A remarkable development of the last 50 years in physics is the unification of statistical mechanics and quantum field theory, into a field sometimes called Statistical Field Theory. The study of planar lattice models, such as the Ising model, allows one to gain a concrete insight into what this means, bringing together beautiful pieces of mathematics (in particular conformal geometry, complex analysis and probability) to get exciting physical results, by following the path of this unification. 
The goal of this mini-course is to give an idea about how we can start with a lattice model, end up with a conformal field theory, and use the symmetries of the latter to get spectacular formulae about the former. 

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.

In this mini-course, I will start by introducing the Brownian loop-soup and describing many of its basic properties. I will then explain the relation between the Brownian loop-soup and many other random objects in the plane such as restriction measures, Schramm-Loewner evolutions (SLE), Conformal loop ensembles (CLE), the Gaussian free field (GFF) and so on. These rich relations enable us to better understand the structure of the Brownian loop-soup. In particular, I will describe a series of results and open questions concerning the decomposition of Brownian loop-soup clusters.

ABSTRACT

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)

ABSTRACT

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)