Sieve Methods with Applications

筛分方法及其应用

• 批准号: 0802246
• 负责人: Henryk Iwaniec
• 金额: $ 37.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0802246&HistoricalAwards=false
• 关键词: Sieve; Methods; Applications

The sieve theory offers tools for selecting subsequences of particular interest from a larger sequence which is typically more accessible by other means. For example in the recent developments prime numbers were captured in polynomial values of degree four. The proposal goes further to solve problems not only concerning prime numbers but also for solving some diophantine equations and estimating the rational points on some cubic surfaces. In more theoretical aspects of sieve theory the goal of the Proposal is to investigate the intrinsic limitations of the methods (parity barrier of sieve) and to find ways to break these limits. Sieve ideas when enhanced with arguments of harmonic analysis (spectral methods) become powerful and versatile tools which can even exceed the capability of the Grand Riemann Hypothesis. The Proposal makes a few suggestions in this direction.Sieve methods turned out to be very attractive for researchers working in cryptography. Although this project does not address such applications directly, it seems likely that advances in the theory of sieves will open new possibilities. The implementation of Fourier analysis to sieve methods creates a lot of demand in modern harmonic analysis and will certainly have valuable impact on shaping the latter. These developments in the interface of combinatorial ideas and analysis will be quite inspiring for graduate students.

筛分理论提供了从更大的序列中选择特别感兴趣的序列的工具，该序列通常更容易通过其他手段获得。例如，在最近的发展中，素数被捕获在四阶多项式值中。该方案不仅解决了素数问题，而且解决了某些丢番图方程的求解问题和某些三次曲面上有理点的估计问题。在筛理论的更多理论方面，提案的目标是调查方法的内在局限性（筛的奇偶性障碍），并找到打破这些限制的方法。当用调和分析（谱方法）的参数增强时，筛的思想成为强大而通用的工具，甚至可以超过大黎曼假设的能力。该提案在这方面提出了一些建议。筛选方法对密码学研究人员来说非常有吸引力。虽然这个项目没有直接解决这些应用，但似乎筛子理论的进步将开辟新的可能性。将傅立叶分析应用到筛分方法中，在现代谐波分析中产生了大量的需求，并且肯定会对形成后者产生有价值的影响。这些发展的接口组合的想法和分析将是相当鼓舞人心的研究生。