Singularities in Complex Analysis and Dynamics

PI将使用估值的代数工具来对复杂动力学和分析中的问题进行详细研究。研究的主要对象是多元解析函数的赋值空间。它已经有了一个复杂的几何结构，在两个变量的情况下，是一棵由无限多个曲线段粘合在一起建立起来的树。在更多的变量中，粘在一起的对象是更高维度的简化。PI将对这一空间进行分析和动力学，并推导出几个问题的结果。在许多提出的具体项目中，有一个项目涉及建立非线性动力学中的奥塞莱德克定理的代数版本。另一个项目涉及Demailly和Koll‘ar关于解析几何中奇点的开放性猜想。在这两个项目中，总的计划是将问题转化为更容易处理的关于估值空间的问题。Favre和PI.早些时候研究过二维情况。奇点出现在整个数学中，即使主要的研究对象是规则对象。几何学的一个例子是一个圆锥体：顶点外的点是规则的，因为当放大时，圆锥体在那里看起来像一个直平面，但顶点是奇点。当应用具有快速收敛速度的迭代算法时，动力系统中会出现另一种类型的奇异性。动力系统和复杂分析是数学的既定领域，与经济学、生物学和工程学有联系。与研究生和本科生合作，PI将对几个不同数学领域的奇点进行统一研究，包括动力系统和复杂分析。

The PI will use the algebraic tool of valuations to undertake a detailed study of problems in complex dynamics and analysis. The key object in the study is a space of valuations on analytic functions in several variables. It has an intricate geometric structure already in the case of two variables, being a tree built up by infinitely many curve segments glued together. In more variables, the objects glued together are higher-dimensional simplices. The PI will do analysis and dynamics on this space, and deduce results on several problems. Among many specific proposed projects, one involves establishing an algebraic version of the Oseledec theorem in nonlinear dynamics. Another project concerns the openness conjecture by Demailly and Koll\'ar on singularities in analytic geometry. In both projects, the general plan is to translate the problem to more tractable questions on valuation space. The two-dimensional cases were studied earlier by Favre and the PI.Singularities appear throughout mathematics, even when the primary objects of study are regular objects. An example from geometry is given by a cone: the points outside the apex are regular, since when magnifying the cone looks like a straight plane there, but the apex is a singularity. Another type of singularity occurs in dynamical systems, when applying iterative algorithms with fast convergence rates. Dynamical systems and complex analysis are established areas of mathematics, with connections to economics, biology and engineering. Working with students at the graduate and undergraduate levels, the PI will undertake a unified study of singularities in several different mathematics fields, including dynamical systems and complex analysis.