Valuations in Dynamics and Analysis

项目编号：DMS-0200614PI: Mattias jonssonabstract首席研究员将利用代数中的估值工具，在两个复杂的维度上研究动力学和分析中的局部问题。赋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，它们在一个点上参数化了所有的本地数据。复杂变量中的局部现象，如多次谐波函数的奇异性或轨道收敛到固定点的速度，可以通过观察沿不同路径的行为来有效地研究。估值是本研究的有力工具。该方法的成功取决于这样一个事实，即二维中一个点上的所有估值的集合是一个可以理解的集合：它是一个树，在分支点上焊接在一起的实间隔的无限集合，因此没有循环出现。这棵树是一个复杂但可管理的对象，它的一维性具有惊人的应用。动力系统是一种数学模型，它被广泛应用于几乎每一门科学中经历时间演化的任何情况。艾萨克·牛顿爵士发明了微积分，即数学分析，用于研究行星系统，这是动力系统的一个主要例子。数学的发展很大程度上是为了回应建模和理解我们周围世界的需要。在这个建议中，首席研究员将使用来自不同数学领域的工具来理解某些动力系统。可以研究的系统的一个例子来自于数值（迭代）算法，本提案中对动态系统的研究将提供有关算法性能的信息。

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.