Loewner Evolutions and Quasiconformal Mappings

主要研究人员将研究由Loewner微分方程生成的共形映射，以及随机共形和拟共形映射。Loewner方程将一个连续递增的平面单连通区域序列与一个实值函数联系起来，实值函数是该方程的驱动项。对应关系是通过保角映射到标准区域(例如盘)上的。通过这种机制，复杂的二维形状可以由看似简单的对象，即实变量的实值函数来编码。形状和它的驾驶术语之间的对应关系很复杂，留下了许多悬而未决的问题。这个项目的目的是为了更好地理解这封信。例如，首席调查员将研究在驱动项变形下集合的连续性。鉴于Oded Schramm的SLE和Lawler、Schramm、Werner、Smirnov等人的精彩工作，随机驾驶项(特别是布朗运动)特别有趣，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。在小比例尺上，保形贴图看起来像旋转和膨胀。因此，物理现象(如布朗运动、渗流、晶体生长、电沉积)的旋转和膨胀不变的数学模型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，并对其中许多模型进行了预测。Oded Schramm对随机Loewner演化(SLE，由一维布朗运动驱动的Loewner方程)的发现和Smirnov关于渗流的工作使这一哲学建立在坚实的数学基础上。近年来取得的成果在数学界和物理界都引起了极大的兴趣。他们还在这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线索。

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.