Loewner Evolutions and Quasiconformal Mappings

主要研究者将研究由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方程将平面单连通域的连续递增序列与方程的驱动项实值函数联系起来。这种对应关系是通过保角映射到标准域（如圆盘）的方式实现的。通过这种机制，复杂的二维形状可以用看似简单的对象，即实变量的实值函数来编码。形状与其驱动项之间的对应关系是复杂的，留下了许多悬而未决的问题。这个项目的目的是更好地理解这种对应关系。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，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，共形图看起来像旋转和膨胀。因此，物理现象（如布朗运动、渗透、晶体生长、电沉积）的旋转和膨胀不变数学模型在保角坐标变化下是不变的，这似乎是合理的。理论物理学家长期以来一直使用这种启发式方法，并对许多模型进行了预测。Oded Schramm对随机洛厄纳演化（SLE，由一维布朗运动驱动的洛厄纳方程）的发现和Smirnov在渗流方面的工作为这一哲学奠定了坚实的数学基础。近年来取得的成果在数学界和物理界都引起了极大的兴奋。他们还在这两个学科之间架起了一座新的桥梁。这项研究的一个目标是揭示这个新兴理论的数学方面的新亮点。

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.