Larry Guth is a 2014 Simons Investigator in Mathematics and professor of mathematics at the Massachusetts Institute of Technology (MIT), where he also received his Ph.D. His research focuses on combinatorial geometry, a branch of mathematics that studies the combinations and organizations of geometric objects, as well as harmonic analysis and metric analysis. He received his undergraduate degree from Yale University and, before returning to MIT to teach, he worked at the University of Toronto and the Courant Institute of Mathematical Sciences at New York University.

Guth learned of combinatorial geometry as a postdoctoral researcher at Stanford University. A colleague told him about some important theorems in the field from the 1980s, and he was impressed to see interesting recent mathematics about lines in the plane. “I felt like these questions really could have been asked in the 1800s, except that nobody had this angle. I thought this was very engaging.”

Guth is also interested in mathematics that can be understood by a younger audience. “There are open problems that are similar to these ones that you can even explain to middle school students.” A friend who runs a summer program for middle school students in New York City invited Guth to teach there about problems related to the intersection pattern you can make with lines in the plane. He taught there for one summer and has visited since.

Recently, Guth began a collaboration with Nets Katz of the California Institute of Technology; the two were able to prove that a complex configuration of lines can be modeled with a low-degree polynomial. “It was counterintuitive to me,” says Guth. “The problem only involves lines, but the structure that appears is algebraic geometry — just polynomials.”

The combinatorial problems related to lines that Guth works with now are ‘overdetermined,’ that is, there are more equations than there are unknowns in those equations. “Your first instinct in seeing an overdetermined situation is that it’s just impossible,” says Guth. “That’s a valid first guess, but there are many situations that come up in math — and perhaps also in other parts of science — where that first guess is wrong. I would say that what we’re trying to find is the right second guess: if you have only ten parameters to play with and you’re trying to make one hundred things happen, what are the situations where that could work, and what should you be looking for?”

]]>By studying the visual cortex of mice and monkeys, they found that neighboring neurons vary in how influenced they are by the firing of the larger population of neurons. Some neurons are strongly coupled to their neighbors, whereas others are weakly coupled, a discovery that may shed light on the global activity of neural populations.

Neurons in the cerebral cortex operate like members of a large orchestra: Some act as the choristers who support the main tune, while others are soloists playing separate melodies, and these roles tend not to change, explain the researchers. Each neuron can be categorized in terms of its ‘population coupling,’ a measurement that characterizes the extent to which the activity of that individual neuron relates to the larger population of neurons, determining whether it is a chorister or a soloist.

“The findings represent a previously unappreciated dimension by which we can characterize neurons,” explains Carandini. “It’s new and simple and it has clear biological backing: Neurons that ‘sing with the choir’ receive more incoming synaptic inputs than ‘soloist’ neurons.”

The discovery that chorister neurons typically receive more synaptic inputs than soloist neurons allows neuroscientists to interpret the joint activity of large populations of neurons without having exact knowledge of the intricate wiring that connects them.

“One might have thought that describing the activity of thousands of neurons would be impossible without computing the millions of correlations between each pair of cells,” says Harris. “This discovery gives us a powerful tool moving forward, to understand complex circuits through simpler analysis.”

The Simons Foundation funds both Carandini and Harris in this collaborative project. Through work together in their joint laboratory, the researchers believe that this one simple principle of population coupling will bring scientists closer to understanding the workings of the brain’s billions of neurons.

“A key goal of neuroscience is to understand how any single neuron relates to the global activity patterns that underlie perception, cognition and action. We discovered that this relationship can be described, at least to a first approximation, in a very simple way,” says Harris.

Now, Carandini and Harris are making new recordings of brain activity to more thoroughly understand the functional consequences of a neuron being a chorister or a soloist. “This is exciting work, but it is just the beginning,” says Harris.

]]>To that end, the Simons Foundation is supporting the launch of the Kaiser Permanente Autism Family Research Bank, a project to gather genetic material from 5,000 families who have a child with autism. The project will give researchers access to detailed genetic, medical and environmental information from ‘trios’ — an individual with autism under 26 years old, plus both of his or her biological parents.

The grant is intended to create this autism research bank over the next three years, with a focus on families in Northern California.

Because autism can be caused by both genetic and environmental factors, advances in understanding the disorder rely on large numbers of families participating in genetic epidemiological studies. Large numbers of participating families will help researchers identify patterns across many individuals that could shed light not only on the causes of autism, but also on potential treatment and prevention strategies.

Information from twin and family studies of autism provides strong evidence for a genetic contribution to autism, and increasing evidence points to a role for environmental factors, particularly during prenatal development and the early post-natal period. With the Autism Family Research Bank, researchers will be able to examine critical factors contributing to autism.

Kaiser Permanente is an integrated managed care consortium, and is the largest private healthcare provider in California.

]]>The golden ratio (1.61803 … ) is greatly hyped, partly for its beautiful mathematical properties but also for nonsensical reasons. Distinguishing between the two requires understanding that mathematics is about structures and relationships, not just numbers in isolation. When the golden ratio truly appears (not just some number in its approximate neighborhood), we can find patterns that account for it.

Two good references for properties of the golden ratio are Clement Falbo, “The Golden Ratio: A Contrary Viewpoint,” *College Mathematics Journal* 36, no. 2 (March 2005): 123-134, and George Markowsky, review of *The Golden Ratio*, by Mario Livio, *Notices of the AMS* 52, no. 3 (March 2005): 344-347.

Related:

More videos from the Mathematical Impressions series.

]]>The faculty scholar grants will provide research support to a significant number of excellent scientists at a time in their careers when funding is difficult to obtain. Talented early-career researchers are often discouraged from pursuing academic research careers by the challenging funding climate in the U.S., which forces them to spend a large portion of their time seeking funding rather than doing science. And pressure to secure funding through federal grant money may lead to ‘safer’ grant proposals rather than to creative and potentially transformative project ideas.

The awards are intended for researchers who have already demonstrated significant research accomplishments and show potential to make unique and important contributions to their fields. It is hoped that the grants will encourage these scientists to pursue research careers and will enable more innovative science.

The three philanthropies will award a total of $148 million over the program’s first five years, awarding up to 70 grants every two and a half to three years. Awardees will receive a five-year, non-renewable grant ranging from $100,000 to $400,000 per year for direct research costs. Additionally, grantees’ institutions will receive 20 percent of the yearly grant for indirect costs.

The competition is open to basic researchers and physician scientists at more than 220 eligible U.S. institutions. Applicants must be using molecular, genetic, computational or theoretical approaches to address fundamental biological or biomedical problems. Applicants must have more than four but fewer than 10 years of post-training professional experience. Once selected, awardees are required to devote at least 50 percent of their effort to research.

The Faculty Scholars Competition marks the first time that HHMI, the Bill & Melinda Gates Foundation and the Simons Foundation have formally undertaken an initiative together. The philanthropies hope that other organizations will join this effort going forward.

“The Simons Foundation is pleased to be a partner in this important effort to support talented early-career scientists,” says Marian Carlson, director of Life Sciences at the Simons Foundation.

For more information about the competition, click here. For information on how to apply, click here. The deadline for application is July 28, 2015 at 3:00 p.m. EDT.

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