The course will introduce students to basic concepts and techniques in geometric function theory and quasiconformal mapping of relevance in random conformal geometry. Topics include: distortion estimates for conformal maps, extremal length, harmonic measure and Brownian motion, the Loewner equation, the Beltrami equation, the geometric and analytic definitions of quasiconformal map, basic properties of quasiconformal maps, applications such as Teichmuller theory and complex dynamics.

The course will introduce students to basic concepts and techniques in geometric function theory and quasiconformal mapping of relevance in random conformal geometry. Topics include: distortion estimates for conformal maps, extremal length, harmonic measure and Brownian motion, the Loewner equation, the Beltrami equation, the geometric and analytic definitions of quasiconformal map, basic properties of quasiconformal maps, applications such as Teichmuller theory and complex dynamics.

The Loewner energy is a conformally invariant quantity that measures the roundness of a Jordan curve, introduced recently, from the study of large deviations of Schramm-Loewner evolutions with vanishing parameter. This perspective naturally connects Loewner energy to determinants of Laplacians and Brownian loop measures.

More surprisingly, we show that a curve has finite Loewner energy if and only if it is a Weil-Petersson quasicircle, a class of Jordan curves studied since the eighties, that has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, and string theory (now we add to the list random conformal geometry for the link to SLE). In these lectures, I will overview the connection of Loewner energy to these probabilistic and analytic concepts.

Main references:
Yilin Wang: Large deviations of Schramm-Loewner evolutions: A survey. ArXiv:2102.07032

Yilin Wang: Equivalent Descriptions of the Loewner Energy.
Invent. Math., Vol. 218. 2, 573-621 (2019)

Leon A. Takhtajan and Lee-Peng Teo: Weil-Petersson metric on the universal
Teichmüller space. Mem. Amer. Math. Soc., 183(861):viii+119 (2006)

The Loewner energy is a conformally invariant quantity that measures the roundness of a Jordan curve, introduced recently, from the study of large deviations of Schramm-Loewner evolutions with vanishing parameter. This perspective naturally connects Loewner energy to determinants of Laplacians and Brownian loop measures.

More surprisingly, we show that a curve has finite Loewner energy if and only if it is a Weil-Petersson quasicircle, a class of Jordan curves studied since the eighties, that has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, and string theory (now we add to the list random conformal geometry for the link to SLE). In these lectures, I will overview the connection of Loewner energy to these probabilistic and analytic concepts.

Main references:
Yilin Wang: Large deviations of Schramm-Loewner evolutions: A survey. ArXiv:2102.07032

Yilin Wang: Equivalent Descriptions of the Loewner Energy.
Invent. Math., Vol. 218. 2, 573-621 (2019)

Leon A. Takhtajan and Lee-Peng Teo: Weil-Petersson metric on the universal
Teichmüller space. Mem. Amer. Math. Soc., 183(861):viii+119 (2006)