Schramm-Loewner Evolution

Introduction.- Iteration of conformal maps.- On stochastic models and connection to statistical physics.- An example: percolation model and Cardy's formula.- On reading this book.- Introduction to stochastic calculus.- Brownian motion.- Stochastic integration.- Ito's formula.- Further topics in stochastic calculus.- Conformal invariance of two-dimensional Brownian motion.- Weak convergence of probability measures.- Introduction to conformal mappings.- Harmonic functions.- Conformal maps.- From Area theorem to distortion.- Conformally invariant tools.- Loewner equation.- Conformal maps of the upper half-plane.- Loewner chains .- Loewner equations in D and Sp.- Schramm-Loewner evolution.-Schramm-Loewner evolution and its elementary properties.- Advanced properties of SLE.- Proofs for some of the advanced properties.- Variants of SLE.- Moments of the derivative of the Loewner map of SLE(k).- Regularity and convergence of random curves.- Continuity properties of the Loewner chains.- Continuity of SLE(k).- Convergence of interfaces in the site percolation model.- Index.

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• Análise complexa, variáveis ??complexas
• Cibernética e teoria dos sistemas
• Física Matemática
• Probabilidade e Estatística
• Estocástica

Conformal Maps;Loewner Equation;Random Curves;Scaling Limits;SLE;Convergence of Random Curves;Area theorem;Cardy-Smirnov formule;Ising model;Brownian motion;Percolation;Cardy formula;Stochastic intergral;Ito's formula;Bessel process;Poisson Kernel;Schwarz-Christoffel mappings;Loewner chains;Schramm's principle