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Abstract
This dissertation gathers five interrelated articles on random conformal geometry. Part I concerns results on Liouville quantum gravity (LQG), which is a conformally covariant model of random two-dimensional geometry. Chapter 2, which is joint work with Ewain Gwynne, demonstrates that the LQG area measure is the Minkowski content measure with respect to the LQG metric in the subcritical phase. Chapter 3, which is joint work with Manan Bhatia and Ewain Gwynne, studies rigorously the heuristic that LQG in the supercritical phase has infinite spikes at a dense set of locations. Concretely, we exhibit that a discrete model of supercritical LQG, when conditioned to be finite, converges to a continuum random tree in the scaling limit. Part II concerns results on Loewner energy, which is a conformally invariant functional on crosscuts of a disk that appears naturally in the semiclassical limit of Schramm–Loewner evolution (SLE). Chapter 4 investigates the reversibility of Loewner energy and gives a deterministic proof of this basic property through commutation relations. Chapter 5, which is joint work with Yilin Wang, studies the first variation of Loewner energy under quasiconformal deformations. Chapter 6, which is joint work with Shuo Fan, shows that the Weil–Petersson Teichmüller space, which consists of loops on the Riemann sphere with finite Loewner energy modulo Möbius transformations, has a quasi-invariant action on the SLE loop measure defined naturally via conformal welding. This gives strong evidence in support of the idea that the Weil–Petersson Teichmüller space is the Cameron–Martin space for SLE loops.