Problems in Conformal Mapping Theory

本文是复杂分析和随机过程中保角映射子领域的一个项目。它的目的是解决与保形映射及其应用有关的几个著名问题。主要研究内容如下：(I)保角焊接，(Ii)调和测度的普适谱，(Iii)扩散限制聚集等具有临界行为的过程。保角映射理论是复变分析的经典领域，它是研究几何不变量的重要工具。共形不变性现象在许多数学结构以及某些物理理论(共形场理论和统计力学)中都很明显。该项目涉及一些关键的共形不变量的度量和普适性质，以及共形不变概率和过程。

ABSTRACTThis is a project in the conformal mapping subfieldof complex analysis and stochastic processes. Its goalis to solve several well-known problems concerning conformal maps and their applications. The main topicsof the study are the following: (i) conformal welding,(ii) universal spectrum of harmonic measure, (iii) diffusion-limited aggregation and other processeswith critical behavior.Conformal mapping theory is a classical area of complex analysis; it is an important tool in the study of geometric invariants. The phenomenon of conformal invariance is manifest in many mathematical structuresas well as in certain physical theories (conformal field theory and statistical mechanics). The project deals with metric and universality properties ofsome key conformal invariants, and with conformally invariant probabilities and processes.