M 09:15 Mikhail Lyubich (SUNY Stony Brook): Mirrors of Conformal Dynamics: Julia sets, Kleinian groups, Schwarz reflections, and correspondences
We will discuss an interplay between four branches of Conformal Dynamics: iterations of (anti-)rational maps, action of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and dynamics of algebraic correspondences. We will show how quasiconformal or David surgery can turn one set on this list to another one (for instance, the classical Apollonian gasket into a Julia set). It also allows one to mate Julia sets with Kleinian limit sets obtaining sets generated by Schwarz reflections or (anti-)algebraic correspondences (for instance, one can mate an anti-holomorphic quadratic polynomial with the modular group). Interesting examples of removable fractals (e.g., limit sets of necklace groups) can be obtained in this way. It also leads to intrinsic relations between the corresponding parameter spaces.
Based on the work with Nick Makarov, Sergiy Merenkov, Sabya Mukherjee, and other people.
M 10:30 Yilin Wang (IHES): Epstein's construction and holography of Loewner energy and Schwarzian action
The group PSL(2, C) acts on the hyperbolic three-space H3 as the group of isometries and on the boundary identified with the Riemann sphere S2 as conformal automorphisms. The holographic principle looks to encode the conformally invariant quantities on the Riemann sphere into geometric quantities in H3 and vice versa. In one lower dimension, we may also study the correspondence between H2 and the circle S1 , where the group PSL(2, R) acts.
The Loewner energy is a conformally invariant quantity that measures the roundness of Jordan curves on the Riemann sphere and has close links to the geometry of universal Teichmuller space and SLE loop measures. We show that the Loewner energy equals the renormalized volume of a submanifold of the hyperbolic H3 constructed using a truncation introduced by Epstein. Applying Epstein's truncation in one lower dimension, we show that the renormalized area of H2 coincides with minus the Schwarzian action of circle diffeomorphisms.
This is based on joint works with Martin Bridgeman, Ken Bromberg, Franco Vargas Pallete and Catherine Wolfram.
M 11:20 Fredrik Viklund (KTH Royal Institute of Technology): Free energy of constrained planar Coulomb gases
Consider a Coulomb gas of charged particles constrained to a set in the complex plane. I will discuss the following question: How does the asymptotic expansion of the free energy depend on the geometry of the set, as the number of particles tends to infinity? When the set is a Jordan domain, curve or arc, this problem is related to Grunsky operators associated to the set, revealing a close connection to the Loewner energy and other interesting domain functionals. Based joint works with K. Courteaut (NYU) and K. Johansson (KTH).
M 15:30 Rostislav Grigorchuk (Texas A&M University): The Collatz 3x + 1 Conjecture, Self-Similar Group, and Ergodic Decomposition
In my talk, based on a joint work with Zoran Sunik, I will explain the relation between three topics listed in the title, present some results and formulate open problems. If time permits, some results obtained with D. Savchuk and Z. Sunik concerning the ergodic decomposition of automorphisms of rooted trees will be presented as well.
M 17:00 Chris Bishop (SUNY Stony Brook): Weil-Petersson curves, traveling salesman theorems and minimal surfaces
Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2006. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knot theory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than twenty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic 3-space that has finite total curvature. I will discuss these and several other characterizations and sketch why they are all equivalent to each other. The lecture will contain many pictures, several definitions, but not too many technical details.
T 09:15 Greg Lawler (University of Chicago): Exploring a paper of Kang and Makarov
I will present a perspective of the Gaussian free field (GFF) as being constructed as it is being explored by a curve. As an example, I will view the operators coming from derivatives of the field as discussed in the paper Gaussian free field and conformal field theory by Kang and Makarov as martingales. If time allows, I will also describe the construction of quantum length in this framework.
T 10:30 Jason Miller (Cambridge): Metrics on non-simple conformal loop ensemble gaskets
We consider the conformal loop ensembles (CLE) for kappa in (4,8), the regime in which the loops intersect themselves, each other, and the domain boundary. We will describe forthcoming work which aims to associate the gasket of such a CLE (i.e., the set of points not surrounded by a loop) with various canonical metrics as well as consequences for discrete models which converge to such a CLE in the scaling limit.
T 11:20 Ewain Gwynne (University of Chicago): Random walk reflected off of infinity
Let G be an infinite graph --- not necessarily one-ended --- on which the simple random walk is transient. We define a variant of the continuous-time random walk on G which reaches ∞ in finite time and "reflects off of ∞" infinitely many times.
We show that the Aldous-Broder algorithm for the random walk reflected off of ∞ gives the free uniform spanning forest (FUSF) on G. Furthermore, Wilson's algorithm for the random walk reflected off of ∞ gives the FUSF on G on the event that the FUSF is connected, but not in general.
We also apply the theory of random walk reflected off of ∞ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge c in (1,25). Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also conjecture that the free uniform spanning forest on these maps is connected when c > 16 (but not when c < 16); and that there is an infinite open cluster for critical percolation on these maps when c < 95/4 (but not when c > 95/4).
Based on joint work with Jinwoo Sung.
T 17:00 Dmitry Chelkak (University of Michigan): Conformal structures in the planar Ising, UST, and dimer models via surfaces in R2,1 and R2,2
Planar Ising model, bipartite dimers, and uniform spanning trees are classical examples of free fermionic lattice models in 2d. Given a sequence of large planar graphs carrying such a model with “mesh size going to zero”, one is interested in finding a relevant complex structure that describes the limit of correlation functions. In recent years, it has been observed that this description most naturally comes from special embeddings of planar graphs into R2,1 or R2,2, the so-called s- and t-surfaces. The aim of the talk is to present recent convergence results obtained in this context for limits of planar random walks (joint work with Basok, Laslier, and Russkikh) and Ising model (joint work with Mahfouf and Park).
T 18:00 Hao Wu (Tsinghua University): Connection probabilities for loop O(n) models and BPZ equations
Critical loop O(n) models are conjectured to be conformally invariant in the scaling limit. In this talk, we focus on connection probabilities for loop O(n) models in polygons. Such probabilities can be predicted using two families of solutions to BPZ equations: Coulomb gas integrals and SLE pure partition functions. The conjecture is proved to be true for the critical Ising model, FK-Ising model, percolation, and uniform spanning tree.
W 09:15 Antti Kupiainen (Helsinki): Wess-Zumino-Witten models and path integrals
The Wess-Zumino-Witten (WZW) model is a 2 dimensional conformal field theory (CFT) where the field takes values in a Lie group G or its coset space. For a compact G this CFT is rational and its cosets G/H include for instance all unitary rational CFTs (e.g. the Ising model). WZW model has a formal path integral representation whose rigorous construction has remained elusive and in fact most of its conjectured properties have been discussed using the representation theory of affine Lie algebras. In this talk I will review the basic facts about the path integral formulation of WZW models and then discuss the coset theory SL(2,C)/SU(2). This theory can be formulated in terms of field taking values in the 3-dimensional hyperbolic space and by the work of Ribault, Teschner, Hikida and Schomerus it has been argued to have a mapping to the Liouville CFT. This map has been argued to provide a “quantum” deformation of the geometric and analytic Langlands correspondence. I will explain briefly how this theory can be constructed probabilistically using the theory of Gaussian Multiplicative Chaos and how on a general Riemann surface the correlation functions of its primary fields can be mapped to those of the Liouville CFT.
W 10:30 Eveliina Peltola (Aalto & Bonn): On geometric properties of conformally invariant curves
How to construct a canonical random conformally invariant path in two dimensions? Motivated by Loewner's classical theory of dynamics of slit domains, Schramm introduced random Loewner evolutions to model canonical random curves via evolutions of conformal maps. While the initial usage of such Schramm-Loewner evolutions (SLEs) was to describe critical interfaces in statistical physics models and their relation to conformal field theory (CFT), SLE type curves quickly turned out to be ubiquitous in various problems in probability theory and mathematical physics, and to have intricate connections to complex geometry and beyond.
This talk highlights some geometric aspects emerging from the study of SLE curves and CFT. As examples, we shall mention versions of Loewner energy (the anticipated action functional of these canonical curve models, or more rigorously, the rate function in large deviations principles for the random curves), classification problems of covering maps with prescribed critical points, and the emergence of the Virasoro algebra (the symmetry algebra of CFTs) from complex deformations of boundaries of bordered Riemann surfaces (i.e., loops).
W 11:20 Tom Alberts (University of Utah & KIAS): Conformal field theory for multiple SLEs
Over the last 10-15 years Kang and Makarov have developed a version of Conformal Field Theory that studies correlation functions of the Gaussian Free Field. Their papers first lay the groundwork for several independent pieces of the theory: the Ward equations (an integration-by-parts formula), the operator product expansion (for subtracting off diverging infinities), the Girsanov theory (for representing shifts of the field), and level two degeneracy (properties of vertex exponentials). Then, in the penultimate stroke, these pieces are combined together to show that a broad class of correlation functions derived from the GFF are martingale observables for an associated SLE type process. Kang and Makarov pay special attention to doing this in a way that is "coordinate free", and ultimately their technique becomes an efficient way of carrying out Ito formula computations with minimal effort.
I will explain how the Kang-Makarov framework gives another viewpoint into the SLE-GFF connection, in a way that is very natural for researchers with a primary background in complex analysis and potential theory. My main goal is to give an idea of how it can be used to explain systems of multiple SLE curves, screening techniques, deterministic limits of SLEs, rational functions, and classical integrable systems.
Th 09:15 Nikolai Nikolski (University of Bordeaux): Cyclic vectors of the dilation semigroup
This is a glimpse into the Dilation Completeness Problem (DCP) in a rather general setting. Four equivalent representations of the multiplicative dilation semigroup f(kx), k = 1, 2, ... are considered on L2 spaces, as well as on more general Banach lattices. A simplest (and historically the tenth in a row) proof of Haar's lemma is presented, still valid for totally multiplicative sequences. We discuss the dilation cyclicity of polynomials for weighted Blp (N) spaces, including for the Drury-Arveson norms ("B" stands for the Bohr-Fourier transform), as well as an elusive "missing link" between the DCP and the RH.
Th 10:30 Sergei Treil (Brown University): The inverse spectral problem for positive Hankel operators
Hankel operators are bounded operators in l2 with matrix with coefficient depending on the sum of indices. They also can be realized as the integral operators on the half-line with kernel of the form h(s+t).
We consider positive semidefinite Hankel operators with simple spectrum and solve the inverse spectral problem for such operators, i.e. reconstruct the kernel h from the appropriately chosen spectral measure.
It turns out that solutions of the spectral problem are drastically different for discrete and continuous realizations: to reconstruct the kernel h one only needs the natural (for the problem) spectral measure. However, in the discrete case (operators in l2) the spectral measure is not enough, and to reconstruct the coefficients one needs an extra spectral invariant.
The talk is based on a joint work with A. Pusnitski.
Th 11:20 Alexander Volberg (MSU and Hausdorff Center for Mathematics): Fourier growth for boolean polynomials
The estimates of the sums of absolute values of level-k Fourier coefficients of boolean polynomials has important applications in Theoretical Computer Science. It is also an exiting harmonic analysis problem. However, good estimates are known only in a few cases. We will demonstrate a couple of such cases.
Th 17:00 Dmitry Beliaev (University of Oxford): Level sets of random functions
We will survey recent progress in understanding the large-scale properties of level sets of smooth Gaussian fields. It is conjectured that the level sets of a large class of fields behave similarly to percolation. We are going to describe this conjecture, the current state-of-the-art, and many open questions.
Th 18:00 Jacek Graczyk (Paris-Saclay): Bloch martingales and dynamics of polynomials
The inspiration for this work is the study of univalent mappings by probability methods presented by N. Makarov over 40 years ago. In complex dynamics, these methods have an immediate bearing on the properties of Riemann mappings for the complement of Julia sets in the phase space or the connectedness locus in parameters. Dyadic martingales can be combined with Yoccoz partitions to get some new information about geometry and global dynamical parameters like Lyapunov exponent or asymptotic variance. In particular, the Mandelbrot set is a Makarov compact.
This approach is based on a recent work with G. Swiatek.
F 09:15 Guy David (Université Paris Saclay): Harmonic measure with a Robin boundary condition
Most of the talk will describe results on the mutual absolute continuity of the natural Hausdorff measure on the boundary of a domain in Euclidean space, with respect to the harmonic measure, but with Robin boundary values, i.e., where we add a multiple of the normal derivative of the solution to the usual Dirichlet condition. We'll discuss a natural setup for this, perhaps some examples in the Dirichlet case, and some absolute continuity results. Joint work with S. Decio, M. Engelstein, M. Filoche, S. Mayboroda, M. Michetti in particular.
F 10:30 Haakan Hedenmalm (KTH Royal Institute of Technology): Gaussian analytic functions and operator symbols of Dirichlet type
We study the correlation structure of two (not independent) copies of the GAF associated with the Dirichlet space. Here, the structure is determined by a contraction acting on area-L2 on the disk. A particularly fascinating instance is that of the Grunsky operator in the theory of conformal mapping. The operator symbol for such a Grunsky operator solves a nonlinear wave equation, and that wave equation determines such symbols uniquely.
F 11:20 Mikhail Sodin (Tel Aviv University): Equivariant Weierstrass theorem
Consider a map Z from the space E of entire functions to the space D of discrete subsets of the complex plane, which assigns to each entire function its zero set. The classical Weierstrass theorem asserts that this map is surjective.
With the natural topologies, both E and D are Polish spaces, and the complex plane acts continuously on each by translation. Moreover, the map Z is equivariant with respect to these actions. Does it admit an equivariant (right) inverse?
In joint work in progress with Konstantin Slutsky and Aron Wennman, we prove the existence of an equivariant Borel inverse if the action of the complex plane on D is free. Such an inverse cannot be made continuous, and no such inverse exists if the action is not free. Our main result is closely related to the notion of a "measurably entire function" introduced by Benjy Weiss.
Similar results hold in several related settings, including the Mittag-Leffler problem and the Poisson and d-bar equations.