
Description
This conference will be held at Landing Jeju Shinhwa World Hotels and Resorts
in Jeju Island
with focus on recent developments in SchrammLoewner evolution and its connections to Gaussian fields and conformal field theory,
the continuum random trees, random surfaces produced by Liouville quantum gravity, and Jordan curves obtained from random conformal weldings.
The conference is by invitation only.
Organizers
Supported by

Samsung Science &
Technology Foundation


Korea Institute
for Advanced Study


Travel Information
 Landing Jeju Shinhwa World Hotels and Resorts
 KIAS will book the rooms at Landing Jeju Shinhwa World Hotels for all invited participants.
 Jeju International Airport  Landing Jeju Shinhwa World Hotels:
 By the Hotel Shuttle,
 from the airport to the hotel, the bus stop is located at C10 zone of the airport parking lot. (No reservation is available. It is on a firstcome, firstserved basis.)
 from the hotel to the airport, please contact the front desk to make reservations.
 By Taxi, it takes about 30 minutes to an hour depending on the traffic and costs about 3040K KRW.
Invited Participants
 Tom Alberts (Utah)
 Morris Jie Jun Ang (Columbia)*
 Ilia Binder (Toronto)*
 Jacopo Borga (Standford)
 SungSoo Byun (KIAS)
 Bertrand Duplantier (CEA)
 Vlad Guskov (KTH)
 Konstantin Izyurov (Helsinki)
 NamGyu Kang (KIAS)
 Ellen Krusell (KTH)
 Antti Kupiainen (Helsinki)*
 Greg Lawler (Chicago)
 Nikolai Makarov (Caltech)
 Jason Miller (Cambridge)
 Pierre Nolin (City U. of Hong Kong)*

 Minjae Park (Chicago)
 S.C. Park (KIAS)
 Eveliina Peltola (Geneva)
 Wei Qian (CNRS)
 Rémi Rhodes (AixMarseille)
 Eero Saksman (Helsinki)*
 Lukas Schoug (Cambridge)
 Scott Sheffield (MIT)
 Stas Smirnov (Geneva)*
 Xin Sun (Columbia)*
 Jinwoo Sung (Chicago)
 Vincent Vargas (ENS)
 Fredrik Viklund (KTH)
 Yilin Wang (IHES)
 Hao Wu (Tsinghua)*


* To be confirmed 
Schedule
 Monday  Tuesday  Wednesday  Thursday  Friday 
09:30  10:30 
Lawler  Sheffield  Miller  Rhodes  Zhan 
   Discussion   
11:00  12:00 
Lupu  Wang  Wu  Duplantier  Kytölä 
12:00  12:40  Lunch  Pfeffer  Lunch 
12:40  13:30  Discussion  Lunch  Discussion 
14:00  14:40 
Powell  Lin  Free Time  Peltola  Schoug 
14:50  15:30 
Byun  Healy  Margarint  Izyurov 
 Coffee Break and Discussion  Coffee Break and Discussion 
16:00  16:40 
Qian  Nolin  Park  Remy 
17:30  19:30   Banquet  
Titles and Abstracts
Monday
Greg Lawler (University of Chicago): Twosided LoopErased Random Walk
The first result I will describe is the existence of the (discrete) looperased random walk in two and three dimensions.
I will then discuss issues about the scaling limit in two dimensions including the relation of the transition probability to random walks with zipper (with Christian Benes and Fredrik Viklund) and twosided annulus SLE (with Mohammad Jahangashahi).
Titus Lupu (ETH): First passage sets of the 2D GFF and Minkowski content
We show that for a class of nonthin local sets of the $2D$ GFF, satisfying a onesided boundedness condition, the restriction of the GFF to this sets is a Minkowski content measure in gauge $log(r)^(1/2) r^2.$
This class of local sets includes the first passage sets of the GFF.
The proof goes through the Gaussian multiplicative chaos.
As a consequence, this gives, via an isomorphism theorem, the gauge to measure the size of clusters in a critical $2D$ Brownian loopsoup.
This is joint work with Avelio Sepulveda (Lyon 1) and Juhan Aru (ETH Zurich).
Ellen Powell (ETH Zurich): A characterisation of the Gaussian free field
Over the past years, the planar Gaussian free field has been conjectured, and in some cases proved, to arise as a universal scaling limit from a broad range of statistical physics models.
I will discuss a recent work with Nathanaël Berestycki and Gourab Ray, showing that any random distribution satisfying conformal invariance and a certain domain Markov property must be a multiple of the Gaussian free field.
This result holds subject only to a fourth moment assumption on the field.
SungSoo Byun (SNU): Annulus SLE partition functions and martingaleobservables
In this talk, I will introduce a version of conformal ﬁeld theory (CFT) and explain its implementations to SLE theory in a doubly connected domain.
The statistical ﬁelds in these implementations are OPE families of central charge modiﬁcations of the Gaussian free ﬁeld with excursion reﬂected/Dirichlet boundary conditions.
After presenting certain equations in CFT including a version of EguchiOoguri and Ward’s equations, I will outline the relation between CFT and SLE theory.
By means of screening, I will explain how to find Euler integral type solutions to the parabolic partial differential equations for the annulus SLE partition functions introduced by Lawler and Zhan and present a class of SLE martingaleobservables associated with these solutions.
This is based on joint work with NamGyu Kang and HeeJoon Tak.
Wei Qian (Cambridge): Uniqueness of the welding problem for SLE and LQG
Fix $\kappa \in (0,8)$ and suppose that $\eta$ is an SLE$_\kappa$ curve in $\mathbb{H}$ from $0$ to $\infty$.
We show that if $\varphi: \mathbb{H} \to \mathbb{H}$ is a homeomorphism which is conformal on $\mathbb{H} \setminus \eta$ and $\varphi(\eta)$, $\eta$ are equal in distribution then $\varphi$ is a conformal automorphism of $\mathbb{H}$.
Applying this result for $\kappa=4$ establishes that the welding operation for critical ($\gamma=2$) Liouville quantum gravity (LQG) is welldefined.
Applying it for $\kappa \in (4,8)$ gives a new proof that the welding of two looptrees of quantum disks to produce an SLE$_\kappa$ on top of an independent $4/\sqrt{\kappa}$LQG surface is welldefined.
These results are special cases of a more general uniqueness result which applies to any nonspacefilling SLEtype curve (e.g., the exotic SLE$_\kappa^\beta(\rho)$ processes).
This is a joint work with Oliver McEnteggart and Jason Miller.
Tuesday
Scott Sheffield (MIT): Scalefree environments: walks, loops, and soups
Yilin Wang (ETH Zurich): Geometric descriptions of the Loewner energy of a simple loop
The Loewner's energy of a simple loop is defined to be the Dirichlet energy of its driving function.
It depends a priori on the parametrization of the simple loop.
However, it was shown in a joint work with Steffen Rohde that there is no such dependence, therefore provides a Moebius invariant quantity for free loops on the Riemann sphere which vanishes only on circles.
In this talk, I will present an intrinsic interpretation of the Loewner energy using the zetaregularizations of determinants of Laplacians and show that the class of finite energy loops coincides with the WeilPetersson class in the universal Teichmueller space.
Peter Lin (University of Washington): Conformal Embedding of the Continuum Random Tree
Every finite combinatorial tree can be canonically embedded in the complex sphere as the solution to a certain conformal welding problem.
We consider the properties of this embedding for large random trees.
In particular we prove the existence, uniqueness and regularity of the limiting random conformal map as the number of edges goes to infinity.
Joint work with Steffen Rohde.
Vivian Healey (University of Chicago): Tree Embedding via the Generalized Loewner Equation
In its most wellknown form, the Loewner equation gives a correspondence between curves in the upper halfplane and continuous real functions (called the "driving function" for the equation).
We consider the generalized Loewner equation, where the driving function has been replaced by a timedependent real measure.
In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper halfplane that are embeddings of trees.
In the second part of the talk, we consider the probabilistic question of finding the scaling limit of these measures for GaltonWatons trees.
We particularly focus on distributions on trees converging to the continuum random tree, and we conclude by describing a connection to the complex Burgers equation.
Pierre Nolin (City University of Hong Kong): No exceptional words for site percolation on $\mathbb{Z}^3$
Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as percolation of words.
We give a positive answer to their Open Problem 2: for site percolation on $\mathbb{Z}^3$ with parameter $p=1/2,$ we prove that almost surely, all words can be embedded.
We also discuss various extensions of this result.
This talk is based on a joint work with Augusto Teixeira (IMPA) and Vincent Tassion (ETH Zurich).
Wednesday
Jason Miller (Cambridge): Is an SLE$_\kappa$ for $\kappa\in(4,8)$ determined by its range?
Hao Wu (Yau Mathematical Sciences Center, Tsinghua University): Hypergeometric SLE and Convergence of Multiple Interfaces in Lattice Models
Conformal invariance and critical phenomena in twodimensional statistical physics have been active areas of research in the last few decades.
This talk concerns conformally invariant random curves that should describe scaling limits of interfaces in critical lattice models.
The scaling limit of the interface in critical planar lattice model with Doburshin boundary conditions (b.c.), if exists, should satisfy conformal invariance (CI) and domain Markov property (DMP).
In 1999, O. Schramm introduced SLE process, and this is the only oneparameter family of random curves with CI and DMP.
In this talk, we first discuss the scaling limit of a pair of interfaces in rectangle with alternating b.c.
The scaling limit of such pair, if exists, should satisfy CI, DMP and symmetry (SYM).
It turns out there is a twoparameter family of random curves satisfying CI, DMP, and SYM, and they are Hypergeometric SLE.
Next, we will explain how to generalize to the more general setting where the notions of multiple SLEs and pure partition functions emerge.
This talk is based on the joint works with V. Beffara and E. Peltola.
Joshua Pfeffer (MIT): Connectivity properties of the adjacency graph of SLE$_\kappa$ bubbles for $\kappa \in (4,8)$
We study the adjacency graph of bubbles  i.e., complementary connected components  of an SLE$_{\kappa}$ curve for $\kappa \in (4,8)$, with two such bubbles considered to be adjacent if their boundaries intersect.
We show that this adjacency graph is a.s. connected for $\kappa \in (4,\kappa_0]$, where $\kappa_0 \approx 5.6158$ is defined explicitly.
This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014).
Our proof in fact yields a stronger connectivity result for $\kappa \in (4,\kappa_0]$, which says that there is a Markovian way of finding a path from any fixed bubble to $\infty$.
We also show that there is a (nonexplicit) $\kappa_1 \in (\kappa_0, 8)$ such that this stronger condition does not hold for $\kappa \in [\kappa_1,8)$.
Our proofs are based on an encoding of SLE$_\kappa$ in terms of a pair of independent $\kappa/4$stable processes, which allows us to reduce our problem to a problem about stable processes.
We will explain in the talk how this encoding comes from the theory of Liouville quantum gravity.
This is joint work with Ewain Gwynne."
Thursday
Rémi Rhodes (University ParisEst Marne La Vallée): Towards quantum Kähler geometry
In this talk, we construct the quantum theory of the coupling of the Liouville action and of Mabuchi's Kenergy.
Both functionals play a prominent role respectively in Riemannian geometry and Kähler geometry.
As an output, we obtain a path integral whose Weyl anomaly presents the standard Liouville anomaly plus an additional Kenergy term.
Motivations come from theoretical physics where these type of path integrals have been proposed by A. Bilal, F. Ferrari, S. Klevtsov and S. Zelditch as a model for fluctuating metrics arising when coupling (small) massive perturbations of conformal field theories to quantum gravity.
Our probabilistic construction relies on a variant of Gaussian multiplicative chaos (GMC), the Derivative GMC (DGMC for short).
The main technical backbone of our construction is twofold and consists in two estimates on (derivative and standard) GMC which are of independent interest.
First, we show that these DGMC random variables possess negative exponential moments; second, we derive optimal small deviations estimates for GMC (associated to a recentered GFF).
Bertrand Duplantier (ParisSaclay University): Complex Generalized Integral Means Spectrum of WholePlane SLE
We describe recent advances in the multifractality of the harmonic measure of SchrammLoewner Evolution (SLE).
A generalized notion of integral means spectrum is introduced, that involves the averaged moments of both the SLE conformal map and its derivative, and provides a unified description of the interior and exterior versions of wholeplane SLE.
In the moment plane, several integrability loci exist, with four analytical forms of the spectrum separated by phase transition lines.
These forms are generalized to complex moments.
Based on joint works with Dmitry Belyaev, Ilia Binder, Xuan Hieu Ho, Thanh Binh Le, and Michel Zinsmeister.
Eveliina Peltola (University of Geneva): Multiple SLEs, pure partition functions, and connection probabilities
Multiple SLEs are conformally invariant measures on families of curves, that naturally correspond to scaling limits of interfaces in critical planar lattice models with alternating ("generalized Dobrushin") boundary conditions (proofs of convergence existing for LERW, Ising, FKIsing, and percolation).
I briefly discuss classification of such measures via two approaches: a "global" classification by a natural cascade property, and a "local" classification as SLE variants with given partition functions.
On the other hand, the partition functions should give scaling limits of connection probabilities in critical planar lattice models (proofs existing for multichordal LERWs, doubledimer pairings, and level lines of the GFF).
The talk is based on joint works with
Vincent Beffara (Université Grenoble Alpes, Institut Fourier),
Kalle Kytölä (Aalto University), and
Hao Wu (Yau Mathematical Sciences Center, Tsinghua University).
Vlad Margarint (Oxford): Using techniques from Stochastic Differential Equations in the study of SchrammLoewner Evolution
In this talk, I will present how abstract results on SDEs can be involved in the context of Loewner Differential Equation in order to study properties of the SLE traces and conformal welding homeomorphisms.
In the first part of the talk, we study a characterization of the law of the SLE tip at a fixed capacity time using a sequence of radius independent SDEs.
This sequence of SDE's is obtained via a random time change in the context of the backward Loewner Differential Equation driven by $\sqrt{\kappa}B_t$.
In the second part, I will focus on the real Bessel SDE, both in the backward and in the forward case.
Here, properties of the Bessel SDEs are applied in the context of SLE in order to study the stability in the parameter $\kappa$ of the conformal welding homeomorphisms and of the SLE traces.
This is a joint project with Prof. Dmitry Belyaev and Prof. Terry Lyons.
Sung Chul Park (EPFL): Local spin correlations in the critical and nearcritical Ising model
We first study the scaling limit of local correlations in the critical Ising model in 2D simply connected domains.
The leading scaling term can be computed in terms of the conformal radius and its derivative, giving further justification to the conjecture that the scaling limit of the 2D Ising model exhibits conformal symmetry.
The proof relies on delicate discrete complex analysis, which naturally generalises to the nearcritical (massive) scaling regime.
Accordingly, we will go on to discuss ongoing work on the $n$point and massive cases and their relevance from the viewpoint of Conformal Field Theory.
Based on joint work with R. Gheissari (first part) and C. Hongler.
Friday
Dapeng Zhan (Michigan State University): Twocurve Green's function for 2SLE
A $2$SLE$_\kappa$ ($\kappa\in(0,8)$), also called bichordal SLE$_\kappa$, is a pair of random curves $(\eta_1,\eta_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_\kappa$ curve in a complement domain.
A twocurve Green's function is the rescaled probability that the two curves of a $2$SLE$_\kappa$ both get near a marked point.
We prove that for an interior point $z_0\in D$, the limit $\lim_{r\to 0^+}r^{\alpha_0} \mathbb{P}[\mbox{dist}(z_0,\eta_j)$ < $r,j=1,2]$, where $\alpha_0=\frac{(12\kappa)(\kappa+4)}{8\kappa}$, exists. We find the convergence rate and the exact formula of this Green's function in terms of a hypergeometric function up to a multiplicative constant. For $\kappa\in(4,8)$, we also prove the convergence of $\lim_{r\to 0^+}r^{\alpha_0} \mathbb{P}[\mbox{dist}(z_0,\eta_1\cap \eta_2)$ < $r]$, whose limit is a constant times the previous Green's function.
In addition, we study the case when the marked point $z_0$ lies on a smooth boundary arc of $D$, and obtain similar results with a differnt exponent $\alpha_0$, which is $\frac 2\kappa(12\kappa)$.
To derive these results, we work on twotimeparameter stochastic processes, and use orthogonal polynomials to derive the transition density of a twodimensional diffusion process that satisfies some system of SDEs.
Kalle Kytölä (Aalto University): Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra
Conjecturally, critical statistical mechanics in two dimensions can be described by conformal field theories (CFT). The CFT description has in particular lead to exact and correct (albeit mostly nonrigorous) predictions of critical exponents and scaling limit correlation functions in many lattice models. The main ingredient of CFT is the Virasoro algebra, accounting for the effect of infinitesimal conformal transformations on local fields. In this talk we show that an exact Virasoro algebra action exists on the probabilistic local fields of two discrete models: the discrete Gaussian free field and the critical Ising model on the square lattice.
The talk is based on joint work with Clément Hongler and Fredrik Viklund.
Lukas Schoug (KTH Royal Institute of Technology): A Multifractal SLE$_\kappa(\rho)$ boundary spectrum
We derive an almost sure multifractal boundary spectrum for SLE$_\kappa(\rho)$ processes, using the socalled imaginary geometry coupling between SLE and GFF to get the correlation estimate.
Konstantin Izyurov (University of Helsinki): Scaling limits of critical Ising correlations: convergence,
fusion rules, applications to SLE
We prove convergence to conformally covariant scaling limits for a family of observables in the critical 2D Ising model, including spins, energies, disorders and fermions.
Combined with a recent work of Hongler, Kytölä and Viklund, this yields convergence of arbitrary local fields.
We also check that the limits satisfy the fusion rules (a. k. a. operator product expansions) as predicted by Conformal Field Theory.
These results also yield a systematic method of proving convergence of Ising interfaces to SLE(3) variants in a general setting.
This is a joint work with D. Chelkak and C. Hongler.
Guillaume Remy (ENS Paris): The FyodorovBouchaud formula and Liouville conformal field theory
Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF.
In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC.
In this talk we will explain how to prove rigorously this formula by using the techniques of conformal field theory.
The key observation is that the moments of the total mass of GMC on the circle are equal to onepoint correlation functions of Liouville conformal field theory in the unit disk.
The same techniques also allow to derive a similar result on the unit interval [0,1] (in collaboration with Tunan Zhu).
Finally we will discuss applications to random matrix theory and to the asymptotics of the maximum of the GFF.
Slides
Monday
Tuesday
Wednesday
Thursday
Friday
Photos
Talks
Registration
 If your hotel room is not being paid for by the conference, then you need to book the hotel yourself and submit the
registration form.
Ask for a "KIAS conference rate" when you book the room.
 If you are supported by the conference, please submit the
registration form at your earliest convenience, but no later than April 22nd.
We will book and cover 6 nights but hope to be able to accommodate 7.
We will book additional nights for you based on your registration form.
 For those who are supported by NSF fund, we will book based on your registration form.
However, we will ask you to pay for the hotel initially and then reimburse later.
NSF Support
We have been tentatively approved for a travel grant from the US National Science Foundation to support the participation of junior researchers at this conference.
Grant funds can be used to support the travel and lodging costs of junior participants with positions at US institutions.
Priority is given to graduate students, postdocs, and junior faculty without other sources of travel funding.
We anticipate being able to cover return airfare from the US to South Korea (that abides by the Fly America act), local travel expenses, and lodging.
Applications for NSF funding should be emailed to Tom Alberts at alberts(at)math(dot)utah(dot)edu.
Deadline for applications is February 28, 2018. Decisions will be announced by March 16, 2018.
Please make the title of your email "NSF Application for Random Conformal Geometry and Related Fields" and include with your application
 a current CV
 if currently a graduate student, the year of your expected PhD and the name and email of your supervisor
 a rough estimate of the cost of round trip airfare to and from Korea, for planning purposes
Questions about the application process can be sent to Tom Alberts at the address indicated above.

