Reducing Coincidence with Mathematics: An Interview with Nets Katz

Raised in Grand Prairie, Texas, Nets Katz began pursuing mathematics at an early age, earning a bachelor's from Rice University in 1990 at the age of 17 and a doctorate from the University of Pennsylvania in 1993 at 20. After completing several postdoctoral fellowships, Katz went on to an assistant professorship at the University of Illinois at Chicago, an associate professorship at Washington University in Saint Louis, and a full professorship at Indiana University before joining the faculty at Caltech in January. Recently, Katz answered a few questions about his move to Southern California and his research in a field of math called combinatorics.

What brought you to Caltech, and why are you excited to be here?

I was offered a job here! It's a great institution; I've always admired it. Tom Wolff, who was a math professor here about 12 years ago, was a major influence in my career, and I flatter myself to think that I'm continuing some of his work.

I'm also really excited about teaching these students. I'm teaching Math 1 in the fall, and I'm really looking forward to the unique opportunity to get across deep and useful ideas to the very best students, including students who aren't in my field. Mathematics has always had a significant impact on the other sciences and engineering, and I think it will continue to do so.

What are your research interests?

I'm interested in showing that you can't have very many coincidences. The problems that I'm interested in are mostly considered to be in combinatorics [a field of math concerned with finding maximum, minimum, and optimum configurations—such as the absolute largest or smallest possible size of an object]. In a problem I worked on a few years ago, called the Erdos distance problem, we wanted to know the minimum number of distinct distances possible between a set number of points in a plane.

For example, say you're playing a game in which you ask an opponent to draw a finite number of n dots on a sheet of paper; the object of the game is for the opponent to position their dots so that the number of distinct distances between the dots is as small as possible. You then determine how well they did by tallying the number of distinct distances between these dots. If the dots are truly positioned randomly, some of the distances between the dots could be the same, but almost all of the distances will be different—meaning your opponent didn't do very well.

Placing the dots in a grid-like lattice pattern would be a relatively good strategy to win the game—this arrangement allows you to position the dots in such a way that a lot of the distances are the same. When working on this problem, we were able to prove that you can't get fewer than n/log n distinct distances—which, surprisingly, means that there isn't a strategy much better than the lattice. If there were a better strategy or design, it would involve a lot of coincidences—and too many coincidences aren't possible. You have to have some really incredibly special design to come close to the lattice arrangement, and what we were able to show is that even the best "incredibly special design" really isn't better than the lattice by very much.

How did you first become interested in math?

My father was a physicist, so we would have conversations about math at the dinner table. Of course, he was very far from a mathematician, but among physicists in his day he was quite well versed in math. And he had criticisms of how math was done, not all of which made sense, so we would have discussions about the foundations of things that were really exciting.

Here we were in Grand Prairie—I, a little kid no one had ever heard of, and he, a physicist largely forgotten—talking about how we might set the foundations of mathematics in a more clever way than all the denizens of the MITs and Caltechs of the world had ever managed to do. It was incredibly empowering. If it weren't constrained by the substantive requirements of mathematics, it might have been megalomaniacal. He made me feel that a person out of nowhere could really change the way people think about things. This was very exciting to me.

How do you like living in Southern California?

Actually, I have to admit I'm much more of an Indiana person—I prefer small towns and rural areas to densely populated cities—so I find I am experiencing a lot of culture shock living in Southern California. But there are nice things about the location; I have a lot of friends at UCLA and being closer is definitely a plus.

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Sky Survey Captures Key Details of Cosmic Explosions

Caltech astronomers report on unique results from the intermediate Palomar Transient Factory

Developed to help scientists learn more about the complex nature of celestial objects in the universe, astronomical surveys have been cataloguing the night sky since the beginning of the 20th century. The intermediate Palomar Transient Factory (iPTF)—led by the California Institute of Technology (Caltech)—started searching the skies for certain types of stars and related phenomena in February. Since its inception, iPTF has been extremely successful in the early discovery and rapid follow-up studies of transients—astronomical objects whose brightness changes over timescales ranging from hours to days—and two recent papers by iPTF astronomers describe first-time detections: one, the progenitor of a rare type of supernova in a nearby galaxy; the other, the afterglow of a gamma-ray burst in July.

The iPTF builds on the legacy of the Caltech-led Palomar Transient Factory (PTF), designed in 2008 to systematically chart the transient sky by using a robotic observing system mounted on the 48-inch Samuel Oschin Telescope on Palomar Mountain near San Diego, California. This state-of-the-art, robotic telescope scans the sky rapidly over a thousand square degrees each night to search for transients.

Supernovae—massive exploding stars at the end of their life span—make up one important type of transient. Since PTF's commissioning four years ago, its scorecard stands at over 2,000 spectroscopically classified supernovae. The unique feature of iPTF is brand-new technology that is geared toward fully automated, rapid response and follow-up within hours of discovery of a new supernova.

The first paper, "Discovery, Progenitor and Early Evolution of a Stripped Envelope Supernova iPTF13bvn," appears in the September 20 issue of Astrophysical Journal Letters and describes the detection of a so-called Type Ib supernova. Type Ib supernovae are rare explosions where the progenitor star lacks an outer layer of hydrogen, the most abundant element in the universe, hence the "stripped envelope" moniker. It has proven difficult to pin down which kinds of stars give rise to Type Ib supernovae. One of the most promising ideas, says graduate student and lead author Yi Cao, is that they originate from Wolf-Rayet stars. These objects are 10 times more massive and thousands of times brighter than the sun and have lost their hydrogen envelope by means of very strong stellar winds. Until recently, no solid evidence existed to support this theory. Cao and colleagues believe that a young supernova that they discovered, iPTF13bvn, occurred at a location formerly occupied by a likely Wolf-Rayet star.

Supernova iPTF13bvn was spotted on June 16, less than a day after the onset of its explosion. With the aid of the adaptive optics system used by the 10-meter Keck telescopes in Hawaii—which reduces the blurring effects of Earth's atmosphere—the team obtained a high-resolution image of this supernova to determine its precise position. Then they compared the Keck image to a series of pictures of the same galaxy (NGC 5806) taken by the Hubble Space Telescope in 2005, and found one starlike source spatially coincident to the supernova. Its intrinsic brightness, color, and size—as well as its mass-loss history, inferred from supernova radio emissions—were characteristic of a Wolf-Rayet star.

"All evidence is consistent with the theoretical expectation that the progenitor of this Type Ib supernova is a Wolf-Rayet star," says Cao. "Our next step is to check for the disappearance of this progenitor star after the supernova fades away. We expect that it will have been destroyed in the supernova explosion."

Though Wolf-Rayet progenitors have long been predicted for Type Ib supernova, the new work represents the first time researchers have been able to fill the gap between theory and observation, according to study coauthor and Caltech alumna Mansi Kasliwal (PhD '11). "This is a big step in our understanding of the evolution of massive stars and their relation to supernovae," she says.

The second paper, "Discovery and Redshift of an Optical Afterglow in 71 degrees squared: iPTF13bxl and GRB 130702A," appears in the October 20 issue of Astrophysical Journal Letters. Lead author Leo Singer, a Caltech grad student, describes finding and characterizing the afterglow of a long gamma-ray burst (GRB) as being similar to digging a needle out of a haystack.

Long GRBs, which are the brightest known electromagnetic events in the universe, are also connected with the deaths of rapidly spinning, massive stars. Although such GRBs initially are detected by their high-energy radiation—GRB 130702A, for example, was first located by NASA's Fermi Gamma-ray Space Telescope—an X-ray or visible-light afterglow must also be found to narrow down a GRB's position enough so that its location can be pinpointed to one particular galaxy and to determine if it is associated with a supernova.

After Fermi's initial detection of GRB 130702A, iPTF was able to narrow down the GRB's location by scanning an area of the sky over 360 times larger than the face of the moon and sifting through hundreds of images using sophisticated machine-learning software; it also revealed the visible-light counterpart of the burst, designated iPTF13bxl. This is the first time that a GRB's position has been determined precisely using optical telescopes alone.

After making the initial correlation between the GRB and the afterglow, Singer and colleagues corroborated their results and gained additional information using a host of other instruments, including optical, X-ray, and radio telescopes. In addition, ground-based telescopes around the world monitored the afterglow for days as it faded away, and recorded the emergence of a supernova five days later.

According to Singer, GRB130702A / iPTF13bxl turned out to be special in many ways.

"First, by measuring its redshift, we learned that it was pretty nearby as far as GRBs go," he says. "It was pretty wimpy compared to most GRBs, liberating only about a thousandth as much energy as the most energetic ones. But we did see it eventually turn into a supernova. Typically we only detect supernovae in connection with nearby, subluminous GRBs, so we can't be certain that cosmologically distant GRBs are caused by the same kinds of explosions."

"The first results from iPTF bode well for the discovery of many more supernovae in their infancy and many more afterglows from the Fermi satellite", says Shrinivas Kulkarni, the John D. and Catherine T. MacArthur Professor of Astronomy and Planetary Science at Caltech and principal investigator for both the PTF and iPTF.

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


Katie Neith
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Let There Be Light: Finding the Earliest Galaxies

Watson Lecture Preview

Richard S. Ellis, the Steele Family Professor of Astronomy, is on the verge of seeing as far back as it is possible to see—not quite back to the dawn of time itself but to the dawn of the first galaxies. He describes the journey at 8 p.m. on Wednesday, October 16, in Caltech's Beckman Auditorium. Admission is free.


Q: What do you do?

A: I study distant galaxies. When we look deep into space with the Hubble Space Telescope and at the W. M. Keck Observatory, we're looking back in time. I'm trying to look back to the moment the very first stars switched on—to the "cosmic dawn." Before then, there was no starlight, and we call that period the Dark Ages.

The universe today is 13.8 billion years old, and we can now look about 97 percent of the way back to the Big Bang. We think we need to look maybe another half-percent farther back, or about another 100 million years, in order to find the very earliest objects. We will do this with a large telescope called the Thirty Meter Telescope, which should begin construction in April, and the James Webb Space Telescope, which, I hope, will be launched in 2018.

The very first stars would only contain hydrogen and helium, and we will be able to recognize them by their colors and spectra. Our models tell us that such objects, free from heavy elements such as magnesium and iron, should be extraordinarily blue.

You might ask, "Why is this important?" And the answer is that, in many ways, we're searching for our origins. Stars produced the elements that make up you and me and everything else around us. We are, in a sense, stardust. The discovery of the microwave background radiation from the Big Bang was a big milestone in cosmic history. The moment when the universe switched on its starlight—cosmic dawn—is another important milestone.


Q: What is the best thing about what you do?

A: Above all, I like observing with large telescopes; it's inspirational to search for objects that are extraordinarily distant. And I like working with graduate students; it's remarkable that Caltech students can accompany me to the telescope and share in the discoveries.


Q: How did you get into this line of work?

A: I became convinced I wanted to be an astronomer when I was six years old. I grew up in Wales, and our public library had a little book called Out Into Space by the late Sir Patrick Moore, the famous popularizer of astronomy. It was about a boy and girl who went to their eccentric uncle's house, and he had a telescope. That hooked me. I never wanted to do anything else. Forty years later, I appeared on his television program, The Sky at Night, and he gave me a signed copy. And when I reread it, I still remembered the relevant chapters.


Q: So, do you now qualify as an eccentric uncle yourself?

A: I do, yeah [laughter]. I am an uncle, and I definitely am eccentric. You can ask my sister's kids [more laughter].


Named for the late Caltech professor Earnest C. Watson, who founded the series in 1922, the Watson Lectures present Caltech and JPL researchers describing their work to the public. Many past Watson Lectures are available online at Caltech's iTunes U site.

Douglas Smith
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Watson Lecture: Let There Be Light: Finding the Earliest Galaxies
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Caltech Named World's Top University in Times Higher Education Global Ranking

For the third year in a row, the California Institute of Technology has been rated the world's number one university in the Times Higher Education global ranking of the top 200 universities.

Harvard University, Oxford University, Stanford University, and the Massachusetts Institute of Technology round out the top five schools in the 2013–2014 rankings.

Times Higher Education compiled the listing using the same methodology as in the 2011–2012 and 2012–2013 surveys. Thirteen performance indicators representing research (worth 30 percent of a school's overall ranking score), teaching (30 percent), citations (30 percent), international outlook (which includes the total numbers of international students and faculty and the ratio of scholarly papers with international collaborators, 7.5 percent), and industry income (a measure of innovation, 2.5 percent) make up the data. The data were collected, analyzed, and verified by Thomson Reuters.

The Times Higher Education site has the full list of the world's top 400 schools and all of the performance indicators.

Kathy Svitil
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Caltech to Offer Online Courses through edX

To expand its involvement in online learning, the California Institute of Technology will offer courses through the online education platform edX beginning this October.

The edX course platform is an online learning initiative launched in 2012 by founding partners Harvard University and the Massachusetts Institute of Technology (MIT). Caltech's rigorous online course offerings will join those of 28 other prestigious colleges and universities in the edX platform's "xConsortium."

This new partnership with edX comes one year after Caltech offered three courses through the online learning platform Coursera in fall 2012. The Institute will now offer courses through both platforms.

"Coursera and edX have some foundational differences which are of interest to the faculty," says Cassandra Horii, director of teaching and learning programs at Caltech. Both organizations offer their courses at no cost to participating students; edX, however, operates as a nonprofit and plans to partner with only a small number of institutions, whereas Coursera—a for-profit, self-described "social entrepreneurship company"—partners with many institutions and state university systems.

The two platforms also emphasize different learning strategies, says Horii. "Coursera has a strong organizational principle built around lectures, so a lot of the interactivity is tied right into the video," she says. Though edX still enables the use of video lectures, a student can customize when he or she would like to take quizzes and use learning resources. In addition, edX allows faculty to embed a variety of learning materials—like textbook chapters, discussions, diagrams, and tables—directly into the platform's layout.

In the future, data collected from both platforms could provide valuable information about how students best learn certain material, especially in the sciences. "Caltech occupies this advanced, really rigorous scientific education space, and in general our interest in these online courses is to maintain that rigor and quality," Horii says. "So, with these learning data, we have some potential contributions to make to the general understanding of learning in this niche that we occupy."

Even before joining edX and Coursera, Caltech had already become an example in the growing trend of Massive Open Online Courses (MOOCs). Yaser Abu-Mostafa, professor of electrical engineering and computer science, developed his own MOOC on machine learning, called "Learning from Data," and offered it on YouTube and iTunes U beginning in April 2012.

Since its debut, Abu-Mostafa's MOOC has reached more than 200,000 participants, and it received mention in the NMC Horizon Report: 2013 Higher Education Edition—the latest edition of an annual report highlighting important trends in higher education. The course will be offered again in fall 2013 on iTunes U, and is now also open for enrollment in edX.

Although Caltech is now actively exploring several outlets for online learning, the Institute's commitment to educational outreach is not a recent phenomenon. In the early 1960s, Caltech physicist Richard Feynman reorganized the Institute's introductory physics course, incorporating contemporary research topics and making the course more engaging for students. His lectures were recorded and eventually incorporated into a widely popular physics book, The Feynman Lectures on Physics, which has sold millions of copies in a dozen languages.

Continuing in the tradition set by Feynman, the MOOCs at Caltech seek to provide a high-quality learning environment that is rigorous but accessible. "No dumbing down of courses for popular consumption . . . no talking over people's heads either; at Caltech, we explain things well because we understand them well," adds Abu-Mostafa.

More information on Caltech's online learning opportunities is available on the Online Education website.

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Friday, October 4, 2013

Undergraduate Teaching Assistant Orientation

Monday, August 12, 2013
Cahill, Hameetman Auditorium – Cahill Center for Astronomy and Astrophysics

Magnetic Fields: A Window to a Planet's Interior and Habitability

Thirty Meter Telescope Project Partners Sign Master Agreement

Document formalizes Caltech’s collaboration among several other international institutions.

Scientific authorities on the Thirty Meter Telescope (TMT) project announced on Friday that they have now signed a master agreement formalizing project goals and providing a governing framework for the international collaboration. TMT is a partnership among Caltech, the University of California, and a number of astronomical observatories and institutions from Canada, China, India, and Japan.

The agreement establishes an official commitment among the partners and delineates the rights and obligations of its global collaborators. These measures are intended to ensure steady progress for the TMT, which is planned to start construction in April 2014 and begin scientific operations in 2022.

"We are pleased with this vote of confidence from the scientific authorities," said Edward Stone, the David Morrisroe Professor of Physics and vice provost for special projects at Caltech, and vice chair of the TMT Board. "Their signing of this master agreement is a key endorsement of TMT's scientific merits as well as the project's overall implementation plan."

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Thirty Meter Telescope Project Partners Sign Master Agreement
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Thursday, September 26, 2013

Graduate TA Orientation & Teaching Conference

Friday, July 19, 2013
Cahill, Hameetman Auditorium – Cahill Center for Astronomy and Astrophysics

"Are We Alone?" Public lecture by Dr. Jill Tarter