Problems in Conformal Mapping Theory

这是复分析和随机过程的保角映射子领域中的一个项目。它的目标是解决与共形映射及其应用有关的几个著名问题。研究的主要内容有：（i）保角焊接，（ii）调和测度的泛谱，（iii）扩散限制聚集和其他具有临界行为的过程。保角映射理论是复分析的经典领域，是研究几何不变量的重要工具。共形不变性的现象在许多数学结构中以及某些物理理论（共形场论和统计力学）中都很明显。该项目涉及的度量和普适性的一些关键的共形不变量，并与共形不变的概率和过程。

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.