Mathematical Sciences: Classical Analysis, Number Theory andSupercomputers

数学科学：经典分析、数论和超级计算机

• 批准号: 8910744
• 负责人: Dennis Hejhal
• 金额: $ 9.35万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1992-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8910744&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Classical; Analysis; Number

Work on problems at the interface of complex analysis and analytic number theory will be the central theme of this mathematical research. To obtain additional insight, supercomputer experimentation will also play a prominent role. From the number-theoretical point of view, the objective of the work is in understanding properties of roots of various kinds of transcendental functions: zeta functions. A particular instance is the Epstein zeta function, a complex function depending on a class of parameters. As the parameters vary, the roots traverse certain paths. This work seeks to understand this motion from the standpoint of a statistical/dynamical analysis. Similar (statistical) analyses have been found useful in studying many other types of zeta functions - though here the emphasis will shift to studying the roots on the critical line where they are believed to reside. Work on complex analysis centers on Schwarzian differential equations on compact Riemann surfaces. Studies of the monodromy group and the conformal mapping properties of the solution as the forcing term of the differential equation - the quadratic differential - diminishes will be carried out. Particular interest focuses on the case where the differential passes through the Bers boundary of Teichmuller space.

解决复杂分析界面上的问题， 解析数论将是本书的中心主题。 数学研究 为了获得更多的洞察力， 超级计算机试验也将发挥突出作用。 从数论的观点来看， 这项工作是为了了解各种根的性质， 超越函数的一个例子：zeta函数。 特定 Epstein zeta函数是一个复杂的函数 取决于一类参数。 随着参数的变化， 根穿过特定的路径。 这项工作试图理解这一点 从统计/动力学分析的角度来看运动。 类似的（统计学）分析在研究 许多其他类型的zeta函数-尽管这里强调的是 将转向研究临界线上的根， 据信居住。 复变函数分析的工作以Schwarzian微分为中心 紧黎曼曲面上的方程 单值性研究 群和解的保角映射性质， 强迫项的微分方程-二次 将进行差异化缩小。 特别 兴趣集中在差分通过的情况下， 通过Teichmuller空间的Bers边界。