Modulus of Continuity of Controlled Loewner-Kufarev Equations and Random Matrices

Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian

We introduce the class of controlled Loewner-Kufarev equations and consider aspects of their algebraic nature. We lift the solution of such a controlled equation to the (Sato)-Segal-Wilson Grassmannian, and discuss its relation with the tau-function. We briefly highlight relations of the Grunsky matrix with integrable systems and conformal field theory. Our main result is the explicit formula which expresses the solution of the controlled equation in terms of the signature of the driving function through the action of words in generators of the Witt algebra.

Univalence and holomorphic extension of the solution to $ω$-controlled Loewner-Kufarev equations

We prove that a solution to the $ω$-controlled Loewner-Kufarev equation, which was introduced by the first two authors, exists uniquely, is univalent and starlike on the unit disk and can be extended holomorphically across the unit circle.

Loewner-Kufarev 能量与 Weil-Petersson 拟圆周的叶状结构

The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles

本文使用 Loewner-Kufarev 方程研究黎曼球面的弦弧 Jordan 曲线叶状结构，证明了局部卷绕函数的 Dirichlet 能量与 Loewner 驱动测度的 Loewner-Kufarev 能量等价，并据此给出了 Weil-Petersson 拟圆周的新定量刻画。

布朗珍珠奶茶

The Brownian bubble tea

该研究利用Loewner-Kufarev演化分解布朗环汤为布朗气泡的泊松点过程，推广了相关结果并给出能量间关系的布朗环解释。

Conformal field theory and Loewner-Kufarev evolution

One of the important aspects in recent trends in complex analysis has been the increasing degree of cross-fertilization between the latter and mathematical physics with great benefits to both subjects. Contour dynamics in the complex plane turned to be a meeting point for complex analysts, specialists in stochastic processes, and mathematical physicists. This was stimulated, first of all, by recent progress in understanding structures in the classical and stochastic Löwner evolutions, and in the Laplacian growth. The Virasoro algebra provides a basic algebraic object in conformal field theory …

Loewner evolution and finite dimensional reductions of integrable systems

The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to some known integrable systems.