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

## Hyperbolic Homogeneous Polynomials, Oh My!

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

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

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

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

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

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

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

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

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

*y*=

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

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

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

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

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

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

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

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

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

## Spring Ombudsperson Training

## Jamie Bock Wins George W. Goddard Award

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

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

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

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

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

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

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

## Teaching & Learning in the American System: Student-Teacher Interactions

## 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.'"

## Dark Matter in Southern California (DaMaSC) Symposium

## Jamie Bock to Speak on BICEP2 Experiment

On Thursday, March 20, 2014, Jamie Bock, professor of physics at Caltech and senior research scientist at the Jet Propulsion Laboratory (JPL), will be giving a talk on the BICEP2 experiment and its recent detection of B-mode polarization indicating the effect of gravitational waves on the cosmic microwave background (CMB). The BICEP experiments began in 2002 as a collaboration between Bock and the late Andrew Lange, the former Marvin L. Goldberger Professor of Physics at Caltech.

Bock will review data from BICEP2 observations at the South Pole conducted over three observing seasons from 2010 to 2012. He will describe the BICEP-2 experiment, discuss how the observations were undertaken and how the data were processed, and will explain why the BICEP2 team is confident that its finding is that of a strong B-mode polarization in the CMB, evidence of cosmic inflation and of primordial gravitational waves.

This talk is part of Caltech's regular weekly Physics Research Conference series. It will be held at 4 p.m. in the Feynman Lecture Hall, 201 East Bridge on the Caltech campus.

## BICEP2 Discovers First Direct Evidence of Inflation and Primordial Gravitational Waves

Astronomers announced today that they have acquired the first direct evidence that gravitational waves rippled through our infant universe during an explosive period of growth called inflation. This is the strongest confirmation yet of cosmic inflation theories, which say the universe expanded by 100 trillion trillion times in less than the blink of an eye.

"The implications for this detection stagger the mind," says Jamie Bock, professor of physics at Caltech, laboratory senior research scientist at the Jet Propulsion Laboratory (JPL) and project co-leader. "We are measuring a signal that comes from the dawn of time."

Our universe burst into existence in an event known as the Big Bang 13.8 billion years ago. Fractions of a second later, space itself ripped apart, expanding exponentially in an episode known as inflation. Telltale signs of this early chapter in our universe's history are imprinted in the skies in a relic glow called the cosmic microwave background. Tiny fluctuations in this afterglow provide clues to conditions in the early universe.

Small, quantum fluctuations were amplified to enormous sizes by the inflationary expansion of the universe. This process created density waves that make small differences in temperature across the sky where the universe was denser, eventually condensing into galaxies and clusters of galaxies. But as theorized, inflation should also produce gravitational waves, ripples in space-time propagating throughout the universe. Observations from the BICEP2 telescope at the South Pole now demonstrate that gravitational waves were created in abundance during the early inflation of the universe.

On Earth, light can become polarized by scattering off surfaces, such as a car or pond, causing the glare that polarized sunglasses are designed to reduce. In space, the radiation of the cosmic microwave background, influenced by the squeezing of gravitational waves, was scattered by electrons, and became polarized, too.

Because gravitational waves have a "handedness"—they can have both left- and right-handed polarizations—they leave behind a characteristic pattern of polarization on the cosmic microwave background known as B-mode polarization. "The swirly B-mode pattern of polarization is a unique signature of gravitational waves," says collaboration co-leader Chao-Lin Kuo of Stanford University and the SLAC National Accelerator Laboratory. This is the first direct image of gravitational waves across the primordial sky."

In order to detect this B-mode polarization, the team examined spatial scales on the sky spanning about one to five degrees (two to 10 times the width of the full moon), which allowed them to gather photons from a broad swath of the cosmic microwave background in an area of the sky where we can see clearly through our own Milky Way galaxy. To do this, the team traveled to the South Pole to take advantage of the cold, dry, stable air. "The South Pole is the closest you can get to space and still be on the ground," says John Kovac of the Harvard-Smithsonian Center for Astrophysics, project co-leader and BICEP2 principal investigator. "It's one of the driest and clearest locations on Earth, perfect for observing the faint microwaves from the Big Bang."

The team also invented completely new technology for making these measurements. "Our approach was like building a camera on a printed circuit board," says Bock. "The circuit board included an antenna to focus and filter polarized light, a micro-machined detector that turns the radiation into heat, and a superconducting thermometer to measure this heat." The detector arrays were made at JPL's Microdevices Laboratory.

The BICEP2 team was surprised to detect a B-mode polarization signal considerably stronger than many cosmologists expected. The team analyzed the data for more than three years in an effort to rule out any errors. They also considered whether dust in our galaxy could produce the observed pattern, but the data suggest this is highly unlikely. "This has been like looking for a needle in a haystack, but instead we found a crowbar," says project co-leader Clem Pryke, of the University of Minnesota.

The prediction that the cosmic microwave background would show a B-mode polarization from gravitational waves produced during the inflationary period was made in 1996 by several theoretical physicists including Marc Kamionkowski, who was a member of the Caltech faculty from 1999 to 2011, and is now on the faculty at Johns Hopkins University. Kamionkowski says this discovery "is powerful evidence for inflation. I'd call it a smoking gun. We've now learned that gravitational waves are abundant, and can learn more about the process that powered inflation. This is a remarkable advance in cosmology."

The BICEP project originated at Caltech in 2002 as a collaboration between Bock and the late physicist Andrew Lange.

BICEP2 is the second stage of a coordinated program with the BICEP and Keck Array experiments, which has a co-PI structure. The four principal investigators are Bock, Kovac, Kuo, and Pryke. All have worked together on the present result, along with talented teams of students and scientists. Other major collaborating institutions for BICEP2 include the University of California at San Diego, the University of British Columbia, the National Institute of Standards and Technology, the University of Toronto, Cardiff University, and Commissariat à l'energie atomique.

BICEP2 is funded by the National Science Foundation. NSF also runs the South Pole Station where BICEP2 and the other telescopes used in this work are located. The W. M. Keck Foundation also contributed major funding for the construction of the team's telescopes. NASA, JPL, and the Gordon and Betty Moore Foundation generously supported the development of the ultrasensitive detector arrays that made these measurements possible.

There are two papers, published March 17, 2014, reporting these results: "BICEP2 I: Detection of B-mode polarization at degree angular scales" and "BICEP2 II: Experiment and Three-Year Data Set."

The journal papers, along with additional technical details, can be found on the BICEP2 release website.