Singularities in Complex Analysis and Dynamics

PI将使用估值的代数工具来详细研究复杂动力学和分析中的问题。研究的主要对象是多元解析函数的赋值空间。它有一个复杂的几何结构已经在两个变量的情况下，是一个树建立了无限多的曲线段粘在一起。在更多的变量中，粘在一起的对象是高维的单形。PI将在这个空间上做分析和动力学，并推导出几个问题的结果。在许多具体的拟议项目中，一个涉及建立非线性动力学中的奥塞莱德克定理的代数版本。另一个项目涉及的开放性猜想Demailly和Koll\'ar奇异解析几何。在这两个项目中，总的计划是将问题转化为估值空间上更容易处理的问题。二维的情况下进行了研究法弗尔和PI。奇异性出现在整个数学，即使当主要研究对象是正常的对象。几何学中的一个例子是圆锥：顶点外的点是规则的，因为当放大圆锥时，它看起来像一个直平面，但顶点是一个奇点。另一种类型的奇异性发生在动力系统中，当应用具有快速收敛速度的迭代算法时。动力系统和复分析是数学的既定领域，与经济学，生物学和工程学有关。PI将与研究生和本科生合作，在几个不同的数学领域进行统一的奇点研究，包括动力系统和复分析。

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.