Everything is Complex

I will survey a variety of problems that involve comparing harmonic measure on two different sides of a domain boundary. First I will discuss harmonic measures on two sides of closed Jordan curve; this involves conformal welding and Makarov's theorem on the dimension of harmonic measure. Next, we will consider finite planar trees where the harmonic measures on the two sides of each edge are equal; such trees are related to Grothendieck's "dessins d'enfants". Which combinatorial trees can have this property? What possible shapes can they have? If time permits I will discuss the analogous questions for infinite planar trees and some applications to holomorphic dynamics.

I will describe a conformally natural way to draw finite planar trees (as "balanced trees" in Chris Bishop's terminology) and will give an overview of what is known about these drawings in both the deterministic and the random setting. Similarly, every planar dendrite has a "combinatorial description" via a conformal map to its complement, encoded by a conformal lamination. I will describe a characterization of the laminations of a large class of dendrites (namely complements of John domains), and will speculate about the Brownian lamination and Collet-Eckman Julia sets.

For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $\gamma$, $|\gamma|=1$. Namely all such unitary perturbations are the operators $U_\gamma:=U+(\gamma-1) b b_1^*$, where $b\in \mathcal H$, $\|b\|=1$, $b_1=U^{-1} b$, $|\gamma|=1$. For $|\gamma|$<1 the operators $U_\gamma$ are contractions with one-dimensional defects.
Restricting our attention on the non-trivial part of perturbation we assume that $b$ is a cyclic vector for $U$, i.e. that $\mathcal H=\overline{\operatorname{span}}\{U^n b : n\in\mathbb Z\}$. In this case the operator $U_\gamma$, $|\gamma|$<1 is a completely non-unitary contraction, and thus unitarily equivalent to its functional model $\mathcal M_\gamma$, which is the compression of the multiplication by the independent variable $z$ onto the model space $\mathcal K_{\theta_\gamma}$, where $\theta_\gamma$ here is the characteristic function of the contraction $U_\gamma$.
The Clark operator $\Phi_\gamma$ is a unitary operator intertwining the contraction $U_\gamma$, $|\gamma|$<1 (in the spectral representation of the operator $U$) and its model $\mathcal M_\gamma$, $\mathcal M_\gamma \Phi_\gamma = \Phi_\gamma U_\gamma$. In the case when the spectral measure of $U$ is purely singular (equivalently, the characteristic function $\theta_\gamma$ is inner) the operator $\Phi_\gamma$ was described from a slightly different point of view by D. Clark. The case when $\theta_\gamma$ is an extreme point of the unit ball in $H^\infty$ was treated by D. Sarason using the sub-Hardy spaces $\mathcal H(\theta)$ introduced by L. de Branges.
We treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator $\Phi^*_\gamma$ which is represented by a singular integral operator, generalizing in a sense the normalized Cauchy transform studied by A. Poltoratskii. We first present a "universal" representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy−Foiaş and the de Branges--Rovnyak transcriptions, and finally obtain the representation of $\Phi_\gamma$.
The talk is based on a joint work with C. Liaw.

This reports on joint work with N. Makarov. We review the macroscopic behavior of Coulomb gas, in the semiclassical limit as the confining potential gets proportionally stronger with the number of particles. The behavior in the bulk of the droplet is now quite well understood in the $\beta=2$ regime (inverse temperature). The boundary behavior is not quite as well understood, but we believe this can be resolved using a new asymptotic expansion for the orthogonal polynomials. Once the boundary behavior can be analyzed to high accuracy, we expect to obtain strong asymptotics for the free energy (the logarithm of the partition function).

Coulomb gas is one of the most celebrated problems of statistical mechanics. The mathematical theory of Coulomb gas has been developed by N. Makarov with friends. In this talk I discuss geometric aspects of the problem. Placing the gas on a Riemann surface one encounter a number of important concepts of Riemann geometry, such as Riemann-Roch theorem, cohomology of Teichmüller space, and more.

Feigenbaum maps are infinitely renormalizable quadratic polynomials with bounded combinatorics. We have analysed the Basic Trichotomy for the Julia sets of this class:
   - Lean Case: HD ( J ) < 2
   - Balanced Case: HD ( J ) = 2 but area J = 0
   - Black Hole Case: area J > 0
and showed that all three cases can be realized (contrary to the intuition suggested by the Dictionary with Kleinian groups).
The black hole Julia sets are locally connected and visible in the parameter plane (making them quite different from the Buff-Cheritat examples of positive area Julia sets). The lean construction is based upon a new recursive estimate for the Poincare series in various renormalization scales.
I can describe the ideas of these constructions (based upon four renormalization theories).
This is based on my joint work with Artur Avila.

Liouville conformal field theory is a basic building bloc of 2d gravity, the scaling limit of discrete random surfaces. We review a probabilistic construction of the theory, discuss its local conformal invariance and speculate about its exact solution. Joint work with David, Rhodes and Vargas.

A quadrature domain is the domain whose Schwarz reflection has the analytic continuation over the domain. The topology of the quadrature domains can be studied by considering the dynamics under the iteration of the Schwarz reflections. The problem is related to the open problem of finding the maximal number of roots for a harmonic polynomial of a given degree. This is a joint work with Nikolai Makarov.

We study continuity properties of dynamical quantities while crossing the Mandelbrot set through typical smooth curves. In particular, we prove that for almost every parameter $c_0$ in the boundary of the Mandelbrot set $M$ with respect of the harmonic measure and every smooth curve $\gamma:[-1,1]\mapsto {\mathbb C}$ with the property that $c_0=\gamma(0)$ there exists a set ${\mathcal A_\gamma}$ having $0$ as a Lebesgue density point and such that $\lim_{x\to 0} \mathrm{HDim}(J_{\gamma(x)}) =\mathrm{HDim}(J_{c_0})$ for the Julia sets $J_c$. This is joint work with Jacek Graczyk.

About 20 years ago, Makarov and Smirnov started their study of the thermodynamic formalism of rational maps and its connection to the integral means spectrum. For an arbitrary rational map, they gave a complete characterization of phase transitions in the negative spectrum. The purpose of this talk is to survey the recent progress on the more involved case of the positive spectrum, leading to the classification of phase transitions for quadratic maps with real coefficients. We will also discuss the occurrence of phase transitions at infinity.

A generalization of Thurston's rigidity theorem yields smoothness and transversality of dynamically defined subloci of natural deformation spaces.

The convergence of statistical mechanics and quantum field theory in two dimensions is one of the most beautiful chapters of mathematical physics, which has unfortunately remained largely conjectural. For the Ising model, we are now close to having a completely rigorous picture. I will discuss some recent results in this direction. Based on joint works with S. Benoist, D. Chelkak, A. Glazman, K. Izyurov, K. Kytölä and S. Smirnov.

We consider network models for quantum Hall transitions, and incorporate a geometric disorder previously overlooked. We argue that in the continuum this leads to an effective description in terms of a conformal field theory coupled to quenched $2D$ quantum gravity. This coupling changes critical behavior at the transition and may explain discrepancy between the values of critical exponents obtained in experiments and in numerical simulations of network models without the geometric disorder.

Backward SLE is defined using the backward chordal or radial Loewner equation, which does not naturally generate a growing curve or increasing family of hulls. One needs to understand the conformal transformation of a backward Loewner chain. For this purpose, a framework is developed to study the effect of analytic perturbations of weldings on the corresponding hulls. With this tool, we were able to describe how two backward Loewner processes interact with each other. Then we used a stochastic coupling technique from the study of forward SLE to construct a commutation coupling between two backward SLEs, and proved the time reversal symmetry of the welding generated by backward chordal $\mathrm{SLE}_\kappa$ for $\kappa\le 4$. The talk is based on the joint work with Steffen Rohde.

We examine several characteristics of conformal maps that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, they agree for domains whose boundaries are regular fractals such as Julia sets or limit sets of quasi-Fuchsian groups. We show that these characteristics have the same universal bounds, and furthermore, the extremals satisfy a central limit theorem. Our method is based on analyzing the local variance of dyadic martingales associated to Bloch functions. Combined with the recent work of Hedenmalm, these ideas show that quasicircles of dimension $1+k^2$ do not exist.

The goal of the talk is to explain how the six vertex model gives rise to models of $(1+1)d$ random growth in the KPZ universality class, and how the Yang-Baxter integrability of the former leads to solvability of the latter.

A complex structure on a two-dimensional real manifold automatically defines a natural measure on Brownian loops with conformal invariance properties. I will survey some features of Brownian loop-soups, the relation between some of these features and those of SLE, CLE and the Gaussian Free Field. This will be partly based on recent joint work with Wei Qian.

I will discuss the construction of the natural (length) parametrization of the Schramm-Loewner evolution as well as a new result showing that the loop-erased random walk (that is, Laplacian growth at the tip) parametrized by the number of steps converges to SLE in the natural parametrization. This will include work with F. Viklund, M. Rezaei, B. Werness, C. Benes.

Given a family of exponentials $\{e^{i\lambda_n x}\}_{\Lambda}$, is it complete in some $L^2(0,a)$ space? This classical problem was the object of study for many analysts in the early 1900s, including Levinson, Koosis and Kahane. The problem was finally resolved in the 1960s by Beurling and Malliavin. In 2005, Makarov and Poltoratski introduced the language of Toeplitz kernels and model spaces to provide a different proof of the B-M theorem, in their paper Meromorphic Inner Functions, Toeplitz kernels and the Uncertainty Principle. In their work, they also set up a general framework to connect completeness problems of certain families of functions to spectral problems of differential operators.
In this talk, we will discuss the connection between Schröodinger operators and model spaces. We will also look at some recent results that use this framework to characterize the solutions of the so-called 'mixed spectral problems'- where part of the potential and part of the spectrum is used to determine the entire potential. This is joint work with Mishko Mitkovski.

A finite positive measure $\mu$ is said to be $a$-determinate if there is no other finite positive measure $\nu$ such that the Fourier transforms of $\mu$ and $\nu$ agree on some interval of length $a$. For a given measure $\mu$ we show how to estimate the largest $a$ for which $\mu$ is $a$-determinate by looking only at the support of $\mu$. Our approach is partly based on the de Branges-Naimark extreme point method. We use the same method to improve the important result of Eremenko and Novikov concerning oscillations of measures with a spectral gap. I will also present some more recent results about the determinacy part of the classical moment problem.

The talk will focus around some results on two-dimensional random matrix models, especially microscopic properties near the edge of the spectrum. Our approach uses rescaling in Ward identities on the mesoscopic scale. This leads to a non-linear equation for the one-point function, which we call "Ward's equation".
In this way, questions about the local distribution of eigenvalues (or more general systems of repelling point charges in a plane) translate to question about unique solution to Ward's equation, under suitable physically plausible conditions. Joint work with Makarov and Kang.

We present Davis decompositions for Hardy martingales with values in Banach spaces, and use them to derive their Davis and Garsia inequalities (DGI). We discuss the relation of (DGI) to the open problem of characterizing complex Banach spaces with non-trivial Hardy-martingale-cotype.
The talk is based on
P.F. X. Müller: A decomposition for Hardy martingales Part II, Math. Proc. Camb. Philos. Scoc (2014), 157, 189 −207
P.F. X. Müller: A decomposition of Hardy martingales Part III, arXiv:1504.06513

This is a survey on the norm control of inverses in terms of the lower spectral parameter for a variety of operators - convolutions/Fourier multipliers, large matrices, $H^\infty$ quotient algebras. Some new integral operators are added to the list.