Monday, December 15, 2014

Mathematicians prove the Umbral Moonshine Conjecture

In theoretical math, the term "moonshine" refers to an idea so seemingly impossible that it seems like lunacy.

By Carol Clark

Monstrous moonshine, a quirky pattern of the monster group in theoretical math, has a shadow – umbral moonshine. Mathematicians have now proved this insight, known as the Umbral Moonshine Conjecture, offering a formula with potential applications for everything from number theory to geometry to quantum physics.

“We’ve transformed the statement of the conjecture into something you could test, a finite calculation, and the conjecture proved to be true,” says Ken Ono, a mathematician at Emory University. “Umbral moonshine has created a lot of excitement in the world of math and physics.”

Co-authors of the proof include mathematicians John Duncan from Case Western Reserve University and Michael Griffin, an Emory graduate student.

“Sometimes a result is so stunningly beautiful that your mind does get blown a little,” Duncan says.

Duncan co-wrote the statement for the Umbral Moonshine Conjecture with Miranda Cheng, a mathematician and physicist at the University of Amsterdam, and Jeff Harvey, a physicist at the University of Chicago.

Ono will present their work on January 11, 2015 at the Joint Mathematics Meetings in San Antonio, the largest mathematics meeting in the world. Ono is delivering one of the highlighted invited addresses.

Ono gave a colloquium on the topic at the University of Michigan, Ann Arbor, in November, and has also been invited to speak on the umbral moonshine proof at upcoming conferences around the world, including Brazil, Canada, England, India, and Germany.

The number of elements in the monster group is larger than the number of atoms in 1,000 Earths.

It sounds like science fiction, but the monster group (also known as the friendly giant) is a real and influential concept in theoretical math.

Elementary algebra is built out of groups, or sets of objects required to satisfy certain relationships. One of the biggest achievements in math during the 20th century was classifying all of the finite simple groups. They are now collected in the ATLAS of Finite Groups, published in 1985.

“This ATLAS is to mathematicians what the periodic table is to chemists,” Ono says. “It’s our fundamental guide.”

And yet, the last and largest sporadic finite simple group, the monster group, was not constructed until the late 1970s. “It is absolutely huge, so classifying it was a heroic effort for mathematicians,” Ono says.

In fact, the number of elements in the monster group is larger than the number of atoms in 1,000 Earths. Something that massive defies description.

“Think of a 24-dimensional doughnut,” Duncan says. “And then imagine physical particles zooming through this space, and one particle sometimes hitting another. What happens when they collide depends on a lot of different factors, like the angles at which they meet. There is a particular way of making this 24-dimensional system precise such that the monster is its symmetry. The monster is incredibly symmetric.”

“The monster group is not just a freak,” Ono adds. “It’s actually important to many areas of math.”

It’s too immense, however, to use directly as a tool for calculations. That’s where representation theory comes in.

The shadow technique is a valuable tool in theoretical math.

Shortly after evidence for the monster was discovered, mathematicians John McKay and John Thompson noticed some odd numerical accidents. They found that a series of numbers that can be extracted from a modular function and a series extracted from the monster group seemed to be related. (One example is the strange and simple arithmetic equation 196884 = 196883 + 1.)

John Conway and Simon Norton continued to investigate and found that this peculiar pattern was not just a coincidence. “Evidence kept accumulating that there was a special modular function for every element in the monster group,” Ono says. “In other words, the main characteristics of the monster group could be read off from modular functions. That opened the door to representation theory to capture and manipulate the monster.”

The idea that modular functions could tame something as unruly as the monster sounded impossible – like lunacy. It was soon dubbed the Monstrous Moonshine Conjecture.

(The moonshine reference has the same meaning famously used by Ernest Rutherford, known as the father of nuclear physics. In a 1933 speech, Rutherford said that anyone who considered deriving energy from splitting atoms was "talking moonshine.”)

In 1998, Richard Borcherds won math’s highest honor, the Fields Medal, for proving the Monstrous Moonshine Conjecture. His proof turned this representation theory for the monster group into something computable.

Fast-forward 16 years. Three Japanese physicists (Tohru Eguchi, Hirosi Ooguri and Yuji Tachikawa) were investigating a particular kind of string theory involving four-dimensional spaces. The appearance of numbers from the Mathieu Group M24, another important finite simple group, was unexpected.

“They conjectured a new way to extract numbers from the Mathieu Group,” Duncan says, “and they noticed that the numbers they extracted were similar to those of the monster group, just not as large.” Mathematician Terry Gannon proved that their observations are true.

It was a new, unexpected analogue that hinted at a pattern similar to monstrous moonshine.

“What I hope is that we will eventually see that everything is unified, that monstrous moonshine and umbral moonshine have a common origin,” Duncan says.

Duncan started investigating this idea with physicists Cheng and Harvey. “We realized that the Mathieu group pattern was part of a much bigger picture involving mock modular forms and more moonshine,” Duncan says. “A beautiful mathematical structure was controlling it.”

They dubbed this insight the Umbral Moonshine Conjecture. Since the final version of the more than 100-page conjecture was published online last June, it has been downloaded more than 2,500 times.

The conjecture caught the eye of Ono, an expert in mock modular forms, and he began pondering the problem along with Griffin and Duncan.

“Things came together quickly after the statement of the Umbral Moonshine Conjecture was published,” Ono says. “We have been able to prove it and it is no longer a guess. We can now use the proof as a completely new and different tool to do calculations.”

Just as modular forms are “shadowed” by mock modular forms, monstrous moonshine is shadowed by umbral moonshine. (Umbra is Latin for the innermost and darkest part of a shadow.)

“The job of a theoretical mathematician is to take impossible problems and make them tractable,” Duncan says. “The shadow device is one valuable tool that lets us do that. It allows you to throw away information while still keeping enough to make some valuable observations.”

He compares it to a paleontologist using fossilized bones to piece together a dinosaur.

The jury is out on what role, if any, umbral moonshine could play in helping to unravel mysteries of the universe. Aspects of it, however, hint that it could be related to problems ranging from geometry to black holes and quantum gravity theory.

“What I hope is that we will eventually see that everything is unified, that monstrous moonshine and umbral moonshine have a common origin,” Duncan says. “And part of my optimistic vision is that umbral moonshine may be a piece in one of the most important puzzles of modern physics: The problem of unifying quantum mechanics with Einstein’s general relativity.”

Images: NASA and Thinkstock.

Mathematicians trace source of Rogers-Ramanujan identities
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Tuesday, December 9, 2014

Birdsong study reveals how brain uses timing during motor activity

Songbirds are one of the best systems for understanding how the brain controls complex behavior.  Image credit: Sam Sober.

By Carol Clark

Timing is key for brain cells controlling a complex motor activity like the singing of a bird, finds a new study published by PLOS Biology.

“You can learn much more about what a bird is singing by looking at the timing of neurons firing in its brain than by looking at the rate that they fire,” says Sam Sober, a biologist at Emory University whose lab led the study. “Just a millisecond difference in the timing of a neuron’s activity makes a difference in the sound that comes out of the bird’s beak.”

The findings are the first to suggest that fine-scale timing of neurons is at least as important in motor systems as in sensory systems, and perhaps more critical.

“The brain takes in information and figures out how to interact with the world through electrical events called action potentials, or spikes in the activity of neurons,” Sober says. “A big goal in neuroscience is to decode the brain by better understanding this process. We’ve taken another step towards that goal.”

Sober’s lab uses Bengalese finches, also known as society finches, as a model system. The way birds control their song has a lot in common with human speech, both in how it’s learned early in life and how it’s vocalized in adults. The neural pathways for birdsong are also well known, and restricted to that one activity.

“Songbirds are the best system for understanding how the brain controls complex vocal behavior, and one of the best systems for understanding control of motor behavior in general,” Sober says.

Researchers have long known that for an organism to interpret sensory information – such as sight, sound and taste – the timing of spikes in brain cells can matter more than the rate, or the total number of times they fire. Studies on flies, for instance, have shown that their visual systems are highly sensitive to the movement of shadows. By looking at the timing of spikes in the fly’s neurons you can tell the velocity of a shadow that the fly is seeing.

An animal’s physical response to a stimulus, however, is much slower than the millisecond timescale on which spikes are produced. “There was an assumption that because muscles have a relatively slow response time, a timing code in neurons could not make a difference in controlling movement of the body,” Sober says.

An Emory undergraduate in the Sober lab, Claire Tang, got the idea of testing that assumption. She proposed an experiment involving mathematical methods that she was learning in a Physical Biology class. The class was taught by Emory biophysicist Ilya Nemenman, an expert in the use of computational techniques to study biological systems.

“Claire is a gifted mathematician and programmer and biologist,” Sober says of Tang, now a graduate student at the University of California, San Francisco. “She made a major contribution to the design of the study and in the analysis of the results.”

Co-authors also include Nemenman; laboratory technician Diala Chehayeb; and Kyle Srivastava, a graduate student in the Emory/Georgia Tech graduate program in biomedical engineering.

The researchers used an array of electrodes, each thinner than a human hair, to record the activity of single neurons of adult finches as they were singing.

“The birds repeat themselves, singing the same sequence of ‘syllables’ multiple times,” Sober says. “A particular sequence of syllables matches a particular firing of neurons. And each time a bird sings a sequence, it sings it a little bit differently, with a slightly higher or lower pitch. The firing of the neurons is also slightly different.”

The acoustic signals of the birdsong were recorded alongside the timing and the rate that single neurons fired. The researchers applied information theory, a discipline originally designed to analyze communications systems such as the Internet or cellular phones, to analyze how much one could learn about the behavior of the bird singing by looking at the precise timing of the spikes versus their number.

The result showed that for the duration of one song signal, or 40 milliseconds, the timing of the spikes contained 10 times more information than the rate of the spikes.

“Our findings make it pretty clear that you may be missing a lot of the information in the neural code unless you consider the timing,” Sober says.

Such improvements in our understanding of how the brain controls physical movement hold many potential health applications, he adds.

“For example,” he says, “one area of research is focused on how to record neural signals from the brains of paralyzed people and then using the signals to control prosthetic limbs. Currently, this area of research tends to focus on the firing rate of the neurons rather than taking the precise timing of the spikes into account. Our work shows that, in songbirds at least, you can learn much more about behavior by looking at spike timing than spike rate. If this turns out to be true in humans as well, timing information could be analyzed to improve a patient’s ability to control a prosthesis.”

The research was supported by grants from the National Institutes of Health, the National Science Foundation, the James S. McDonnell Foundation and Emory’s Computational Neuroscience Training Program.

Bird graphic courtesy of Sam Sober.

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Thursday, December 4, 2014

Scientists zeroing in on psychosis risk factors

The onset of schizophrenia and other psychotic disorders typically occurs at about 21 years of age, with warning signs beginning around age 17, on average.

By Carol Clark

During the first phase of a major national study, scientists have uncovered a new cluster of preclinical symptoms linked to a significant increase in the risk that a young person will go on to develop a psychotic illness, including schizophrenia. The consortium of researchers, from Emory and seven other universities, has also discovered several biological processes tied to the transition from subtle symptoms to clinical psychosis.

The onset of schizophrenia and other psychotic disorders typically occurs at about 21 years of age, with warning signs, known as a prodromal syndrome, beginning around age 17, on average. About 30 to 40 percent of youth who meet current criteria for a prodromal syndrome will develop schizophrenia or another psychotic disorder.

“We are moving at an unprecedented pace towards identifying more precise predictors,” says Elaine Walker, an Emory professor of psychology and neuroscience. “By increasing our understanding of the factors that give rise to psychosis, we hope to ultimately improve the ability to provide preventive intervention.”

Walker is one of the principal investigators in the North American Prodrome Longitudinal Study (NAPLS). The National Institute of Mental Health (NIMH) funded the ongoing study, which unites the efforts of Emory, the University of North Carolina, Yale, Harvard, the University of Calgary, UCLA and UC San Diego, and the Feinstein Institute at Hillside Hospital.

“The only way we can do this research is by having a large consortium, combining a range of expertise, from genetics to neuro-endocrinology, psychology and psychiatry,” Walker says. “It is also difficult to identify individuals who are at risk for psychosis and in order to have enough statistical power, we need a large sample of study subjects.”

The consortium has published a flurry of 60 papers during the past four years, involving more than 800 adolescents and young adults who were showing clinical warning signs of impending psychosis and a group of 200 healthy youth.

Among the key findings: Prodromal youth with elevated levels of the stress hormone cortisol and indicators of neuro-inflammation are more likely to become psychotic within a year.

“We’ve developed a risk-prediction algorithm, including measures of symptoms as well as biomarkers, that we have made available for clinicians,” Walker says. “In the future, they can take saliva samples from at-risk patients to check cortisol levels, and to monitor those levels over time. As we get more information, we keep adding to the algorithm to improve the sensitivity and specificity of prediction. It’s important because anti-psychotic medications have a lot of side effects. You don’t want to give them to young people unless you are fairly confident that they are on the way to a psychotic disorder.”

The researchers are now working on refining a blood-biomarker algorithm that clinicians could use to monitor at-risk patients for signs of neuro-inflammation, oxidative stress, hormones and metabolism.

In addition to medication, cognitive therapy and other interventions that reduce stress may be used to help an individual safely make it through the high-risk period, Walker says. “We’ve found that youth with the highest risk tend to be both exposed to more stress and more reactive to stress.”

The consortium of researchers also discovered that the brains of at-risk patients who later develop psychosis show a dramatic decline of gray matter in the year leading up to the diagnosis. And they found that the elevation in a patient’s cortisol level predicts the magnitude of the decline in brain volume.

In 2008, the NIMH awarded $25 million to the consortium for the first phase of the NAPLS project, which lasted five years.

The NAPLS project is now entering its second five-year phase through additional NIMH funding, including a $2.4 million award to Emory.

“We have better technology than ever before for studying the human brain and changes in the brain over time,” Walker says. “We’re at an especially fruitful time in terms of discoveries we can make.” 

During phase two, “we will be looking more closely at hormones, especially stress hormones and indicators of neuro-inflammatory processes,” Walker says. “And we’re going to be looking much more closely at changes in brain structure and function over time. We’re hoping to identify in real-time, with much greater clarity, what is causing what. In other words, the chain of neural mechanisms.” 

Schizophrenia, the most extreme psychosis, affects about 1 percent of the population and can have devastating consequences. Most people diagnosed with schizophrenia are unable to hold a job or live independently for most of their lives. Preventing the onset of schizophrenia and other psychoses has become a major area of emphasis at the NIMH.

“Psychosis is extremely complex, there is no doubt about it, and we’re learning that it’s even more complex than we previously realized,” Walker says. “But if we’re ever going to make progress in prevention and treatment, we’re going to have to come to grips with that complexity and fully understand it.”

For more information about the project, contact the Mental Health and Development Program at Emory: 404-727-7547.

Thinkstock photos by Brian McEntire (top) and Michael Blann (bottom).

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Tuesday, December 2, 2014

'A beautiful find' by Emory mathematician vies for top science story of 2014

Much of the work of number theorist Ken Ono, above, involves solving long-standing mysteries stemming from the work of Indian math genius Srinivasa Ramanujan. (Emory Photo/Video)

A find by Emory mathematician Ken Ono and collaborators ranked 15th in Discover Magazine’s top 100 science stories for 2014.

That makes the discovery of the “Mother Lode of Mathematical Identities” eligible for the magazine’s “people’s choice” awards for the top science story of the year. You can cast your vote for the “Math Breakthrough” by clicking here. The Emory math discovery made it through the first two rounds of voting and is now among the four finalists.

Last summer, Ono and his collaborators Michael Griffin and Ole Warnaar found a framework for the celebrated Rogers-Ramanujan identities and their arithmetic properties, yielding a treasure trove of algebraic numbers and formulas to access them.

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

In the people’s choice awards, the math discovery is vying against stories of cosmic inflation, cybersecurity leaks, the collapse of the West Antarctic ice sheet, the battle against the Ebola outbreak, the genomes of the first Americans, entangled photons and the Rosetta spacecraft’s rendezvous with a comet. It was another big year for science news, showcasing a wide range of disciplines. In fact, one of the only things all of these science advances have in common is their reliance on math.

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

Friday, November 21, 2014

Athletes' testosterone surges not tied to winning, study finds

Kathleen Casto, number 1931 in the center, shown competing in cross country as an undergraduate in North Carolina. She is now a graduate student in psychology at Emory, studying the hormonal correlates of competition in women.

By Carol Clark

A higher surge of testosterone in competition, the so-called “winner effect,” is not actually related to winning, suggests a new study of intercollegiate cross country runners.

The International Journal of Exercise Science published the research, led by David Edwards, a professor of psychology at Emory University, and his graduate student Kathleen Casto.

“Many people in the scientific literature and in popular culture link testosterone increases to winning,” Casto says. “In this study, however, we found an increase in testosterone during a race regardless of the athletes’ finish time. In fact, one of the runners with the highest increases in testosterone finished with one of the slowest times.”

The study, which analyzed saliva samples of participants, also showed that testosterone levels rise in athletes during the warm-up period. “It’s surprising that not only does competition itself, irrespective of outcome, substantially increase testosterone, but also that testosterone begins to increase before the competition even begins, long before status of winner or loser are determined,” Casto says.

Cross country is "an intense experience."
Casto was a Division I cross country runner as an undergraduate at the University of North Carolina, Wilmington. She majored in psychology and chemistry and became interested in the hormonal correlates of competition in women. She applied as a graduate student in psychology at Edwards’ lab when she learned about his work.

Edwards has been collecting data since 1999 on hormone levels of Emory sports teams that have volunteered to participate. The research has primarily involved women athletes. Edwards’ lab also developed a questionnaire to measure the status of an athlete. Members of the team rate the leadership ability of other individuals on the team, to provide a combined rating score for each of the participating athletes.

Many of the labs’ previous studies involved sports such as volleyball and soccer that require team coordination, intermittent physical exertion and only overall team outcomes of win or loss. Casto wanted to investigate how hormones relate to individual performance outcomes in cross country racing.

Cross country racing is both a team and individual sport. Teams are evaluated through a points-scoring system, but runners are also judged on their individual times, clearly ranking their success in an event.

“Cross country running is a unique sport. It’s associated with a drive to compete and perseverance against pain over a relatively long period of time,” Casto says. “It’s an intense experience.”

Participants in the study were consenting members of the 2010 and 2011 Emory varsity men’s and women’s cross country teams. Each participant provided three saliva samples: One before warming up (to serve as a baseline), one after warming up, and a third immediately after crossing the finish line.

Testosterone went up from the baseline for both men and women during the warm-up, while levels of cortisol – a hormone related to stress – did not.

At the end of the race, both men and women participants showed the expected increases in cortisol and surges in testosterone. Neither hormone, however, was related to finish time.

This research follows on the heels of a 2013 study of women athletes in a variety of sports by Edwards and Casto, published in Hormones and Behavior. They found that, provided levels of the stress hormone cortisol were low, the higher a woman’s testosterone, the higher her status with teammates.

The body uses cortisol for vital functions like metabolizing glucose. “Over short periods, an increase in cortisol can be a good thing, but over long periods of chronic stress, it is maladaptive,” Casto says. “Among groups of women athletes, achieving status may require a delicate balance between stress and the actions or behaviors carried out as a team leader.”

Higher baseline levels of testosterone have been linked to long-term strength and power, such as higher status positions in companies.

“Although short-term surges of testosterone in competition have been associated with winning, they may instead be indicators of a psychological strength for competition, the drive to win,” Casto says.

Photos courtesy Kathleen Casto.