Valuations in Dynamics and Analysis

建议编号：DMS-0200614PI：Mattias Jonsson ABSTRACT首席研究员将使用代数中的工具赋值来研究动力学中的局部问题，并从两个复杂的维度进行分析。估值可以具体地可视化为一系列点收敛到原点的不同方式。事实上，它们在某一点上将所有本地数据参数化。复变量中的局部现象，如多次调和函数的奇异性或轨道收敛到某一固定点的速度，可以通过观察不同途径到该点的行为来有效地研究。估值构成了这项研究的一个非常强大的工具。这种方法的成功取决于这样一个事实，即两维中一点上所有赋值的集合是一个可以理解的集合：它是一棵树，它是在分支点上焊接在一起的无限实数区间集合，因此不会出现循环。这棵树是一个复杂但可管理的对象，它的单维性有着惊人的应用。动力系统是几乎所有科学中广泛使用的数学模型，用于经历时间演化的任何情况。艾萨克·牛顿爵士发明了微积分或数学分析，用于研究行星系统，这是动力系统的一个主要例子。许多数学是为了对我们周围的世界建模和理解的需要而发展起来的。在这个建议中，主要的研究人员将使用不同数学领域的工具来理解某些动力系统。一个可以研究的系统的例子来自于数值(迭代)算法，在本提案中对动力系统的研究将提供关于算法性能的信息。

Proposal Number: DMS-0200614PI: Mattias JonssonABSTRACTThe principal investigator will use valuations, a tool from algebra, to study local problems in dynamics and analysis in two complex dimensions. Valuations can be visualized concretely as thedifferent ways that a sequence of points can convergeto the origin. In fact, they parameterize all thelocal data at a point. Local phenomena in complex variables, such as the singularity of a plurisubharmonic functions or the rate of convergence of an orbit to a fixed point,can be effectively studied by observing the behavior alongdifferent approaches to the point. Valuations constitute avery powerful tool for this study. The success of theapproach depends on the fact that the set of all valuationsat a point in two dimensions is a set that can be understood:it is a tree in the sense of an infinite collection of real intervals welded together at branch points so that no cyclesappear. This tree is a complicated but manageable objectand its one-dimensionality has striking applications.Dynamical systems are mathematical models that are widely used in virtually every science for any situation thatundergoes a time-evolution. Sir Isaac Newton inventedcalculus, or mathematical analysis, for the study of theplanetary system, a prime example of a dynamical system.Much of mathematics has developed as a respond to the needto model and understand the world around us.In this proposal the principal investigator will use a toolfrom a different field of mathematics in order to understandcertain dynamical systems. One example of a systemthat can be studied arises from numerical (iterative) algorithms, and the study of the dynamical systems in this proposal will give information on the performance of the algorithms.