
Notices
 The conference will take place in Halla Room on the ground floor of Landing Jeju Shinhwa World Hotels and Resorts.
 Based on the registration forms, KIAS has booked the rooms at
Marriott Jeju Shinhwa World Hotels and Resorts and will cover six nights (June 04  June 10).
The conference rate for an extra night is 187,000 Korean Won per night.
If you have changed your itinerary or the number of accompanying persons after submitting the registration form, please let us know the updated information. Otherwise, you will not be able to get the conference rate.
If there are accompanying persons, you are requested to pay for their additional breakfast costs of 33,000 Korean Won per meal.
 Your rooms have been upgraded from Landing to Marriott. See the map.
 Students without accompanying persons will share their rooms.
 The Conference Banquet will take place on Monday evening at JEJUSEON, Marriott Resort GF.
 If you would like to join the Wednesday Excursion to Jeju Jungmun Tourist Complex and Yakcheonsa Temple, please sign up for it on Monday (at the registration desk) with the excursion fee (10,000 Korean Won + Donation). Please pay this fee on Monday or Tuesday. The Excursion Dinner will take place at Gasdon (Jeju Island Korean Barbecue).
 If you plan to hike Mount Hallasan, you must make a reservation for the top courses, Seongpanak Trail and Gwaneumsa Trail, at the Reservation Site.
You can explore other courses without reservation.
Description
This conference will be held at Jeju Shinhwa World 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


Moving Around
Incheon International Airport  Kimpo International Airport
 More information will be shared.
 See our presentation that you may find helpful when you are traveling.
Jeju International Airport  Marriott Jeju Shinhwa World Hotels
 After the pandemic, the hotel does not offer a shuttle service between the airport and the hotel.
 By Taxi, it takes about 40 minutes to an hour depending on the traffic and costs about 3040K KRW.
 By Bus, it takes about one hour to one and a half hours. Get on a public bus (151, 152, 600) at the 4th GATE bus stop on the 1st floor → Transfer a bus (255, 8202,7522, 7712) at Donggwang transfer station2 (GEC districts) → Get off at the Jeju Shinhwa World.
Hiking
 If you plan to hike Mount Hallasan, you must make a reservation for the top courses, Seongpanak Trail and Gwaneumsa Trail, at the Reservation Site.
You can explore other courses without reservation.
 The Yeongsil Trail is a moderately strenuous route and it takes about 3 hours to complete.
Invited Participants
 Osama Abuzaid (Aalto)
 Tom Alberts (Utah)
 Valeria Ambrosio (Cambridge)
 Manan Bhatia (MIT)
 Ilia Binder (Toronto)
 Jacopo Borga (Standford)
 SungSoo Byun (KIAS)
 Gefei Cai (Beijing)
 Bertrand Duplantier (ParisSaclay)
 Yuyang Feng (Chicago)
 Yifan Gao (City U. of Hong Kong)
 Jiongji Guo (Geneva)
 Vlad Guskov (KTH)
 Liam Hughes (Cambridge)
 Konstantin Izyurov (Helsinki)
 Janne Junnila (Helsinki)
 NamGyu Kang (KIAS)
 Konstantinos Kavvadias (Cambridge)
 Seonwoo Kim (SNU)
 Ellen Krusell (KTH)
 JungKyoung Lee (KIAS)
 Yongwoo Lee (SNU)
 Xinyi Li (Beijing)
 Marcin Lis (Vienna Univ. of Tech.)
 Sid Maibach (Bonn)

 Vlad Dumitru Margarint (Colorado)
 Jason Miller (Cambridge)
 Kyeongsik Nam (KAIST)
 Pierre Nolin (City U. of Hong Kong)
 Minjae Park (Chicago)
 S.C. Park (KIAS)
 Eveliina Peltola (Geneva)
 Wei Qian (CNRS)
 Rémi Rhodes (AixMarseille)
 Steffen Rohde (U. of Washington)
 Lukas Schoug (Cambridge)
 Insuk Seo (SNU)
 Chen Shang (Columbia)
 Scott Sheffield (MIT)
 Jinwoo Sung (Chicago)
 Fredrik Viklund (KTH)
 Yilin Wang (IHES)
 Baojun Wu (AixMarseille)
 Hao Wu (Tsinghua)
 Aoteng Xia (Beijing)
 Shengjing Xu (UPenn)
 Pu Yu (MIT)
 Yizheng Yuan (Cambridge)
 Dapeng Zhan (Michigan State)
 Zijie Zhuang (UPenn)

Schedule
 Monday  Tuesday  Wednesday  Thursday  Friday 
09:30  10:10 
Duplantier  Rohde  Sheffield  Binder  Zhan 
10:30  11:10 
Alberts  Wang  Rhodes  H. Wu  Schoug 
11:20  12:00 
Lis  Qian  Miller  Peltola  Izyurov 
12:00  13:00   
Lunch   
13:00  18:00  Discussion  Excursion  Discussion 
18:00  20:00  Banquet  Dinner  Dinner  Dinner  Dinner 
20:00  20:20   Borga   Yu  S.C. Park 
20:30  20:50   M. Park   Margarint  B. Wu  21:10  21:30   Yuan   Junnila  Sung 
Notices

The conference will take place in Halla Room (A + B + C) on the ground floor of the Landing Convention Centre.
No food or drinks are allowed in the conference room except for bottled water.
The conference room is open from 9 AM to 10 PM (except for Monday and the excursion day).
 Breakfast is served at SKY ON 5 DINING  Marriott Resort 5F.
If there are accompanying persons, you are requested to pay for their additional breakfast costs of 33,000 Korean Won per meal.
 Lunch is served at Yeongsil Room and Eorimok Room, two of eight Meeting Rooms on the ground floor of the Landing Convention Centre.
Vegetarian lunch boxes will be available in the Eorimok Room for those who have signed up for it on the registration forms.
 The Conference Banquet will take place on Monday evening at JEJUSEON, Marriott Resort GF.
A vegetarian menu will be available for those who have signed up for it on the registration forms.
 The Excursion Dinner will take place at Gasdon (Jeju Island Korean Barbecue).
A vegetarian menu will be available for those who have signed up for it on the registration forms.
Titles and Abstracts
Monday
Bertrand Duplantier (ParisSaclay): Hamiltonian paths on bicolored random planar maps and KPZ
We consider various configuration exponents of Hamiltonian
paths drawn on bicubic random planar maps. Estimates from exact finite size results
are compared with predictions based on the KPZ relations,
as applied to regular exponents on the hexagonal lattice.
Surprisingly, a naive use of KPZ does not reproduce all the measured exponents
but a certain Ansatz may possibly account for the observed discrepancies.
We further study Hamiltonian cycles on various families of bicolored
planar maps, which fall into two universality classes, with
central charges $c = 1$ or $c = 2$. The first group comprises
$p$regular maps of fixed vertex valency $p$ greater or equal to $3$, and the second,
maps of mixed vertex valencies, and a socalled rigid case. For
each class, a universal configuration exponent, as well as a novel
critical exponent associated with longdistance contacts
along a Hamiltonian cycle are obtained from KPZ and the corresponding
exponent on regular (hexagonal or square) lattices. This time, the
predictions are numerically confirmed by enumeration results
for $p$regular maps with $p = 3; 4; : : : ; 7,$ and for maps with mixed valencies
$(2; 3), (2; 4)$ and $(3; 4)$.
Based on joint work with Ph. Di Francesco, O. Golinelli and E. Guitter.
Tom Alberts (Utah): TBA
Marcin Lis (TU Vienna): Conformal invariance of critical double random currents
The double random current (DRC) model is a natural percolation model whose geometric properties are intimately related to spin correlations of the Ising model.
In two dimensions, it moreover carries an integer valued height function on the graph, called the nesting field. We study the critical DRC model on bounded domains of the square lattice.
We fully describe the joint scaling limit of the (primal and dual) DRC clusters and the nesting field as the lattice mesh size vanishes.
We prove that the nesting field becomes the Dirichlet Gaussian free field (GFF) in this limit, and that the outer boundaries of the DRC clusters
with free boundary conditions are the conformal loop ensemble with $\kappa=4$ (CLE4) coupled to that GFF. Moreover, we also show that the
inner boundaries of the DRC clusters form a twovalued local set with values ${\mp 2\lambda, (2\sqrt2 \mp 2) \lambda}$ for the field restricted to a CLE4 loop
with boundary value $\pm 2\lambda$. Our proof is a combination of exact solvability of the Ising model, new crossing estimates for the DRC model
(which does not posses the FKG property), and a careful analysis of the structure of twovalued local sets of the continuum GFF.
This is joint work with Hugo DuminilCopin and Wei Qian.
Tuesday
Steffen Rohde (University of Washington): Piecewise geodesic Jordan curves
Given a finite number of points on the sphere, which simple closed curves minimize the Loewner energy among all curves that pass through these points? We will discuss several characterizations of such curves in terms of hyperbolic geometry, conformal welding, and differential equations. This is joint work with Mario Bonk, Janne Junnila, Don Marshall and Yilin Wang.
Yilin Wang (IHES): Quasiconformal deformation of the Loewner driving function
We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformation with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of the driving function of the curve. This gives another explanation of the identity between the Loewner energy and universal Liouville action, introduced by Takhtajan and Teo. Considering the quasiconformal deformation allows us to identify the holomorphic stressenergy tensor of the Loewner energy as the Schwarzian derivative of the uniformizing conformal map of the complement of the curve, which is at the heart of many identities of the Loewner energy, e.g., with renormalized volume, a Fredholm determinant involving the Grunsky operator, etc. This is a joint work with Jinwoo Sung.
Wei Qian (City University of Hong Kong): Conformally invariant fields out of Brownian loop soups
For each central charge $c\in (0,1]$, we construct a conformally invariant field which is a measurable function of the local time field $\mathcal{L}$ of the Brownian loop soup with intensity $c$ and i.i.d. signs given to each cluster. This field is canonically associated to $\mathcal{L}$, in a sense which is similar to the isomorphism theory that associates the Gaussian free field to the loop soup with critical intensity. Isomorphisms between Brownian motions and random fields were previously developed by Symanzik, BrydgesFröhlichSpencer, Dynkin and Le Jan in several different settings.
In a key intermediate step, we obtain the crossing exponent for the event that a cluster in the subcritical loop soup passes near a given point. Among other things, it allows us to deduce that the Minkowski gauge function of a cluster in a loop soup with intensity $c\in (0,1)$, which is $t^2 f(t)$ for some $f(t)=\log t^{1c/2+o(1)}$.
I will also present several conjectures about this family of fields. This talk is based on joint works with Antoine Jego (EPFL) and Titus Lupu (CNRS).
Jacopo Borga (Stanford): Meanders and meandric systems
A meandric system is a collection of loops obtained from two noncrossing perfect matchings on $\{1,\ldots,2n\},$ one drawn above the real line and the other below the real line. A meander is a meandric system with exactly one loop. We conjectured that the scaling limit of meanders (resp. meandric systems) should be described by $\sqrt2$Liouville quantum gravity sphere decorated by two SchrammLoewner evolutions with parameter $\kappa=8$ (resp. decorated by a SchrammLoewner evolution with parameter $\kappa=8$ and a conformal loop ensemble with parameter $\kappa=6$).
In this talk, we will introduce these conjectures and discuss recent progress in the study of meanders and their conjectural scaling limit. The case of meandric systems will be addressed in Park’s talk.
Based on joint works with Ewain Gwynne, Xin Sun, and Minjae Park.
Minjae Park (University of Chicago): Geometry of uniform meandric systems
In Borga's talk, the largescale geometry of a uniformly sampled meandric system of size n is conjectured to be described by an independent triple consisting of a Liouville quantum gravity (LQG) sphere with parameter $\gamma=\sqrt2$, a SchrammLoewner evolution (SLE) curve with parameter $\kappa=8$, and a conformal loop ensemble (CLE) with parameter $\kappa=6$. I will further discuss the geometry of uniform meandric systems and present several rigorous results consistent with this conjecture. In particular, a uniform meandric system admits macroscopic loops, and the halfplane version of the meandric system has no infinite paths. Based on joint work with Jacopo Borga and Ewain Gwynne.
Yizheng Yuan (University of Cambridge): Regularity of SLE trace
I will present a few results on the regularity of the SLE trace. There are several notions of regularity such as variation and modulus of continuity. The optimal leading exponents (a.k.a. $p$variation resp. Hölder exponent) have been previously identified by several authors. Here, we will focus on logarithmic refinements. In the case of variation regularity (which is a notion that depends only on the curve modulo parametrisation), we can prove the optimal (up to constants) result. The modulus of continuity depends on the parametrisation; for the natural parametrisation, we have the optimal (up to constants) result, whereas for the capacity parametrisation, we have a fairly good upper bound. Finally, I will comment on the regularity of the uniformising maps for SLE.
Part of this talk is joint work with Nina Holden.
Wednesday
Scott Sheffield (MIT): TBA
Rémi Rhodes (AixMarseille university): Path integral for Generalized Minimal Models
I will present a probabilistic formulation for a path integral based on the Liouville action functional with imaginary parameters. This lays the foundation of a Conformal Field Theory (CFT), which turns out to be non unitary. Our construction is restricted to real values of the central charge $c<2.$ When the central charge corresponds to minimal models, the Hamiltonian has Jordan cells, hence this model is a concrete prototype of log CFT.
In physics, this CFT is conjecturally related to the scaling limit of Potts or $O(N)$ models. Joint work with C. Guillarmou and A. Kupiainen.
Jason Miller (University of Cambridge): Conformal removability of SLE($\kappa$) for $\kappa \in [4,8)$
We consider the SchrammLoewner evolution (SLE$_\kappa$) with $\kappa=4$, the critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve and above which it is selfintersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical ($\gamma=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\mathbf C}$ to be conformally removable which applies in the case that $X$ is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLE$_\kappa$ curves for $\kappa \in (4,8)$ in the case that the adjacency graph of connected components of the complement is a.s. connected.
Based on joint work with Konstantinos Kavvadias and Lukas Schoug.
Thursday
Ilia Binder (University of Toronto): Quasisymmetric images of Brownian graph: minimal dimension
Conformal dimension of a set is the minimal Hausdorff dimension of its quasisymmetric image. In the talk, I will discuss the conformal dimensions of various deterministic and stochastic sets, such as BedfordMcMullen sets and selfaffine Fractal Percolation Clusters. I will show that the Brownian graph is minimal, i.e. its conformal dimension is equal to 3/2, its Hausdorff dimension. The talk is based on joint works with Hrant Hakobyan (Kansas State) and Wenbo Li (Toronto).
Hao Wu (Tsinghua University): Connection probabilities for randomcluster model and uniform spanning tree
Conformal invariance of critical lattice models in twodimensional has been vigorously studied for decades. The first example where the conformal invariance was rigorously verified was the planar uniform spanning tree (together with looperased random walk), proved by Lawler, Schramm and Werner around 2000. Later, the conformal invariance was also verified for Bernoulli percolation (Smirnov 2001), level lines of Gaussian free field (SchrammSheffield 2009), and Ising model and FKIsing model (ChelkakSmirnov et al 2012). In this talk, we focus on connection probabilities of these critical lattice models in polygons with alternating boundary conditions.
Eveliina Peltola (Geneva): TBA
Pu Yu (MIT): Multiple SLE via conformal welding of Liouville quantum gravity disks
Liouville quantum gravity (LQG) is a natural model for a random fractal surface, and one canonical finite volume LQG surface is the LQG disk. A powerful tool in the study of LQG is conformal welding, where multiple LQG surfaces are combined into a single LQG surface, with the interfaces typically being variants of SchrammLoewner evolutions (SLE). In this talk, we will present our recent result on conformal welding of multiple LQG disks. In particular, the LQG disks can be conformally welded together according to any given topological link pattern. The interface is multiple chordal SLE, while the random conformal structure of the marked points is encoded by the $\kappa<4$ multiple SLE pure partition function. Based on joint work with Ang and Sun.
Vlad Margarint (University of Colorado at Boulder): Multiple SLE, Bessel processes and Random Matrices
In this talk, I will present some preliminary recent results on the Multiple SLE model with a Dyson Brownian Motion driver. Dyson Brownian motion is a very effective tool in the study the Universality of Random Matrix Ensembles. In my presentation, I will investigate this Multiple SLE model focusing on two particular cases. In the first case, the analysis of Bessel processes is effective and in the second one tools from Random Matrices are useful. Both of the works study the behaviour of the corresponding Loewner chains under perturbations of parameters. In the last part of the talk, I will present some future directions that I plan to study. The result in the first case is obtained jointly with J. Chen, and the one in the second case is part of a recent study performed jointly with A. Campbell, and K. Luh from the Random Matrix Theory community.
Janne Junnila (University of Helsinki): Multiplicative chaos measures from thick points of logcorrelated fields
I will discuss the problem of constructing Gaussian multiplicative chaos measures from level sets of the thick points of the field, where the thickness measured using some fixed approximation scheme of the field. In a recent joint work with G. Lambert and C. Webb we showed convergence in distribution of the level set measures to GMC assuming only certain asymptotic behaviour of the exponential moments of the approximations used. In particular the approximations can be nonGaussian and as an application we were able to prove a conjecture of Fyodorov and Keating on the fluctuations of the mass of thick points of the CUE characteristic polynomial.
Friday
Dapeng Zhan (Michigan State University): Boundary Green's function, Minkowski content, and bubble measure for SLE$_\kappa(\rho)$
We prove the existence of the intersection of an SLE$_\kappa(\rho)$ curve with a real interval using the standard Green’s function approach. Then we show the existence of Minkowski content of the intersection of an SLE$_\kappa(\rho)$ curve with real intervals, which further implies the existence of Minkowski content measure. SLE$_\kappa(\rho)$ bubble measures resemble SLE$_\kappa$ loop measures. The rooted SLE$_\kappa(\rho)$ measures exist for all $\kappa>0$ and $\rho>2$, which satisfies some SLE$_\kappa(\rho)$related domain Markov property, and is the weak limit of the usual SLE$_\kappa(\rho)$ curves with the two endpoints tending to the same point. For $\kappa\in(0,8)$ and $\rho\in ((2)\vee(\frac\kappa 24),\frac\kappa 22)$, we derive a decomposition formula for the rooted SLE$_\kappa(\rho)$ bubble with respect to the boundary Minkowski content, and use it to further construct unrooted SLE$_\kappa(\rho)$ bubble measures.
Lukas Schoug (Helsinki): TBA
Konstantin Izyurov (University of Helsinki): Properties of critical Ising correlations: bosonization, BPZ equations, OPE to all orders
We check several properties that were predicted by physicists about the correlations in the scaling limits of the critical Ising model on planar domains. Among them are the bosonization, i.e., the equality between the square of an Ising correlation and a correlation for the (compactified) Gaussian free field, the secondorder PDE known as the BelavinPolyakovZamolodchikov (BPZ) equations, and the structure of operator product expansions (OPE) to all orders, as predicted by conformal field theory. Joint work with Baran Bayraktaroglu, Tuomas Virtanen, and Christian Webb.
Sung Chul Park (Korea Institute for Advanced Study): Field Theory of the Massive Ising Model
We prove the convergence of correlations between the spin, disorder, fermion, and energy fields in the planar Ising model under massive scaling limit. Massive analogues of the critical operator product expansions (OPE's) are explicitly given. The correlation functions in the limit may be bosonized and identified with correlations in the massless SineGordon model. Based on joint works with Chelkak, Izyurov, Webb, and others.
Baojun Wu (AixMarseille university): Integrability result of boundary Liouville CFT
During this presentation, I will cover several topics related to boundary Liouville CFT, including the current understanding of structure constants, the BPZ equation, the conformal bootstrap method, and how these concepts relate to 2D statistical models.
This talk has two parts.
 In the first part, we consider critical randomcluster model with cluster weight $q\in (0,4)$ and give conjectural formulas for connection probabilities of multiple interfaces. The conjectural formulas are proved for $q=2$, i.e. the FKIsing model.
 In the second part, we consider uniform spanning tree (UST) and give formulas for connection probabilities of multiple Peano curves. UST can be viewed as the limit of randomcluster model as $q$ goes to 0. Its connection probabilities turn out to be related to logarithmic CFT.
This talk is based on joint works with Yu Feng, Mingchang Liu, and Eveliina Peltola.
Jinwoo Sung (University of Chicago): SLEdecorated LQG as an infinitely divisible law on metric spaces
The matingoftrees theory of Duplantier, Miller, and Sheffield states that a $\gamma$LQG surface explored by an independent spacefilling SLE($16/\gamma^2$) curve has an infinitely divisible law, where the relevant observables are the LQG measures in bulk and on the boundary of SLE segments. We extend this formulation to the local LQG metric within SLE cells and show that, almost surely, the $\gamma$LQG diameter of a finite segment of the spacefilling SLE($16/\gamma^2$) curve is finite if and only if $\gamma<\sqrt{8/3}$. This fact resolves a conjecture of Gwynne, Holden, and Sun on the distance exponent for matedCRT maps. This is joint work with Yuyang Feng (University of Chicago).
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.

