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This lecture explores the embeddability properties of Liouville quantum gravity (LQG) metrics, a canonical model of random fractal Riemannian surfaces introduced by Polyakov in the 1980s. Discover how LQG can be defined either as a path integral over fields corresponding to the Liouville action or as a random metric measure space that describes the scaling limit of various two-dimensional discrete objects. Learn about how discrete conformal embeddings of random planar maps converge to canonical embeddings of LQG surfaces into 2D Euclidean space. Understand the surprising result that no embedding of an LQG surface into ℝⁿ can be quasisymmetric, which generalizes Troscheit's finding for √8/3-LQG (corresponding to uniform random planar maps). The lecture also touches on future research directions in the study of metric embeddability for LQG.
