Mathematicians trace source of Rogers-Ramanujan identities, find algebraic gold

Nautilus shell spirals are among the many forms in nature that can be related to the golden ratio, the most famous algebraic number of them all. Now mathematicians have discovered a new treasure trove of algebraic numbers and formulas to access them.

By Carol Clark

Mathematicians have found a framework for the celebrated Rogers-Ramanujan identities and their arithmetic properties, solving another long-standing mystery stemming from the work of Indian math genius Srinivasa Ramanujan.

The findings, by mathematicians at Emory University and the University of Queensland, yield a treasure trove of algebraic numbers and formulas to access them.

“Algebraic numbers are among the first numbers you encounter in mathematics,” says Ken Ono, a number theorist at Emory “And yet, it’s surprisingly difficult to find functions that return them as values in a uniform and systematic way.”

Ono is the co-author of the new findings, along with S. Ole Warnaar of the University of Queensland and Michael Griffin, an Emory graduate student.

Ono announced the findings in April as a plenary speaker at the Applications of Automorphic Forms in Number Theory and Combinatorics conference at Louisiana State University. He will also present them as a plenary speaker at the 2015 Joint Mathematics Meetings, the largest mathematics meeting in the world, set for January in San Antonio. Warnaar, Griffin and others will give additional talks on the findings during an invited special session to accompany Ono’s plenary address.

Ramanujan had "a Midas touch."

The most famous algebraic number of all is the golden ratio, also known by the Greek letter phi. Many great works of architecture and art, such as the Parthenon, are said to embody the pleasing proportions of the golden ratio, which is also seen in beautiful forms in nature. Mathematicians, artists and scientists, from ancient times to today have pondered the qualities of phi, which is approximately equal to 1.618, although its digits just keep on going, with no apparent pattern.

“People studied the golden ratio before there was a real theory of algebra,” Ono says. “It was a kind of prototype for algebraic numbers.”

Although no other algebraic units are as famous as the golden ratio, they are of central importance to algebra. “A fundamental problem in mathematics is to find functions whose values are always algebraic numbers,” Ono says. “The famous Swiss mathematician Leonhard Euler made some progress on this problem in the 18th century. His theory of continued fractions, where one successively divides numbers in a systematic way, produces some very special algebraic numbers like the golden ratio. But his theory cannot produce algebraic numbers which go beyond the stuff of the quadratic formula that one encounters in high school algebra.”

Ramanujan, however, could produce such numbers, and he made it look easy.

“Ramanujan has a very special, almost mythic, status in mathematics,” says Edward Frenkel, a mathematician at the University of California, Berkeley. “He had a sort of Midas touch that seemed to magically turn everything into gold.”

And the Rogers-Ramanujan identities are considered among Ramanujan’s greatest legacies, adds Frenkel, a leading expert on the identities.

“They are two of the most remarkable and important results in the theory of q-series, or special functions,” says Warnaar, who began studying the Rogers-Ramanujan identities shortly after he encountered them while working on his PhD in statistical mechanics about 20 years ago.

Although originally discovered by L. J. Rogers in 1894, the identities became famous through the work of Ramanujan, who was largely self-taught and worked instinctively.

The Rogers-Ramanujan identities are among Ramanujan's greatest legacies.

In 1913, Ramanujan sent a letter from his native India to the British mathematician G. H. Hardy that included the two identities that Rogers discovered and a third formula that showed these identities are essentially modular functions and their quotient has the special property that its singular values are algebraic integral units. That result came to be known as the Rogers-Ramanujan continued fraction. 

Hardy was astonished when he saw the formulas. “I had never seen anything in the least like this before,” Hardy wrote. “A single look at them is enough to show they could only be written down by a mathematician of the highest class. They must be true because no one would have the imagination to invent them.”

“Ramanujan seemed to produce this result out of thin air,” Ono says.

Ramanujan died in 1920 before he could explain how he conjured up the formulas. “They have been cited hundreds of times by mathematicians,” Ono says. “They are used in statistical mathematics, conformal field theory and number theory. And yet no one knew whether Ramanujan just stumbled onto the power of these two identities or whether they were fragments of a larger theory.”

For nearly a century, many great mathematicians have worked on solving the mystery of where Ramanujan’s formulas came from and why they should be true.

"Ramanujan has a very special, almost mythic, status among mathematicians," says Frenkel. Above is a still photo from an upcoming film, "Ramanujan," a biography of the math genius by Camphor Cinema.

Ono uses the analogy of going for a walk in a creek bed and discovering a piece of gold. Had Ramanujan accidentally found a random nugget? Or was he drawn to that area because he knew of a rich seam of gold nearby?

Warnaar was among those who pondered these questions. “Just like digging for gold, in mathematics it’s not always obvious where to look for a solution,” he says. “It takes time and effort, with no guarantee of success, but it helps if you develop a lot of intuition about where to look.”

Finally, after 15 years of focusing almost entirely on the Rogers-Ramanujan identities, Warnaar found a way to embed them into a much larger class of similar identities using something known as representation theory.

“Ole found the mother lode of identities,” Ono says.

When Ono saw Warnaar’s work posted last November on arXiv.org, a mathematics-physics archive, his eyes lit up.

“It just clicked,” Ono recalls. “Ole found this huge vein of gold, and we then figured out a way to mine the gold. We went to work and showed how to come full circle and make use of the formulas. Now we can extract infinitely many functions whose values are these beautiful algebraic numbers.” 

“Historically, the Rogers-Ramanujan identities have tantalized mathematicians,” says George Andrews, a mathematician at Penn State and another top authority on the identities. “They have played an almost magical role in many areas of math, statistical mechanics and physics.”

The collaboration of Warnaar, Ono and Griffin “has given us a big picture of the general setting for these identities, and deepened our theoretical understanding for many of the breakthroughs in this area of mathematics during the past 100 years,” Andrews says. “They’ve given us a whole new set of tools to be able to attack new problems.”

“It’s incredibly exciting to solve any problem related to Ramanujan, he’s such an important figure in mathematics,” Warnaar says. “Now we can move on to more questions that we don’t understand. Math is limitless, and that’s fantastic.”

Related:
Math formula gives new glimpse into the magical mind of Ramanujan
New theories reveal the nature of numbers
How a hike in the woods led to a math 'Eureka!'

Image credits: Top, iStockphoto.com; center, Wikipedia Commons; bottom, Camphor Cinema

Lack of respect for insects bugs a biologist

"Hey, watch where you step! Just because I'm an insect doesn't mean you should crush me!" Emory biologist Jaap de Roode will give students a bug's-eye view of the world in his Maymester course "Insect Biology." (Photo by CaPro/Wikipedia Commons.)

“More than half of all animal species alive today are insects and it’s believed that there’s at least a million species of beetles alone,” says Emory biologist Jaap de Roode. “Yet most people will just see an insect and call their pest control agency and move on, and that’s it. What they’re missing is this beautiful understanding of these creatures.”

De Roode runs one of the few labs in the world focused on monarch butterflies. He’s discovered that monarchs actually use medication they find in nature to treat their young for parasites.

He hopes to inspire more students to appreciate the world of insects in his Maymester course “Insect Biology.” Through lectures, labs and field trips, students will learn to distinguish the major groups of insects and to analyze the importance of insects for ecology, human food production and health.

“We’re going to address a lot of questions,” de Roode says. “Why is it that only female mosquitoes bite and female bees sting? And how do these insects live in these amazing societies where there is specialization of different tasks?”

The course will also cover insects and human culture, including questions such as: Why are there so many beetle depictions in Egyptian art?

“Insects have amazing sex lives,” de Roode adds. “In some cases, it’s flowers and chocolate and romance. But not so in other cases. During the mating of praying mantises the female will often bite off the head of the male -- if he’s lucky, during mating, and if he’s unlucky, before it. We’ll talk about why that is and how that behavior evolved.”

Watch the video above to hear more about the Insect Biology course.

Click here to learn more about other Maymester and summer courses. 

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'Math detective' analyzes odds for suspicious lottery wins

Emory mathematician Skip Garibaldi (above in a classroom) helped do the math for a Palm Beach Post investigation of suspicious wins in the Florida Lottery. Garibaldi has since started eyeing data from the Georgia Lottery. (Emory Photo/Video)

By Carol Clark

When investigative reporter Lawrence Mower decided to dig into public records for the winners of the Florida Lottery, he noticed an intriguing pattern. Over a decade, a few names kept popping up as winners of all kinds of games. The most prolific of these winners, according to the lottery data, was a man who claimed an incredible 252 prizes during six years, for a total take of $719,000.

“If you’re winning like this in Las Vegas, they’re going to take you into a back room and find out how you’re cracking the game,” Mower said.

But when Mower asked a Florida lottery official about what seemed like a suspicious number of repeat wins by some players he was told that they could just be lucky.

Mower and his colleagues at the Palm Beach Post wanted to find out exactly how lucky these dominant players were, but they needed help calculating the odds.

“I starting looking for a mathematician who had dealt with odds and the lottery,” Mower says. “That’s how I found Skip.”

Skip Garibaldi, a professor in Emory’s Department of Mathematics and Computer Science and associate director of UCLA’s Institute for Pure and Applied Mathematics, was happy to work on the project.

“It was like a dream come true,” says Garibaldi, whose previous research on lotteries received the Lester R. Ford Award for mathematics and is the subject of a chapter in the popular book “Brain Trust” by Garth Sundem. Garibaldi enjoys breaking down complex math for the general public and has appeared on 20/20, CNN and Fox & Friends.

“I get this call from Lawrence, and he says, ‘I have this huge database I’d like to feed you of everyone who has won more than a $600 prize in the Florida Lottery over a decade,” Garibaldi recalls.

Philip Stark, a statistician from the University of California, Berkeley, and Richard Arratia, a probabilist from the University of Southern California, were also recruited to work with Garibaldi on the project. The three mathematicians are now writing an academic paper that will explain their lottery findings in more technical detail.

Two of the underlying principles for the lottery analysis were probability theory and the law of large numbers, which both trace their beginnings to a 16th-century Italian mathematician, Gerolamo Cardano. In fact, it was his love of dice games and other forms of gambling that sparked Cardano to work on probability questions.

“The subtle point about probability or quantum mechanics,” Garibaldi says, “is that there are things that we know are possible and we can calculate the probability of them happening, but they are so unlikely that no one has seen them happen and likely no one ever will. For this lottery question, something that happens to fewer than one-in-20-trillion lottery gamblers is one of these utterly implausible events.” 

Using this generous bar for random luck, the analysis identified winners who were defying the odds during the past decade, and would have had to lose prodigious amounts of money to win so many times. The most prolific winner, for instance, would have had to spend an estimated minimum of $2.07 million to have a one-in-20-trillion shot at his 252 wins and winnings of $719,051, for a net loss of about $2.35 million.

“But even if every single citizen in the state of Florida spent $2 million on lottery tickets,” Garibaldi says, “the odds are less than one in a million that anybody would have won that many times.”

These kinds of figures “put us in pretty safe territory with our suspicions that something was amiss,” Garibaldi says.

The mathematical analysis was also able to identify winners of multiple prizes who appeared to be legitimate gamblers. “The fact that a person claims multiple prizes does not necessarily indicate that they are doing anything suspicious,” Garibaldi says. “A finer analysis and more inspection can show that some people are likely just spending a lot of money on tickets and occasionally getting lucky.”

“The math was the critical part of this story,” Mower says of the package of investigative pieces published on Sunday, March 30. “It’s been really valuable working with Skip because he’ll see something that I won’t.”

On Monday, March 31, the state legislature called for more oversight of the Florida Lottery. On Tuesday, the lottery announced that it would adopt safeguards, such as software to track frequent winners. By Wednesday, lottery officials began raiding stores and seizing lottery equipment associated with some of the top prize winners. (Six of the 10 top winners in the lottery records were store clerks and owners who sold lottery tickets.)

Meanwhile, Garibaldi started eyeing data for winners of the Georgia Lottery. “The data from the Georgia Lottery is not as good as from Florida, because Georgia only lists winners for prizes of $5,000 or more,” he says. “But just casually looking at the Georgia data since 2003, I see what may be suspicious numbers for repeat winners.”

Related:
Lottery study zeroes in on risk
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Seeing the tropical forest amid disease

"Insights over the past three decades have clarified how the health and persistence of tropical forest systems depend on critical ecosystem services provided by wildlife," Emory disease ecologist Thomas Gillespie told Liz Kimbrough in a recent intreview for the conservation news site Mongabay. "We've lost pollinators—honeycreeper declines in Hawaii due to introduced malaria; seed dispersers—lowland gorilla declines due to Ebola; and indicator species—frog declines and extinctions due to chytrid fungus."

Gillespie at work in Uganda

Gillespie, an associate professor in Emory's Department of Environmental Sciences and the Rollins School of Public Health, was featured as one of the 12 top innovators in tropical conservation. Below is an excerpt from the interview:

Mongabay: What's the next big thing in forest conservation? What approaches or ideas are emerging or have recently emerged? What will be the catalyst for the next big breakthrough?

Thomas Gillespie: In the past decade, technological advances in non-invasive pathogen surveillance have allowed us to make great strides in understanding how infectious diseases may threaten endangered species (i.e., fecal assays to examine respiratory pathogens and blood-borne pathogens like malaria and immunodeficiency viruses). Similarly, an exciting innovation that's just taking off is letting mosquitos, leeches, and carrion flies do the work for us! All of these invertebrates seek out wildlife for a meal. By trapping them and analyzing their gut contents, we can determine which wildlife species they fed on, and in some cases, the pathogens that infected those individuals. This is especially interesting in regards to carrion flies, since they feed more or less indiscriminately on dead or dying animals. As a result, screening large numbers of carrion flies in a selected tropical forests could provide an inventory of faunal and pathogen diversity including data on the presence or absence of cryptic species. In addition, this method may alert wildlife managers to major mortality events (i.e., based on anomalies in the frequency and amount of DNA from a specific vertebrate species being recovered).

Read the whole interview at mongabay.com.

Related:
In Madagascar, a health crisis of people and their ecosystem
Primate disease ecologist tracks germs in the wild
How germs jump species

Dengue study to focus on asymptomatic carriers

A NASA satellite image shows the metropolitan area of Iquitos, Peru, nestled in the Amazon Basin, on the banks of the Amazon River (lower left) and surrounded by smaller rivers, lakes and lagoons.

By Carol Clark

Dengue fever is a leading cause of illness and death in the tropics and subtropics, infecting as many as 400 million people annually, according to the CDC.

“Currently, the most effective way to control dengue outbreaks is to spray for mosquitoes and help people to avoid getting bit by them,” says Gonzalo Vazquez-Prokopec, a disease ecologist in Emory’s Department of Environmental Sciences.

Vazquez-Prokopec is a co-principal investigator on a major dengue research project ongoing in Iquitos, Peru, which is honing in on ways to control outbreaks of the disease and more effectively treat infections. The National Institutes of Health recently awarded $7 million to the project team, led by the University of California, Davis, and also including the U.S. Navy, North Carolina State University, the University of Iowa, Tulane University and San Diego State.

Emory’s portion of the grant – $1.3 million – will be used to study how people who are infected with a dengue virus, but not showing symptoms, may contribute to the spread of dengue.

Infections can spread like wildfire through urban areas of the developing world where many people live in close quarters in substandard housing. Mosquitoes are the vectors of the disease, transmitting the four viruses that cause dengue between people.

“It’s a complex disease, made even more complicated by the fact that four different species of dengue viruses can interact with one another,” Vazquez-Prokopec says. “And yet, we actually know more about these viruses than we do about the behaviors of the people who get infected with them.”

The Emory research team also includes Uriel Kitron, chair of Emory’s Department of Environmental Sciences, and Lance Waller, chair of the Department of Biostatistics and Bioinformatics at the Rollins School of Public Health.

Dengue is endemic in Iquitos, a city on the edge of the Amazon rainforest. A large portion of people who get infected do not experience the usual debilitating symptoms of dengue and continue going to work or visiting friends and relatives.

“We want to determine if these people are significant spreaders of the disease,” Vazquez-Prokopec says.

The experiment will require participants to undergo blood tests and wear a device that uses a GPS to track their movements and a body temperature monitor to record any bouts of fever. The data from asymptomatic people who are infected will be analyzed along with spatial-temporal data on infected people who become sick during an outbreak. Ultimately, the project will infer the contribution of asymptomatic people by linking those data with mathematical models simulating virus spread within the city.

Previously, Vazquez-Prokopec and colleagues used GPS technology to quantify the movement and contact dynamics of nearly 600 residents of Iquitos, where most people are self-employed or hold several jobs to try and make ends meet. The results showed that the participants visited an average of six locations per day overall, compared to cities in North America and Europe where urbanites visit an average of two to four locations daily.

The researchers have also conducted spatial-temporal analyses of dengue outbreak patterns through two large neighborhoods of Iquitos. When a case of dengue was confirmed through a blood test, social workers would interview the patient, recording all the places he patient went during the 15 days leading up to the onset of fever. That study found that one of the main drivers for infection was people visiting friends and relatives in nearby homes, as opposed to large gathering places like schools.

“This large project will shed light onto something that hasn’t been explored before: The role of people who are undetected by the health system in propagating dengue,” says Vazquez-Prokopec.

Related:
How the dengue virus makes a home in the city
Human mobility data may help curb urban epidemics

Sharing the 'wow' of science

Emory chemist Doug Mulford blows a fireball at the Atlanta Science Festival expo March 29. Emory Photo/Video.

By Megan Terraso, Emory Report

Rows of children sat with rapt attention, their mouths agape at what they were seeing at the Atlanta Science Festival.

"That look of wonder is why we do it," says Douglas Mulford, director of undergraduate studies of chemistry and senior lecturer at Emory. "We wanted to show the 'wow' of science and show how fascinating it can be."

Mulford was one of many Emory faculty, staff and students who participated in 25 Emory demonstrations and exhibits at the Atlanta Science Festival March 22-29. The massive weeklong festival included more than 100 events at nearly 35 venues and everything from tours and film screenings to trivia nights and flashy science demonstrations.

At the festival's science expo March 29 at the Georgia World Congress Center, Mulford and several Emory students put on a science stage show with bubbling beakers and exploding balloons to a packed room with an audience of around 1,000.

Mulford also oversaw the very popular "cornstarch dance pit," a three foot by three foot pit where visitors could dance or sink in the cornstarch and water mix. "As long as you dance, you stay up. When you stop, you sink. That was a lot of fun," Mulford says.

Other Emory exhibits at the expo, which attracted tens of thousands of visitors, included monarch butterflies, the opportunity to touch a real brain and a booth that allowed visitors to swab the bottom of their shoe or their ear and follow the growth of the bacteria they'd swabbed over a few hours or days via a website.

Read more at Emory Report.