Loewner微分方程

In 2000，SLE was introduced by Schramm in order to consider the scaling limits of some models in statistical physics. SLE widely attracted the focus of the people since it's issued. It led to new development in the study of the scaling limit in statistical physics, and to the proof of Mandelbrot's conjecture, and achieved two Fields Medals respectively in 2006 and 2010. As an important branch of SLE, Loewner differential equation aroused wide public concern. The main aim of this project is to study the (chordal) Loewner differential equation in the following three parts: the combination of Loewner chains and fractal geometry figures, the asymptotic relationship between the driving function and its corresponding slit, the relationship between the smoothness of the driving function and the smoothness of its corresponding slit. Using some special vertical and tangential slits, we will investigate a precise asymptotic relationship when the half-plane capacity tends to zero, and consider the relationship between the asymptotic behaviors and the smoothness. Therefore the research of this project will be very important for the development of the Loewner differential equation.

2000年，Schramm为研究统计物理中某些模型的Scaling极限，引入了SLE。SLE一经出现就引起了数学、物理学界的广泛关注。它促进了统计物理中Scaling极限新的发展，引导了Mandelbrot猜想的解决，并分别于2006年和2010年成就了两位菲尔兹奖得主。作为SLE的一个重要分支，Loewner微分方程重新唤起了人们的关注与重视。本项目主要在以下三个方面研究(弦的)Loewner微分方程：(1).分形图与Loewner链的结合；(2).驱动函数与其裂纹之间的渐近关系；(3).驱动函数与其裂纹的光滑性之间的关系。利用某些特殊的垂直和切向裂纹，我们将给出明确的渐近表达式，并考虑其渐近行为与光滑性之间的关系。由此可见，本项目的研究对Loewner微分方程的发展将起到重要的作用。

2000年，Schramm为研究统计物理中某些模型的Scaling极限，引入了SLE。作为SLE的一个重要分支，Loewner微分方程重新唤起了人们的关注与重视。项目负责人和参与人在此项目的资助下，研究Loewner微分方程及相关问题，取得了如下主要研究进展：1.证明了切向裂纹在共形映射下所对应的驱动函数保持渐近速度不变的性质；2.给出了一列垂直裂纹所对应的驱动函数的具体表达式；3.给出了Stephenson问题成立的两个充分条件；4.给出了一类具有特殊形式的自相似测度是谱测度的充要条件；5.给出了2维Jordan矩阵和连续共线数字集生成的Moran测度为谱测度的一个充分条件。