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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...

AIM is pleased to announce that Melanie Matchett Wood and Kirsten Graham Wickelgren have been named the 2009 AIM Five-Year Fellows.

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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

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A new mathematical object was revealed on March 12, 2008, during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics.

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The news caused excitement at the AIM workshop attended by 25 of the world's leading analytic number theorists. The work is a joint project between graduate student Ce Bian and his adviser, Andrew Booker. Booker commented that, "This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.

"This breakthrough opens a door to the study of higher degree L-functions," said Dennis Hejhal, Professor of Mathematics at the University of Minnesota and Uppsala University. "It's a big advance" added Harold Stark of the University of California, San Diego, who, 30 years ago was the first to accurately calculate second degree transcendental L-functions. "I thought we were years away from doing this. The geometry of what you have to do and the scale of the computation are orders of magnitude harder."

There are two types of L-functions: algebraic and transcendental, and these are classified according to their degree. The Riemann zeta-function is the grand-daddy of all L-functions. It holds the secret to how the prime numbers are distributed, and is a first degree algebraic L-function. The Riemann Hypothesis, announced in 1859 and today the most important of all unsolved math problems, is an example of something that should be true for every L-function. Michael Rubinstein from the University of Waterloo, a participant at the workshop, quickly tested and confirmed the Riemann Hypothesis for the first few zeroes of this newly minted L-function. Rubinstein, along with William Stein of the University of Washington, will direct a new initiative to systematically chart L-functions; this project has been recommended for funding by the National Science Foundation. "The techniques developed by Bian and Booker open up whole new possibilities for experimenting with these powerful and mysterious functions and are a key step towards making our group project a success." Rubinstein added.

"It's a big step toward our understanding the 'world of L,' which is where most of the secrets of number theory are kept." said Brian Conrey, Director of AIM.

Dorian Goldfeld, Professor of Mathematics at Columbia University summarized the excitement, saying "This discovery is analogous to finding planets in remote solar systems. We know they are out there, but the problem is to detect them and determine what they look like. It gives us a glimpse of new worlds."