Mathematical Sciences: Classical Analysis, Number Theory andSupercomputers

在复分析和解析数论的界面上的问题的工作将是这个数学研究的中心主题。为了获得更多的见解，超级计算机实验也将发挥重要作用。从数论的角度来看，这项工作的目标是理解各种超越函数的根的性质：ζ函数。一个特殊的例子是Epstein zeta函数，一个依赖于一类参数的复函数。随着参数的变化，根会遍历特定的路径。这项工作试图从统计/动力学分析的角度来理解这种运动。类似的（统计）分析在研究许多其他类型的zeta函数中被发现是有用的——尽管这里的重点将转移到研究它们被认为存在的临界线上的根。复分析的工作集中在紧致黎曼曲面上的Schwarzian微分方程上。本文将研究二阶微分方程作为强迫项时解的单群和保角映射性质。特别关注微分通过Teichmuller空间的Bers边界的情况。

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.