimal resistance.

On top of the TiO2, the researchers deposited 100-nanometer-thick "islands" of an abundant, inexpensive nickel oxide material that successfully catalyzed the oxidation of water to form molecular oxygen.

The work appears to now make a slew of choices available as possible light-absorbing materials for the oxidation side of the water-splitting equation. However, the researchers emphasize, it is not yet known whether the protective coating would work as well if applied using an inexpensive, less-controlled application technique, such as painting or spraying the TiO2 onto a semiconductor. Also, thus far, the Caltech team has only tested the coated semiconductors for a few hundred hours of continuous illumination.

"This is already a record in terms of both efficiency and stability for this field, but we don't yet know whether the system fails over the long term and are trying to ensure that we make something that will last for years over large areas, as opposed to weeks," says Lewis. "That's the next step."

The work, titled "Amorphous TiO2 Coatings Stabilize Si, GaAs, and GaP Photoanodes for Efficient Water Oxidation," was supported by the Office of Science of the U.S. Department of Energy through an award to JCAP, a DOE Energy Innovation Hub. Some of the work was also supported by the Resnick Sustainability Institute and the Beckman Institute at Caltech. Additional coauthors on the paper are graduate students Matthew Shaner, Joseph Beardslee, and Michael Lichterman, as well as Bruce S. Brunschwig, director of the Molecular Materials Resource Center at Caltech.

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Stabilizing Semiconductors for Solar Fuels Generation
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Miniature Truss Work

Fancy Erector Set? Nope. The elaborate fractal structure shown at right (with a close-up below) is many, many times smaller than that and is certainly not child's play. It is the latest example of what Julia Greer, professor of materials science and mechanics, calls a fractal nanotruss—nano because the structures are made up of members that are as thin as five nanometers (five billionths of a meter); truss because they are carefully architected structures that might one day be used in structural engineering materials.

Greer's group has developed a three-step process for building such complex structures very precisely. They first use a direct laser writing method called two-photon lithography to "write" a three-dimensional pattern in a polymer, allowing a laser beam to crosslink and harden the polymer wherever it is focused. At the end of the patterning step, the parts of the polymer that were exposed to the laser remain intact while the rest is dissolved away, revealing a three-dimensional scaffold. Next, the scientists coat the polymer scaffold with a continuous, very thin layer of a material—it can be a ceramic, metal, metallic glass, semiconductor, "just about anything," Greer says. In this case, they used alumina, or aluminum oxide, which is a brittle ceramic, to coat the scaffold. In the final step they etch out the polymer from within the structure, leaving a hollow architecture.

Taking advantage of some of the size effects that many materials display at the nanoscale, these nanotrusses can have unusual, desirable qualities. For example, intrinsically brittle materials, like ceramics, including the alumina shown, can be made deformable so that they can be crushed and still rebound to their original state without global failure.

"Having full control over the architecture gives us the ability to tune material properties to what was previously unattainable with conventional monolithic materials or with foams," says Greer. "For example, we can decouple strength from density and make materials that are both strong (and tough) as well as extremely lightweight. These structures can contain nearly 99 percent air yet can also be as strong as steel. Designing them into fractals allows us to incorporate hierarchical design into material architecture, which promises to have further beneficial properties."

The members of Greer's group who helped develop the new fabrication process and created these nanotrusses are graduate students Lucas Meza and Lauren Montemayor and Nigel Clarke, an undergraduate intern from the University of Waterloo.

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Supernova Caught in the Act by Palomar Transient Factory

Supernovae—stellar explosions—are incredibly energetic, dynamic events. It is easy to imagine that they are uncommon, but the universe is a big place and supernovae are actually fairly routine. The problem with observing supernovae is knowing just when and where one is occurring and being able to point a world-class telescope at it in the hours immediately afterward, when precious data about the supernova's progenitor star is available. Fortunately the intermediate Palomar Transient Factory (iPTF) operated by Caltech scans the sky constantly in search of dramatic astrophysical events. In 2013, it caught a star in the act of exploding.

The iPTF is a robotic observing system mounted on the 48-inch Samuel Oschin Telescope on Palomar Mountain. It has been scanning the sky since February 2013. The iPTF (and its predecessor experiment, the Palomar Transient Factory [PTF], which operated between 2009 and 2012) regularly observes a wide swath of the night sky looking for astronomical objects that are moving and developing quickly, such as comets, asteroids, gamma-ray bursts, and supernovae. Both the earlier PTF and the current iPTF collaborations are led by Shrinivas Kulkarni, the John D. and Catherine T. MacArthur Professor of Astronomy and Planetary Science and director of the Caltech Optical Observatories.

Last year the iPTF discovered an object of special interest: a supernova with a spectral signature suggesting that its progenitor star was a Wolf-Rayet star. Massive stars are typically structured like an onion, with the heaviest elements in the core, while lighter elements are layered over them and then frosted, if you will, by a layer of hydrogen gas on the stellar surface. Wolf-Rayet stars, which are unusually large and hot, are exceptions to this rule, being relatively deficient in hydrogen and characterized by strong stellar winds. Astronomers have long wondered if Wolf-Rayet stars are the progenitors of certain types of supernovae, and according to a recent paper published in Nature this is just what the iPTF found in May 2013.

This supernova, SN2013cu, was picked up on a routine sky scan by the iPTF. The on-duty iPTF team member in Israel promptly sounded an alert, asking colleagues at the W. M. Keck Observatory on Mauna Kea to take a spectral image of the supernova before the sun rose in Hawaii.

When supernovae explode, they briefly ionize the sky immediately around them. The ionized materials rapidly recombine, producing unique spectral features that enable astronomers to get a full picture of the ambient material of a supernova event. This process lasts from minutes to a few days and hence is called a "flash spectrum" of the event. Flash spectrography is a novel observational method developed by Avishay Gal-Yam of the Weizmann Institute of Science in Israel, leader of the team that published the Nature paper.

In the case of SN2013cu, the flash spectrum showed relatively less hydrogen and relatively more nitrogen, suggesting that perhaps the progenitor of the supernova was a nitrogen-rich Wolf-Rayet star. This finding will enable astronomers to better understand the evolution of massive stars and identify potential progenitors of supernovae.

"I could not believe my eyes when I saw those high-ionization features perfectly matching emission lines from a Wolf-Rayet star," says Yi Cao, a graduate student from Caltech who works with Kulkarni. "Our software pipeline efforts were paying off. Now we are working even harder so that we can get flash spectra of many more supernova flavors to probe their progenitor stars."

Above all, the observation of SN2013cu highlights the success of the intermediate Palomar Transient Factory at catching the universe in the act of doing something interesting, something that might merit a second look. Though especially intriguing, SN2013cu is only one of over 2,000 supernovae that PTF/iPTF has detected during its four and a half years of observations. As Kulkarni remarks, "I am proud of how the global iPTF network is working together to invent new techniques enabling entirely new science."

The iPTF is a collaboration between Caltech, Los Alamos National Laboratory, the University of Wisconsin–Milwaukee, the Oskar Klein Centre, the Weizmann Institute of Science, the TANGO Program of the University System of Taiwan, and the Kavli Institute for the Physics and Mathematics of the Universe.

Coauthors on the paper, "A Wolf-Rayet-like progenitor of supernova SN 2013cu from spectral observations of a wind," include Kulkarni, Cao, Mansi Kasliwal, Daniel Perley, and Assaf Horesh of Caltech; Gal-Yam, I. Arcavi, E. O. Ofek, S. Ben-Ami, A. De Cia, D. Tal, P. M. Vreeswijk, and O. Yaron of the Weizmann Institute of Science; S. B. Cenko of NASA's Goddard Space Flight Center; J. C. Wheeler and J. M. Silverman of the University of Texas at Austin; F. Taddia and J. Sollerman of Stockholm University; P. E. Nugent of the Lawrence Berkeley National Laboratory; and A. V. Filippenko of UC Berkeley.

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Tricking the Uncertainty Principle

Caltech researchers have found a way to make measurements that go beyond the limits imposed by quantum physics.

Today, we are capable of measuring the position of an object with unprecedented accuracy, but quantum physics and the Heisenberg uncertainty principle place fundamental limits on our ability to measure. Noise that arises as a result of the quantum nature of the fields used to make those measurements imposes what is called the "standard quantum limit." This same limit influences both the ultrasensitive measurements in nanoscale devices and the kilometer-scale gravitational wave detector at LIGO. Because of this troublesome background noise, we can never know an object's exact location, but a recent study provides a solution for rerouting some of that noise away from the measurement.

The findings were published online in the May 15 issue of Science Express.

"If you want to know where something is, you have to scatter something off of it," explains Professor of Applied Physics Keith Schwab, who led the study. "For example, if you shine light at an object, the photons that scatter off provide information about the object. But the photons don't all hit and scatter at the same time, and the random pattern of scattering creates quantum fluctuations"—that is, noise. "If you shine more light, you have increased sensitivity, but you also have more noise. Here we were looking for a way to beat the uncertainty principle—to increase sensitivity but not noise."

Schwab and his colleagues began by developing a way to actually detect the noise produced during the scattering of microwaves—electromagnetic radiation that has a wavelength longer than that of visible light. To do this, they delivered microwaves of a specific frequency to a superconducting electronic circuit, or resonator, that vibrates at 5 gigahertz—or 5 billion times per second. The electronic circuit was then coupled to a mechanical device formed of two metal plates that vibrate at around 4 megahertz—or 4 million times per second. The researchers observed that the quantum noise of the microwave field, due to the impact of individual photons, made the mechanical device shake randomly with an amplitude of 10-15 meters, about the diameter of a proton.

"Our mechanical device is a tiny square of aluminum—only 40 microns long, or about the diameter of a hair. We think of quantum mechanics as a good description for the behaviors of atoms and electrons and protons and all of that, but normally you don't think of these sorts of quantum effects manifesting themselves on somewhat macroscopic objects," Schwab says. "This is a physical manifestation of the uncertainty principle, seen in single photons impacting a somewhat macroscopic thing."

Once the researchers had a reliable mechanism for detecting the forces generated by the quantum fluctuations of microwaves on a macroscopic object, they could modify their electronic resonator, mechanical device, and mathematical approach to exclude the noise of the position and motion of the vibrating metal plates from their measurement.

The experiment shows that a) the noise is present and can be picked up by a detector, and b) it can be pushed to someplace that won't affect the measurement. "It's a way of tricking the uncertainty principle so that you can dial up the sensitivity of a detector without increasing the noise," Schwab says.

Although this experiment is mostly a fundamental exploration of the quantum nature of microwaves in mechanical devices, Schwab says that this line of research could one day lead to the observation of quantum mechanical effects in much larger mechanical structures. And that, he notes, could allow the demonstration of strange quantum mechanical properties like superposition and entanglement in large objects—for example, allowing a macroscopic object to exist in two places at once.

"Subatomic particles act in quantum ways—they have a wave-like nature—and so can atoms, and so can whole molecules since they're collections of atoms," Schwab says. "So the question then is: Can you make bigger and bigger objects behave in these weird wave-like ways? Why not? Right now we're just trying to figure out where the boundary of quantum physics is, but you never know."

This work was published in an article titled "Mechanically Detecting and Avoiding the Quantum Fluctuations of a Microwave Field." Other Caltech coauthors include senior researcher Junho Suh; graduate students Aaron J. Weinstein, Chan U. Lei, and Emma E. Wollman; and Steven K. Steinke, visitor in applied physics and materials science. The work was funded by the Institute for Quantum Information and Matter, the Defense Advanced Research Projects Agency, and the National Science Foundation. The device was fabricated in Caltech's Kavli Nanoscience Institute, of which Schwab is a codirector.

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Research Update: An Autism Connection

Caltech neuroscientists find link between agenesis of the corpus callosum and autism

Building on their prior work (see "Bridging the Gap"), a team of neuroscientists at Caltech now report that rare patients who are missing connections between the left and right sides of their brain—a condition known as agenesis of the corpus callosum (AgCC)—show a strikingly high incidence of autism. The study is the first to show a link between the two disorders.

The findings are reported in a paper published April 22, 2014, in the journal Brain.

The corpus callosum is the largest connection in the human brain, connecting the left and right brain hemispheres via about 200 million fibers. In very rare cases it is surgically cut to treat epilepsy—causing the famous "split-brain" syndrome, for whose discovery the late Caltech professor Roger Sperry received the Nobel Prize. People with AgCC are like split-brain patients in that they are missing their corpus callosum—except they are born this way. In spite of this significant brain malformation, many of these individuals are relatively high-functioning individuals, with jobs and families, but they tend to have difficulty interacting with other people, among other symptoms such as memory deficits and developmental delays. These difficulties in social behavior bear a strong resemblance to those faced by high-functioning people with autism spectrum disorder.

"We and others had noted this resemblance between AgCC and autism before," explains Lynn Paul, lead author of the study and a lecturer in psychology at Caltech. But no one had directly compared the two groups of patients. This was a challenge that the Caltech team was uniquely positioned to do, she says, since it had studied patients from both groups over the years and had tested them on the same tasks.

"When we made detailed comparisons, we found that about a third of people with AgCC would meet diagnostic criteria for an autism spectrum disorder in terms of their current symptoms," says Paul, who was the founding president of the National Organization for Disorders of the Corpus Callosum.

The research was done in the laboratory of Ralph Adolphs, Bren Professor of Psychology and Neuroscience and professor of biology at Caltech and a coauthor of the study. The team looked at a range of different tasks performed by both sets of patients. Some of the exercises that involved certain social behaviors were videotaped and analyzed by the researchers to assess for autism. The team also gave the individuals questionnaires to fill out that measured factors like intelligence and social functioning.

"Comparing different clinical groups on exactly the same tasks within the same lab is very rare, and it took us about a decade to accrue all of the data," Adolphs notes.

One important difference between the two sets of patients did emerge in the comparison. People with autism spectrum disorder showed autism-like behaviors in infancy and early childhood, but the same type of behaviors did not seem to emerge in individuals with AgCC until later in childhood or the teen years.

"Around ages 9 through 12, a normally formed corpus callosum goes through a developmental 'growth spurt' which contributes to rapid advances in social skills and abstract thinking during those years," notes Paul. "Because they don't have a corpus callosum, teens with AgCC become more socially awkward at the age when social skills are most important."

According to Adolphs, it is important to note that AgCC can now be diagnosed before a baby is born, using high-resolution ultrasound imaging during pregnancy. This latest development also opens the door for some exciting future directions in research.

"If we can identify people with AgCC already before birth, we should be in a much better position to provide interventions like social skills training before problems arise," Paul points out. "And of course from a research perspective it would be tremendously valuable to begin studying such individuals early in life, since we still know so little both about autism and about AgCC."

For example, the team would like to discern at what age subtle difficulties first appear in AgCC individuals, and at what point they start looking similar to autism, as well as what happens in the brain during these changes.

"If we could follow a baby with AgCC as it grows up, and visualize its brain with MRI each year, we would gain such a wealth of knowledge," Adolphs says.

The Brain paper, "Agenesis of the Corpus Callosum and Autism: A Comprehensive Comparison," also includes as coauthors Daniel Kennedy, assistant professor of psychology at Indiana University, and Christina Corsello, a member of the research staff at Rady Children's HospitalSan Diego. The research was funded by the Simons Foundation, Autism Speaks, and the Brain and Behavior Research Foundation.

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Unlocking a Mystery of Human Disease . . . in Space

An experiment just launched into orbit by a team of Caltech researchers could be an important step toward understanding a devastating neurodegenerative disease.

Huntington's disease is a grim diagnosis. A hereditary disorder with debilitating physical and cognitive symptoms, the disease usually robs adult patients of their ability to walk, balance, and speak. More than 15 years ago, researchers revealed the disorder's likely cause—an abnormal version of the protein huntingtin; however, the mutant protein's mechanism is poorly understood, and the disease remains untreatable.

Now, a new project led by Pamela Bjorkman, Max Delbrück Professor of Biology, will investigate whether the huntingtin protein can form crystals in microgravity aboard the International Space Station (ISS)—crystals that are crucial for understanding the molecular structure of the protein. The experiment was launched from Cape Canaveral in Florida on Friday, April 18 aboard the SpaceX CRS-3 cargo resupply mission to the ISS. On Sunday, April 20 the station's robotic arm captured the mission's payload, which included the proteins for Bjorkman's experiment—which is the first Caltech experiment to take place aboard the ISS.

In the experiment, the researchers hope to grow a crystal of the huntingtin protein—the crystal would be an organized, latticelike arrangement of the protein's molecules—which is needed to determine the molecular structure of the protein. However, molecules of the huntingtin protein tend to aggregate, or clump together, in Earth's gravity. And this disordered arrangement makes it incredibly hard to parse the protein's structure, says Gwen Owens, a graduate student in Bjorkman's lab and a researcher who helped design the study.

"We need crystals for X-ray crystallography, the technique we use to study the protein, in which we shoot an X-ray through the protein crystal and analyze the organized pattern of radiation that scatters off of it," Owens says. "That pattern is what we depend on to identify the location of every carbon, nitrogen, and sulfur atom within the protein; if we shoot an X-ray beam at a clumped, aggregate protein—like huntingtin often is—we can't get any data from it," she says.

Researchers have previously studied small fragments of crystallized huntingtin, but because of its large size and propensity to clumping, no one has ever successfully grown a crystal of the full-length protein large enough to analyze with X-ray crystallography. To understand what the protein does—and how defects in it lead to the symptoms of Huntington's disease—the researchers need to study the full-length protein.

Looking for a solution to this problem, Owens was inspired by a few previous studies of protein formation on space shuttles and the ISS—studies suggesting that proteins can form crystals more readily in a condition of near-weightlessness called microgravity. "The previous studies looked at much simpler proteins, but we thought we could make a pretty good case that huntingtin would be an excellent candidate to study on the ISS," Owens says.

They proposed such an experiment to the Center for the Advancement of Science in Space (CASIS), which manages U.S. research on the ISS, and it was accepted, becoming part of the first Advancing Research Knowledge, or ARK1, mission.

Because Owens and Bjorkman cannot travel with their proteins, and staff and resources are limited aboard the ISS, the crystal will be grown with a Handheld High-Density Protein Crystal Growth device—an apparatus that will allow astronauts to initiate growth of normal and mutant huntingtin protein crystals from a solution of protein molecules with just the flip of a switch.

As the crystals grow larger over a period of several months, samples will come back to Earth via the SpaceX CRS-4 return mission. The results of the experiment are scheduled to drop into the ocean just off the coast of Southern California—along with the rest of the return cargo—sometime this fall. At that point, Owens will finally be able to analyze the proteins.

"Our ideal result would be to have large crystals of the normal and mutant huntingtin proteins right away—on the first try," she says. After analyzing crystals of the full-length protein with X-ray crystallography, the researchers could finally determine huntingtin's structure—information that will be crucial to developing treatments for Huntington's disease.

Owens, a joint MD/PhD student at Caltech and UCLA's David Geffen School of Medicine, has also had the opportunity to work with Huntington's disease patients in the clinic, adding a human connection to her experiment in the sky. "The patients and families I have met who are affected by Huntington's disease are excited to see something big like this. It's inspiring for them—and hopefully it will inspire new research, too."

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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.

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Caltech Researchers Discover the Seat of Sex and Violence in the Brain

As reported in a paper published online today in the journal Nature, Caltech biologist David J. Anderson and his colleagues have genetically identified neurons that control aggressive behavior in the mouse hypothalamus, a structure that lies deep in the brain (orange circle in the image at right). Researchers have long known that innate social behaviors like mating and aggression are closely related, but the specific neurons in the brain that control these behaviors had not been identified until now.

The interdisciplinary team of graduate students and postdocs, led by Caltech senior research fellow Hyosang Lee, found that if these neurons are strongly activated by pulses of light, using a method called optogenetics, a male mouse will attack another male or even a female. However, weaker activation of the same neurons will trigger sniffing and mounting: mating behaviors. In fact, the researchers could switch the behavior of a single animal from mounting to attack by gradually increasing the strength of neuronal stimulation during a social encounter (inhibiting the neurons, in contrast, stops these behaviors dead in their tracks).

These results suggest that the level of activity within the population of neurons may control the decision between mating and fighting.  

The neurons initially were identified because they express a protein receptor for the hormone estrogen, reinforcing the view that estrogen plays an important role in the control of male aggression, contrary to popular opinion. Because the human brain contains a hypothalamus that is structurally similar to that in the mouse, these results may be relevant to human behavior as well.

The results of the study were published in journal Nature on April 16. David J. Anderson is the Seymour Benzer Professor of Biology and an investigator with the Howard Hughes Medical Institute.

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Katie Neith
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For Cells, Internal Stress Leads to Unique Shapes

From far away, the top of a leaf looks like one seamless surface; however, up close, that smooth exterior is actually made up of a patchwork of cells in a variety of shapes and sizes. Interested in how these cells individually take on their own unique forms, Caltech biologist Elliot Meyerowitz, postdoctoral scholar Arun Sampathkumar, and colleagues sought to pinpoint the shape-controlling factors in pavement cells, which are puzzle-piece-shaped epithelial cells found on the leaves of flowering plants. They found that these unusual shapes were the cell's response to mechanical stress on the microtubule cytoskeleton—protein tubes that act as a scaffolding inside the cells. These microtubules guide oriented deposition of cell-wall components, thus providing structural support.

The researchers studied this supportive microtubule arrangement in the tissue of pavement cells from the first leaves—or cotyledons—of a young Arabidopsis thaliana plant (right). By fluorescently marking the cells' microtubules (yellow, top surface of cell; purple, bottom surface of cell), the researchers could image the cell's structural arrangement—and watch how this arrangement changed over time. They could also watch the microtubule modifications that occurred due to changes in the mechanical forces experienced by the cells.

Microtubules strengthen a cell's structure by lining up in the direction of stress or pressure experienced by the cell and guiding the deposition of new cell-wall material, providing a supportive scaffold for the cell's shape. However, Meyerowitz and colleagues found that this internal stress is also influenced by the cell's shape. The result is a feedback loop: the cell's shape influences the microtubule arrangement; this arrangement, in turn, affects the cell's shape, which modulates the microtubules, and so on. Therefore, the unusual shape of the pavement cell represents a state of balance—an individual cell's tug-of-war to maintain structural integrity while also dynamically responding to the pushes and pulls of mechanical stress.

The results of the study were published in the journal eLife on April 16. Elliot Meyerowitz is George W. Beadle Professor of Biology and an investigator with the Howard Hughes Medical Institute.

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Antennae Help Flies "Cruise" In Gusty Winds

Caltech researchers uncover a mechanism for how fruit flies regulate their flight speed, using both vision and wind-sensing information from their antennae.

Due to its well-studied genome and small size, the humble fruit fly has been used as a model to study hundreds of human health issues ranging from Alzheimer's to obesity. However, Michael Dickinson, Esther M. and Abe M. Zarem Professor of Bioengineering at Caltech, is more interested in the flies themselves—and how such tiny insects are capable of something we humans can only dream of: autonomous flight. In a report on a recent study that combined bursts of air, digital video cameras, and a variety of software and sensors, Dickinson and his team explain a mechanism for the insect's "cruise control" in flight—revealing a relationship between a fly's vision and its wind-sensing antennae.

The results were recently published in an early online edition of the Proceedings of the National Academy of Sciences.

Inspired by a previous experiment from the 1980s, Dickinson's former graduate student Sawyer Fuller (PhD '11) wanted to learn more about how fruit flies maintain their speed in flight. "In the old study, the researchers simulated natural wind for flies in a wind tunnel and found that flies maintain the same groundspeed—even in a steady wind," Fuller says.

Because the previous experiment had only examined the flies' cruise control in gentle steady winds, Fuller decided to test the limits of the insect's abilities by delivering powerful blasts of air from an air piston in a wind tunnel. The brief gusts—which reached about half a meter per second and moved through the tunnel at the speed of sound—were meant to probe how the fly copes if the wind is rapidly changing.

The flies' response to this dynamic stimulus was then tracked automatically by a set of five digital video cameras that recorded the fly's position from five different perspectives. A host of computers then combined information from the cameras and instantly determined the fly's trajectory and acceleration.

To their surprise, the Caltech team found that the flies in their experiments, unlike those in the previous studies, accelerated when the wind was pushing them from behind and decelerated when flying into a headwind. In both cases the flies eventually recovered to maintain their original groundspeed, but the initial response was puzzling, Fuller says. "This response was basically the opposite of what the fly would need to do to maintain a consistent groundspeed in the wind," he says.

In the past, researchers assumed that flies—like humans and most other animals—used their vision to measure their speed in wind, accelerating and decelerating their flight based on the groundspeed their vision detected. But Fuller and his colleagues were also curious about the in-flight role of the fly's wind-sensing organs: the antennae.

Using the fly's initial response to strong wind gusts as a marker, the researchers tested the response of each sensory mode individually. To investigate the role of wind sensation on the fly's cruise control, they delivered strong gusts of wind to normal flies, as well as flies whose antennae had been removed. The flies without antenna still increased their speed in the same direction as the wind gust, but they only accelerated about half as much as the flies whose antennae were still intact. In addition, the flies without antennae were unable to maintain a constant speed, dramatically alternating between acceleration and deceleration. Together, these results suggested that the antennae were indeed providing wind information that was important for speed regulation.

In order to test the response of the eyes separately from that of the antennae, Fuller and his colleagues projected an animation on the walls of the fly-tracking arena that would trick the eyes into thinking there was no speed increase, even though the antenna could feel the increased windspeed. When the researchers delivered strong headwinds to flies in this environment, the flies decelerated and were unable to recover to their original speed.

"We know that vision is important for flying insects, and we know that flies have one of the fastest visual systems on the planet," Dickinson says, "But this response showed us that as fast as their vision is, if they're flying too fast or the wind is blowing them around too quickly, their visual system reaches its limit and the world starts getting blurry." That is when the antennae kick in, he says.

The results suggest that the antennae are responsible for quickly sensing changes in windspeed—and therefore are responsible for the fly's initial deceleration in a headwind. The information received from the fly's eyes—which is processed much more slowly than information from the wind sensors on the antenna—is responsible for helping the fly regain its cruising speed.

"Sawyer's study showed that the fly can take another sensor—this little tiny antenna, which doesn't require nearly the amount of processing area within the brain as the eyes—and the fly is able to use that information to compensate for the fact that the information coming out of the eyes is a bit delayed," Dickinson says. "It's kind of a neat trick, using a cheap little sensor to compensate for the limitations of a big, heavy, expensive sensor."

Beyond learning more about the fly's wind-sensing capabilities, Fuller says that this information will also help engineers design small flying robots—creating a sort of man-made fly. "Tiny flying robots will take a lot of inspiration from flies. Like flies, they will probably have to rely heavily on vision to regulate groundspeed," he says.

"A challenge here is that vision typically takes a lot of computation to get right, just like in flies, but it's impossible to carry a powerful processor to do that quickly on a tiny robot. So they'll instead carry tiny cameras and do the visual processing on a tiny processor, but it will just take longer. Our results suggest that little flying vehicles would also do well to have fast wind sensors to compensate for this delay."

The work was published in a study titled "Flying Drosophila stabilize their vision-based velocity controller by sensing wind with their antennae." Other coauthors include former Caltech senior postdoc Andrew D. Straw, Martin Y. Peek (BS '06), and Richard Murray, Thomas E. and Doris Everhart Professor of Control and Dynamical Systems and Bioengineering at Caltech, who coadvised Fuller's graduate work. The study was supported by the Institute for Collaborative Biotechnologies through funding from the U.S. Army Research Office and by a National Science Foundation Graduate Fellowship.

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