Mar 28, 2022
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Geometry, Universality and Beltrami Complex Structure for Scaling Limits of Random Dimer Coverings
Kari Astala (University of Helsinki)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Under suitable boundary conditions, scaling limits of random tilings present surprising geometric features: one observes definite deterministic and disordered (or frozen and liquid) limit configurations with interesting geometric properties.
In this talk we will show how with properties of the Beltrami equation it is possible to understand, for all dimer models and even beyond, the geometry of the boundary between the frozen/deterministic and liquid/random phase. It turns out that in this class the geometry frozen boundaries is universal, i.e. independent of the specific model.
The talk is based on a joint work with E. Duse, I. Prause and X. Zhong.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Integrability of SLE via Conformal Welding of Random Surfaces
Nina Holden (ETH Zurich)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
 
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


The Loewner Equation with ComplexValued Driving Functions
Joan Lind (University of Tennessee)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The chordal Loewner equation provides a correspondence between realvalued functions, called driving functions, and certain growing 2dimensional sets, called hulls. In this talk, we will consider the generalization to complexvalued driving functions, which was first studied by Huy Tran. We will discuss some key differences between the hulls in the complexvalued setting and those in the realvalued setting, including the question of the phase transition from simplecurve hulls. This is joint work with Jeffrey Utley.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Some Progress on 3D YangMills
Sourav Chatterjee (Stanford University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will talk about some recent progress on the problem of constructing 3D Euclidean YangMills theories. This is based on joint work with Sky Cao.
 Supplements



Mar 29, 2022
Tuesday

09:30 AM  10:30 AM


Random Weierstrass ZetaFunctions
Mikhail Sodin (Tel Aviv University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
 
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Random Clusters in the Villain Model
Julien Dubedat (Columbia University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In the Ising and Potts model, random cluster representations provide a geometric interpretation to spin correlations. We discuss similar constructions for the Villain model, where spins take values in the circle.
Work in progress with Hugo Falconet.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


On LogCFT for Uniform Spanning Trees and SLE(8)
Eveliina Peltola (Rheinische FriedrichWilhelmsUniversität Bonn)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I discuss the emergence of logarithmic CFT content associated to SLE(8) and nonlocal observables in the planar uniform spanning tree (UST) model, constructed via scaling limits of Peano curves and their crossing probabilities. In particular, with explicit correlation functions and their fusion thus obtained, one sees that any CFT describing the geometry of UST must be nonunitary (thus not reflection positive). This is of course no surprise  we give a systematic construction directly from the lattice model via its scaling limit, together with immediate relation to SLE(8).
Joint work with Mingchang Liu and Hao Wu.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Critical Liouville Quantum Gravity and Brownian HalfPlane Excursions
Ellen Powell (University of Durham)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with SchrammLoewner evolutions (SLE) can be described by the mating of two continuum random trees. In this talk I will discuss the counterpart of their result for critical LQG and SLE. More precisely, I will explain how, as we approach criticality from the subcritical regime, the spacefilling SLE degenerates to the uniform CLE_4 exploration introduced by Werner and Wu, together with a collection of independent coin tosses indexed by the branch points of the exploration. Furthermore, although the pair of continuum random trees collapse to a single continuum random tree in the limit we can apply an appropriate affine transform to the encoding Brownian motions before taking the limit, and get convergence to a Brownian halfplane excursion. I will try to explain how observables of interest in the critical CLE decorated LQG picture are encoded by a growth fragmentation naturally embedded in the Brownian excursion. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun.
 Supplements

Slides
4.12 MB application/pdf


04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Mar 30, 2022
Wednesday

09:30 AM  10:30 AM


Loewner Evolution Driven by Complex Brownian Motion
Ewain Gwynne (University of Chicago)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We consider the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of SchrammLoewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither $B^1$ nor $B^2$ is identically equal to zero, then the complements of the Loewner hulls are not connected. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls, and a phase where the hulls are spacefilling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero. Based on joint work with Josh Pfeffer, with simulations by Minjae Park.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Conformally Invariant Random Geometry on Riemannian Manifolds of Even Dimension
KarlTheodor Sturm (Universität Bonn)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We construct and study conformally invariant, logcorrelated Gaussian random fields on compact Riemannian manifolds of general even dimension uniquely defined through its covariance kernel given as inverse of the GrahamJenneMasonSparling (GJMS) operator. The corresponding Gaussian Multiplicative Chaos is a generalization to the ndimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly.
 Supplements



Mar 31, 2022
Thursday

09:30 AM  10:30 AM


Stability of Regularized HastingsLevitov Aggregation in the Subcritical Regime
Amanda Turner (University of Lancaster)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In 1998, Hastings and Levitov proposed a family of models for random growth, indexed by a parameter alpha, which includes versions of diffusionlimited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. They predicted a change at alpha = 1 (which corresponds to the Eden model), from stable to turbulent behaviour. In this talk, I will show that the limit dynamics of the HL model, as the particle size goes to zero, follow solutions of a certain LoewnerKufarev equation, where the driving measure is made to depend on the solution and on the parameter alpha. The fluctuations around the scaling limit are shown to be Gaussian, with independent OrnsteinUhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if alpha is less than or equal to 1, consistent with Hastings and Levitov's prediction.
This talk is based on the paper https://arxiv.org/abs/2105.09185 which is joint work with James Norris and Vittoria Silvestri.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Plumbing Liouville Theory
Antti Kupiainen (University of Helsinki)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Riemann surfaces can be constructed by gluing ( “plumbing”) three holed spheres along their boundary circles. This operation is postulated by the physicists to lift to Conformal Field Theory (CFT) under the name conformal bootstrap. I will explain how this works probabilistically in the case of Liouville CFT. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Conformal Structures on Random Fractal Surfaces Arising from Subdivision Rules
Peter Lin (State University of New York, Stony Brook)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
We consider "conformal" parameterizations of random fractal spaces $X$ arising as limits of certain stochastic subdivision rules.
One motivation comes from the field of random geometry, where it is an important and difficult problem to understand this parameterization when $X$ arises from limits of random planar maps.
Deterministic versions of our model, and the analogous questions relating to conformal parameterizations, are closely related to W. Thurston's topological characterization of rational maps.
In all the settings mentioned above, a key difficulty is in proving that the fractal approximations do not degenerate in the complex analytic sense.
We will overcome this difficulty in our setting by proving a contraction inequality for probabilistic iteration on a variant of the universal Teichmuller space.
This inequality also provides a different perspective on random quasiconformal map models considered by AstalaRohdeSaksmanTao and IvriiMarkovic.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Scattering of Harmonic Functions and Forms in Quasicircles
Eric Schippers (University of Manitoba)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Given a harmonic function h of bounded Dirichlet energy on a quasidisk, there exists a harmonic function of bounded Dirichlet energy on the interior of its complement with the same boundary values (except on a negligible set). We call this the overfare of h. The existence of a bounded overfare with respect to the Dirichlet seminorm characterizes quasicircles among Jordan curves. A similar process is obtained for L^2 harmonic oneforms by conjugating with differentiation. We extend this overfaring process for oneforms to collections of quasicircles separating a Riemann surface, and give an explicit expression in terms of integral operators of Schiffer related to the PlemeljSokhotski jump decomposition. We use this to elucidate the geometric and analytic meaning of the Grunsky and Faber operators and their generalizations. This process is further related to the geometry of the specific surface and moduli space, for example providing index theorems for the Schiffer operators. If time allows we also discuss analogies with scattering theory. Joint work with Wulf Staubach.
 Supplements



Apr 01, 2022
Friday

09:30 AM  10:30 AM


Approximations of Liouville Brownian Motion
ZhenQing Chen (University of Washington)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Liouville Brownian motion was introduced as a canonical diffusion process under Liouville quantum gravity. It is constructed as a time change of 2dimensional Brownian motion by the continuous additive functional associated with a Liouville measure, through a regularizing approximation procedure of the Gaussian free field. In this talk, we are concerned with the question whether one can construct Liouville Brownian motion directly from the Liouville measure. We will present a discrete approximation scheme that in fact works for more general time changed Brownian motions.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


The Moduli of Annuli in Random Conformal Geometry
Xin Sun (University of Pennsylvania)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk and the disk boundary. The formula is as conjectured from the partition function of the $O(n)$ loop model on the annulus derived by Cardy (2006). The third is for the annulus partition function of the $\SLE_{8/3}$ loop introduced by Werner (2008). It again confirms a prediction of Cardy (2006). The physics principle underlying our proofs is that 2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the Liouville CFT. We expect our method to be applicable to a variety of questions related to the random moduli of nonsimplyconnected random surfaces. Joint work with Morris Ang and Guillaume Remy.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Conformal Field Theory for Multiple SchrammLoewner Evolutions
NamGyu Kang (Korea Institute for Advanced Study (KIAS))

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Multiple SLEs describe several random interfaces consistent with conformal symmetries. I will implement a version of conformal field theory constructed from background charge modifications of Gaussian free field and insertion of Nleg operators with screening to show that this version produces a collection of martingaleobservables for commuting multiple SLEs. I will explain how this theory is related to its classical limit counterpart, "Loewner Dynamics for Real Rational Functions and the Multiple SLE(0) Process," presented by Tom Alberts in "Introductory Workshop: The Analysis and Geometry of Random Spaces."
Based on joint work with Tom Alberts and Nikolai Makarov.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Jordan Curves with the Piecewise Geodesic Property
Wang Yilin (Massachusetts Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We consider Jordan curves passing through a given set of n points with the property such that each edge is the hyperbolic geodesic in the complement of the rest of the curve. Those curves are related to the minimizers of the Loewner energy and give a specific complex projective structure on the npunctured sphere with real and parabolic holonomy around the punctures. Similar to a result of TakhtajanZograf, we also obtain a new type of accessory parameters and show that they can be expressed as differentials of the Loewner energy.
This is based on the joint work with Don Marshall and Steffen Rohde.
 Supplements



