09:00 to 09:50 Registration 09:50 to 09:55 Welcome from Christie Marr (INI Deputy Director) 09:55 to 10:00 Welcome from Organisers 10:00 to 11:00 W Werner (ETH Zürich)Some news from the loop-soup front Co-author: Wei Qian (ETH Zürich) We will present some new features of loop-soups. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Conformal representations of Random Maps and Surfaces Motivated by the quest to find a somewhat conformal map from the Riemann sphere to a Liouville quantum-gravity sphere, I will talk about uniformization of discrete maps (based on joint work with Don Marshall), and discuss an analog of Mario Bonk's carpet uniformization in the setting of the Conformal Loop Ensemble CLE (based on joint work with Brent Werness). INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 J Miller (Massachusetts Institute of Technology)Liouville quantum gravity and the Brownian map Co-author: Scott Sheffield (MIT) Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics. We show that the Brownian map is equivalent to Liouville quantum gravity with parameter $\gamma = \sqrt{8/3}$. INI 1 15:00 to 15:30 Afternoon Tea 15:30 to 16:30 Essential spanning forests on periodic planar graphs The laplacian on a periodic planar graph has a rich algebraic and integrable structure, which we usually don't see when we do standard potential theory. We discuss the combinatorial, algebraic and integrable features of the laplacian, and in particular interpret combinatorially the points of the "spectral curve" of the laplacian in terms of probability measures on spanning trees and forests. INI 1 16:30 to 17:30 Welcome Wine Reception
 10:00 to 11:00 SLE Quantum Multifractality INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Radial SLE martingale-observables Co-author: Nikolai Makarov (Caltech) After implementing a version of radial conformal field theory in the OPE family of statistical fields generated by background charge modification of the Gaussian free field, I present an analytical and probabilistic proof of a well-known statement in physics that the correlation functions of such fields under the insertion of one-leg operator form a collection of radial SLE martingale-observables. In the construction of one-leg operator, the so-called neutrality conditions on the charges play an important role. I explain two neutrality conditions: first, the linear combination of bosonic fields is to be a well-defined Fock space field; second, the Coulomb gas correlation function is to be conformally invariant. To reconcile these two neutrality conditions, one needs to place the background charge at the marked interior point, the target point of SLE. This is a joint work with Nikolai Makarov. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Scaling limit of the probability that loop-erased random walk uses a given edge Co-authors: Christian Benes (CUNY), Greg Lawler (University of Chicago) I will discuss a proof of the following result: The probability that a loop-erased random walk (LERW) uses a given edge in the interior of a lattice approximation of a simply connected domain converges in the scaling limit to a constant times the SLE(2) Green's function, an explicit conformally covariant quantity. I will also indicate how this result is related to convergence of LERW to SLE(2) the natural parameterization. This is based on joint work with Christian Benes and Greg Lawler and work in progress with Greg Lawler. INI 1 15:00 to 15:20 Afternoon Tea 15:20 to 16:20 Welding of the Backward SLE and Tip of the Forward SLE Co-author: Steffen Rohde (University of Washington) Let $\kappa\in(0,4]$. A backward chordal SLE$_\kappa$ process generates a conformal welding $\phi$, which is a random auto-homeomorphism of $\mathbb R$ that satisfies $\phi^{-1}=\phi$ and has a single fixed point: $0$. Using a stochastic coupling technique, we proved that the welding $\phi$ satisfies the following symmetry: Let $h(z)=-1/z$. Then $h\circ \phi\circ h$ has the same law as $\phi$. Combining this symmetry result with the forward/backward SLE symmetry and the conformal removability of forward SLE curve, we then derived some ergodic property of the tip of a forward SLE$_\kappa$ curve for $\kappa\in(0,4)$. INI 1 16:20 to 16:30 Break 16:30 to 17:30 Boundary Measures and Natural Time Parameterization for SLE Among Greg's many important contributions to the Schramm-Loewner evolution, he and his co-authors have in recent years been largely responsible for the construction and analysis of the natural time parameterization. The term natural'' refers to the parameterization favored by probabilists in the discrete curve setting, one unit of time per lattice site, and although it has not yet been shown that this discrete time parameterization converges it is still possible to define and study what should be the corresponding time parameterization for the continuum SLE curve. I will review many of the results in this direction that are due to Greg et al, along with some related results of my own in the boundary case, and then discuss some work in progress for extending the results on the boundary. INI 1