Valuations in Dynamics and Analysis

提案编号：DMS-0200614 PI：Mattias Jonsson摘要首席研究员将使用估值，从代数的工具，在两个复杂的维度研究动力学和分析的局部问题。赋值可以具体地看作是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局部数据。复变函数中的局部现象，如多重次调和函数的奇异性或轨道向不动点的收敛速度，可以通过观察沿不同方向的行为来有效地研究。估值构成了本研究的一个非常有力的工具。这种方法的成功取决于这样一个事实，即二维空间中一点上的所有赋值的集合是一个可以理解的集合：它是一棵树，在这个意义上，它是一个由真实的区间组成的无限集合，这些区间在分支点上焊接在一起，这样就不会出现循环。这棵树是一个复杂但可管理的对象，它的一维性有着惊人的应用。动力系统是一种数学模型，被广泛用于几乎每一门科学中的任何经历时间演化的情况。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，动力系统的一个主要例子。许多数学的发展是为了响应建模和理解我们周围世界的需要。在这个提议中，首席研究员将使用一个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systemthat的一个例子来自数值（迭代）算法，在这个建议中的动力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.