Loewner Evolutions and Quasiconformal Mappings

主要研究者将研究由Loewner微分方程产生的共形映射，以及随机共形和拟共形映射。Loewner方程将连续增加的平面单连通域序列与一个实值函数（方程的驱动项）联系起来。对应是通过保角映射到一个标准域（如一个磁盘）。通过这种机制，复杂的二维形状可以由看似简单的对象编码，即真实的变量的实值函数。形状与其驱动项之间的对应关系是复杂的，并留下许多悬而未决的问题。这个项目的目的是提供一个更好地了解这种对应关系。例如，首席研究员将研究驱动项变形下集合的连续性。鉴于Oded Schramm的SLE和Lawler、Schramm、Werner、Smirnov等人的出色工作，随机驱动项（特别是布朗运动）特别有趣，并将成为研究的重点。保角映射通常用于将坐标从一个区域更改为更简单的区域，例如圆盘。它们在数学的许多领域和物理学的几个分支中都有应用。在小尺度上，共形映射看起来像旋转和膨胀。因此，物理现象的旋转和膨胀不变的数学模型（例如，布朗运动、渗流、晶体生长、电沉积等）在共形坐标变化下是不变的。理论物理学家长期以来一直使用这种启发式方法，并获得了许多这些模型的预测。奥德施拉姆的发现随机Loewner演变（SLE，Loewner方程驱动的一维布朗运动）和斯米尔诺夫的工作渗流把这一哲学的坚实的数学基础。近年来所取得的成果在数学界和物理界都引起了很大的轰动。他们还在两个学科之间建立了一座新的桥梁。这项研究的一个目标是为这一新兴理论的数学方面提供新的见解。

The principal investigator will study conformal mappings generated by the Loewner differential equation, and random conformal and quasiconformal mappings. The Loewner equation relates a continuously increasing sequence of planar simply connected domains to a real-valued function, the driving term of the equation. The correspondence is by means of conformal maps onto a standard domain (such as a disc). Through this mechanism complicated two-dimensional shapes can be encoded by seemingly simpler objects, namely, real-valued functions of a real variable. The correspondence between a shape and its driving term is complicated and leaves many open questions. The aim of this project is to provide a better understanding of this correspondence. For instance, the principal investigator will study the continuity of the sets under deformations of the driving terms. In light of Oded Schramm's SLE and the spectacular work of Lawler, Schramm, Werner, Smirnov and others, random driving terms (in particular, Brownian motion) are especially interesting and will be a focus of the research. Conformal mappings are often used to change coordinates from one region to a simpler region, such as a disc. They have applications in many areas within mathematics and to several branches of physics. On small scale, conformal maps look like rotations and dilations. Hence it is plausible that rotation- and dilation-invariant mathematical models of physical phenomena (e.g., Brownian motion, percolation, crystal growth, electrodeposition) are invariant under conformal coordinate changes. Theoretical physicists have long used this heuristic and obtained predictions for many of these models. Oded Schramm's discovery of the stochastic Loewner evolution (SLE, the Loewner equation driven by one-dimensional Brownian motion) and Smirnov's work on percolation have put this philosophy on a firm mathematical basis. The results obtained in recent years have generated a lot of excitement in both the mathematics and the physics communities. They have also created a new bridge between the two disciplines. A goal of this research is to shed new light on the mathematical side of this emerging theory.