If you want to read Grigori Perelman’s proof of the Poincaré conjecture, arguably the most important mathematical result of the 21st century so far, you won’t find it in a book or an academic journal. The iconoclastic mathematician, who famously turned down a Fields Medal and a million-dollar prize for his work, published his articles only on the preprint server arXiv, where anyone with an Internet connection can download and read them.

The site is celebrating its 25th anniversary this year. Read more about the site’s growth and importance to science here.

]]>One second after the Big Bang, an unfathomable number of neutrinos were liberated from the surrounding chaos and started traveling through the universe at nearly the speed of light. Cosmologists believe a person is bombarded with something on the order of a quadrillion of these ‘relic neutrinos’ every second, dwarfing, unbelievably, the number of neutrinos that come from other relatively nearby sources, such as nuclear fusion in the sun or radioactive decay.

“You hear this, and you’re driven to ask whether it’s really true,” says Princeton University physics professor Christopher Tully. “How can we know it’s true?”

]]>A new algorithmic approach to quantifying language production in individuals with autism has the potential to fill a vexing gap in autism research: the shortage of objective measures of whether a proposed therapy truly improves behavioral outcomes. Mark Clements, an engineering professor at the Georgia Institute of Technology, is building mathematical tools that can analyze audio recordings obtained using a wearable recording device known as the LENA (Language Environment Analysis) System.

]]>Subhash Khot, a professor of computer science at New York University’s Courant Institute of Mathematical Sciences, conducts research on hardness of approximation and probabilistically checkable proofs. He is a 2015 Simons Investigator in Theoretical Computer Science and one of 13 principal investigators in the Simons Collaboration on Algorithms and Geometry (A&G), which brings together experts in mathematics and theoretical computer science.

According to Khot, the A&G collaboration has prompted some ambitious long-term projects. By investing energy in understanding each other’s areas and ideas, investigators gain a new perspective for their own work. “Usually people are expert in a specific area and are well aware of the most challenging questions as well as the limitations of current techniques in that area,” he says. “They are often thinking about these problems for years, perhaps decades, and unable to make further progress. By interacting with people with different backgrounds and research interests, they can discover connections that might exist. Once in a while, one stumbles upon unexpected connections between two areas that greatly benefit both.”

Much of Khot’s own work lies at the interface of mathematics and computer science. Recently, in collaboration with Muli Safra, a professor of computer science and Dor Minzer, a graduate student, both at Tel Aviv University, Khot studied a computer science problem called monotonicity testing that might be stated as follows: Given a function on the Boolean hypercube, you want to test whether this function is monotone, that is, entirely increasing or entirely decreasing, by looking at as few values of the function as possible. It turns out that this problem, quite unexpectedly, is closely connected to isoperimetric problems that are central to geometry. In Euclidean space — the space we live in — we know that if we have an object with a fixed volume, its surface area is at least equal to the surface area of a sphere with this volume. But in other spaces, where we have a notion of volume and of surface area, what object of fixed volume has the smallest surface area? The question can be asked in many different geometric spaces, including the Boolean hypercube, which is of particular importance in theoretical computer science. By proving new variants of known isoperimetric results on the hypercube, Khot and his collaborators were able to design the best possible algorithm for the monotonicity testing problem.

“This is one question that I recently worked on, and we solved the problem,” Khot says. “We were very happy with it, and it shows the connections between computer science and mathematics really, really closely. In the context of this problem, and there are more examples, computer science and mathematics go hand in hand — it’s difficult to say which inspired which.”

]]>Over the past six years, gene-sequencing studies have definitively linked spontaneous (or de novo) mutations to autism in children who are the only members of their immediate family to have the condition. Studies of the Simons Simplex Collection (SSC) — a repository of data from families in which one child has autism, but the parents and siblings are unaffected — have consistently shown that the children with autism have more de novo mutations than their siblings.

In 2014, a landmark sequencing study of the protein-coding regions of the genomes of 2,515 families from the SSC identified about 400 loss-of-function mutations — ones that indisputably disrupt the function of a gene — in some of the children with autism. This doesn’t mean that all 400 mutations are in fact connected to autism, however; about 200 of the children’s siblings also have de novo loss-of-function mutations. The siblings’ mutations, though, are presumably benign, at least with respect to autism.

Assuming, as researchers believe, that the children with autism have benign mutations at about the same rate as their siblings, these numbers suggest that only about half of the 400 loss-of-function mutations in the children with autism are true autism risk genes, while the other half are red herrings. But which genes fall into which category?

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