The American Institute of Mathematics (AIM) is pleased to introduce a new tool to the world of web pages - the knowl. Like the familiar hyperlink, knowls can be used to provide relevant, supplementary information and are referenced from within the body of a web page. But unlike the hyperlink, which simply takes you to a new web page, the knowl conveniently serves up the information at your original location with the click of a mouse button. With a similar click, the knowl then disappears.

"The self-contained nature of knowls makes them useful in many different settings. They can be thought of as 'building blocks' that can be called upon when needed. I envision a time when the Internet has a repository of such knowls, reliable and ready to be referenced anywhere." says David Farmer, Director of Programs at AIM.

Knowls were developed to support research projects in advanced mathematics, but they can be easily incorporated to enhance any website. Knowls creator Harald Schilly noted, "The technology for knowls is at least 10 years old, and the theoretical basis, known as transclusion, is more than 50 years old. Our contribution is to provide an interface that fits the way people browse the Web today."

In addition to the examples in the paragraphs above, AIM has prepared a demonstration page with knowls for Rob Beezer's free textbook on Linear Algebra.

The mathematics research projects which led to the development of knowls were funded by the Division of Mathematical Sciences at the National Science Foundation.

Read about how to add knowls to your own website

In June at AIM, teams of middle school teachers and mathematicians from around the U.S. participated in a workshop on "How to Run a Math Teachers' Circle." Math Teachers' Circles are groups of teachers and research mathematicians who meet regularly to work on mathematically rich problems. The intent is to develop teachers' mathematical problem-solving skills and confidence in approaching difficult problems while connecting them with the larger mathematical community.

One example of a Math Teachers' Circle problem is featured in this week's Numberplay blog by Gary Antonick on NYTimes.com. This particular type of problem, called a "Mad Vet" scenario, can be used to explore questions in abstract algebra and graph theory, as described in an article by Gene Abrams and Jessica K. Sklar that appeared in the June 2010 issue of Mathematics Magazine.

During this workshop, teams participated in example Math Teachers' Circle sessions and developed detailed plans for starting and sustaining their own Math Teachers' Circle. The participating teams were from Columbia, South Carolina; Eau Claire, Wisconsin; Greeley, Colorado; Richmond, Kentucky; San Diego, California; and Winston-Salem, North Carolina. Their new Math Teachers' Circles will begin meeting by Summer 2012.

The first Math Teachers' Circle began at AIM in 2006, and since then the series of "How to Run a Math Teachers' Circle" workshops has helped develop a network of 31 Math Teachers' Circles in 20 states. The Math Teachers' Circle Network, based at AIM, provides mathematical and logistical resources to this growing community.

In January, 2011, AIM brought together mathematicians, graduate students, and industry and public agency representatives to work on a variety of sustainability problems, including renewable energy, air quality, water management, and other environmental issues.

For more details, please see the announcement page.

The American Institute of Mathematics (AIM) has been awarded a grant from the Tellabs Foundation to support a workshop for middle school math teachers. The teachers will join the AIM Math Teachers' Circle, which connects teachers with mathematicians to work on mathematical problem solving.
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Mathematicians from North America, Europe, Australia, and South America have resolved the first one trillion cases of an ancient mathematics problem. The advance was made possible by a clever technique for multiplying large numbers. The numbers involved are so enormous that if their digits were written out by hand they would stretch to the moon and back. The biggest challenge was that these numbers could not even fit into the main memory of the available computers, so the researchers had to make extensive use of the computers' hard drives.
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The seven NSF Mathematical Sciences Research Institutes announce the creation of 45 new two-year positions for young, highly-trained mathematical scientists across the country. In addition to furthering research in all areas of the mathematical sciences, these positions will allow recent PhDs to teach at community colleges and other higher-education institutions or to participate in projects tied to business and industry. This new initiative is a result of a partnership among the National Science Foundation-supported mathematics institutes. Read more...

Update, January 26: Soundararajan has proven the original version of the QUE conjecture, completing the missing step in Lindenstrauss' program for noncompact arithmetic surfaces. His paper is available on the ArXiv.

October 10, 2008: In a seminar co-organized by Stanford University and the American Institute of Mathematics, Soundararajan announced that he and Roman Holowinsky have proven a significant version of the quantum unique ergodicity (QUE) conjecture. "This is one of the best theorems of the year," said Peter Sarnak, a mathematician from Princeton who along with Zeev Rudnick from the University of Tel Aviv formulated the conjecture fifteen years ago in an effort to understand the connections between classical and quantum physics. "I was aware that Soundararajan and Holowinsky were both attacking QUE using different techniques and was astounded to find that their methods miraculously combined to completely solve the problem," said Sarnak. Both approaches come from number theory, an area of pure mathematics which recently has been found to have surprising connections to physics.

The motivation behind the problem is to understand how waves are influenced by the geometry of their enclosure. Imagine sound waves in a concert hall. In a well-designed concert hall you can hear every note from every seat. The sound waves spread out uniformly and evenly. At the opposite extreme are "whispering galleries" where sound concentrates in a small area.

The mathematical world is populated by all kinds of shapes, some of which are easy to picture, like spheres and donuts, and others which are constructed from abstract mathematics. All of these shapes have waves associated with them. Soundararajan and Holowinsky showed that for certain shapes that come from number theory, the waves always spread out evenly. For these shapes there are no "whispering galleries."		Uniformly distributed points in a fundamental domain for SL(2,Z). Image courtesy of Fredrik Stromberg

Read more..., including articles by Peter Sarnak and Zeev Rudnick.