09:00 to 10:00 Parafermionic observables and order of the phase transition in planar random-cluster models Co-authors: Vincent Tassion (Université de Genève), Vladas Sidoravicius (IMPA) This talk will be devoted to the study of the parafermionic observable for planar random-cluster models. We will present a few applications of such observables, including the determination of the order of the phase transition for cluster-weights q between 1 and 4, as well as the computation of the critical value. INI 1 10:00 to 11:00 Liouville Quantum Gravity on the Riemann sphere Co-authors: David (), Kupiainen (), Vargas () In this talk, I will explain how to rigorously construct Liouville quantum field theory on the Riemann sphere and precise conjectures relating this object to the scaling limit of random planar maps conformally embedded onto the Riemann sphere. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 tba Schramm Loewner Evolution (SLE), Conformal Loop Ensemble (CLE), and Gaussian Free Field (GFF) are three important planar objects that arise from Statistical physics and quantum field theory. In the first part of this talk, I will introduce these three objects. In the second part, I will explain the construction of a conformally invariant growing mechanism on CLE4. In the last part, I will discuss two couplings between GFF and CLE4. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 A conformally invariant metric on CLE(4) Co-authors: Scott Sheffield (MIT), Hao Wu (MIT) Werner and Wu introduced a conformally invariant way of exploring the loops in a CLE$_4$ $\Gamma$ in a simply connected domain. Using the relationship between CLE$_4$ and the Gaussian free field, we show that the dynamics of this exploration process are a deterministic function of the CLE$_4$ loops, and we use this fact to construct a conformally invariant metric on $\Gamma$ for which a ball of radius $t$ coincides with the set of loops explored up to time $t$ by the exploration process. It is conjectured that this metric space is related in the $\epsilon \to 0$ limit to the contact graph metric on CLE$_{4+\epsilon}$ as well as the contact graph metric on $\epsilon$-neighborhoods of CLE$_4$ loops. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 Conformal invariance of boundary touching loops of FK Ising model Co-author: Stanislav Smirnov (University of Geneva and St. Petersburg State University) I will present a result showing the full conformal invariance of Fortuin-Kasteleyn representation of Ising model (FK Ising model) at criticality. The collection of all the interfaces, which in a planar model are closed loops, in the FK Ising model at criticality defined on a lattice approximation of a planar domain is shown to converge to a conformally invariant scaling limit as the mesh size is decreased. More specifically, the scaling limit can be described using a branching SLE(?,?-6) with ?=16/3, a variant of Oded Schramm's SLE curves. We consider the exploration tree of the loop collection and the main step of the proof is to find a discrete holomorphic observable which is a martingale for the branch of the exploration tree. This is a joint work with Stanislav Smirnov (University of Geneva and St. Petersburg State University) INI 1
 09:00 to 10:00 SLE correlations and singular vectors We discuss relations between SLE correlations, in particular the (higher order) differential equations they satisfy, and highest weight representations of the Virasoro algebra. INI 1 10:00 to 11:00 Almost sure multifractal spectrum of SLE Co-authors: Jason Miller (Massachusetts Institute of Technology), Xin Sun (Massachusetts Institute of Technology) Suppose that $\eta$ is an SLE$_\kappa$ in a smoothly bounded simply connected domain $D \subset \mathbb C$ and that $\phi$ is a conformal map from the unit disk $\mathbb D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \rightarrow [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial \mathbb D$ such that $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ as $\epsilon \rightarrow 0$. I will present a rigorous computation of the a.s. multifractal spectrum of SLE (joint with J. Miller and X. Sun), which confirms a prediction due to Duplantier. The proof makes use of various couplings of SLE with the Gaussian free field. As a corollary, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 From the critical Ising model to spanning trees We will show an explicit mapping between the squared partition function of the critical Ising model defined on isoradial graphs and the partition function of critical spanning trees, thus proving a strong relation between two classical models of statistical mechanics. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 GFF with SLE and KPZ This talk concerns three objects that remain in fashion: the Gaussian free field (GFF), the Schramm-Loewner Evolution (SLE) and the Liouville measure. We first discuss a way to derive the exponential moments of the winding for the chordal SLE curves and then show how this enters into determining the quantum fractal dimension for the SLE curves in their flow line coupling with the GFF. It comes out that the usual KPZ relation does not hold for the SLE flow lines of the GFF, yet it is not clear whether one should be actually surprised. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 Generalized Multifractality of Whole-Plane SLE Co-authors: H Ho (Orléans University), B Le (Orléans University), M Zinsmeister (Orléans University) We introduce a generalized notion of integral means spectrum for unbounded conformal maps, depending on two moments, giving access to logarithmic coefficients. We study this (average) generalized integral means spectrum for unbounded whole-plane SLE. The usual SLE multifractal spectrum, predicted by the author in 2000 and proved by Beliaev and Smirnov in 2005 and Gwynne, Miller and Sun in 2014, crosses over to a novel spectrum along phase transition lines in the plane of moment orders. A conjecture is proposed for the universal generalized multifractal spectrum, which is proved for a certain range of moments. INI 1 19:30 to 22:00 Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co.