Problems in Conformal Mapping Theory

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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.