Workshop: Analytic Number Theory and Diophantine Approximation

摘要：本奖学金将资助美国学生参加于2008年6月30日至7月11日在渥太华大学举办的解析数论和丢番图分析暑期学校。暑期学校的目的是让年轻的研究人员接触到分析数论和丢番图分析的一些最新成果和技术。近年来，这些领域取得了很大的进展，其中一些亮点是Goldston， Pintz和Yildirim关于素数之间的小间隙的工作，以及Rivoal关于黎曼ζ函数奇值的无理性的工作。此外，其他突破，如格林和陶关于素数长等差数列的定理，依赖于解析数论领域的进步。暑期学校的第一周将提供几门入门课程，作为研究生的背景。第二周将提供有关最新发展的更高级课程。也希望这些不同但相关领域的研究人员能够将他们的想法和见解结合起来，从而取得进一步的突破。暑期学校紧接在安大略省滑铁卢附近举行的大型国际会议（加拿大数论协会第十次会议）。在暑期学校的训练也应该有助于研究生从这次研究会议中更充分地受益。这个由国家科学基金资助的暑期学校是在数论领域。数论是数学的一个领域，有着非常古老且易于表述的问题，其解决方案非常深入，并将许多不同的数学领域联系在一起。数论的进步有时会导致实际应用，如密码学和编码理论。

Abstract: This grant will help fund American students to attend a summer school in Analytic number theory and Diophantine Analysis held at the University of Ottawa, June 30-July 11, 2008. The object of the summer school is to expose young researchers to some of the latest results and techniques in analytic number theory and diophantine analysis. These areas have seen a lot of progress in recent years, with some highlights being the work of Goldston, Pintz and Yildirim on small gaps between prime numbers, and the work of Rivoal on irrationality of odd values of the Riemann zeta-function. Further, other breakthroughs such as Green and Tao's theorem on long arithmetic progressions of prime numbers relied on advances in the area of analytic number theory. The first week of the summer school will offer several introductory courses as background for beginning graduate students. The second week will then offer more advanced courses on recent developments. It is also hoped that researchers in these different, but related fields would be able to combine their ideas and insights, leading to further breakthroughs.The summer school immediately precedes a large international conference (the tenth meeting of the Canadian Number Theory Association) to be held in nearby Waterloo, Ontario. The training at the summer school should help graduate students benefit more fully from this research conference as well.The summer school supported by this NSF grant is in the area of number theory. Number theory is an area of mathematics with very old and easy-to-state problems whose solutions lie very deep, and connect very many disparate areas of mathematics. Progress in number theory has sometimes led to practical applications, such as in cryptography and in coding theory.