Applications of the Geometric Function Theory to Mathematical Physics

几何函数论是一个经典的分析领域，由黎曼证明他著名的映射定理。1936年，第一届菲尔兹奖的两个奖项之一授予了L。阿尔福斯在该领域的研究。1985年，著名的比伯巴赫系数猜想被L.德布兰日。证明中使用的主要技术工具之一是所谓的Loewner微分方程。后者是C. Loewner在1923年专门为解决比伯巴赫猜想的目的。Loewner演化在几何函数论之外发现了一个意想不到的应用，在统计物理的共形不变普适性现象的研究中。这一现象最早出现在六十年代后期的L. Kadanoff和K.威尔逊（后来因其研究获得诺贝尔奖），并在八十年代由A。直观地说，该原理指出各种格模型的长程行为只依赖于“普适类”，并且在共形映射下是局部不变的。对这一现象的严格数学理解仍然遥不可及，直到一个随机版本的Loewner方程，施拉姆Loewner演变（SLE），是由O。1998年的施拉姆。该族被证明是不同普适类格点模型界面的标度极限。G. Lawler，O. Schramm和W.沃纳（他在2006年因这项工作获得菲尔兹奖）验证了循环擦除随机游走的猜想。S.斯米尔诺夫，谁也获得了菲尔兹奖（2010年），验证了猜想的临界Hisconal渗流和伊辛模型。在O. Schramm和S.谢菲尔德提出的关于高斯自由场能级线的猜想。然而，对于大多数格模型，这个猜想仍然是开放的。我们项目的目标之一是促进对猜想的理解，以及建立相关的收敛速度。SLE曲线本身的精细几何性质可以用多重分形谱来表示。在物理学文献中，特别是在我以前的工作中，预测了光谱的值。我们将严格执行这些标准 **。另一个重要的物理现象是安德森局域化。非正式地说，它指出，在某些条件下，无序可以抑制波。安德森定域化物理理论是上个世纪凝聚态物理学的主要成就之一，而它的数学对应理论仍在积极发展之中。我打算研究两个数学模型的障碍：雅可比运营商与准周期和随机潜力。在第一种情况下，我将主要集中在逆谱问题，从它的频谱了解潜在的性质。我计划使用几何函数论的经典工具之一调和测度的估计。在随机设置中，我将致力于改进本地化的定量估计。**几何函数论本身的一些未回答的问题可以用平面域的多重分形谱的最优边界来表述。其中之一是著名的布伦南猜想及其推广。我打算用最近开发的方法来解决这些问题。我还打算获得更深入的了解共形映射的可计算性及其边界值。

Geometric Function Theory is a classical area of Analysis, initiated by Riemann's proof of his famous mapping theorem. One of the two first Fields medals was awarded in 1936 to L. Ahlfors for his research in the field. In 1985, one of the main conjectures in the Theory, the famous Bieberbach coefficients conjecture, which remained open since 1916, was successfully verified by L. de Branges. One of the main technical tools used in the proof was the so-called Loewner Differential Equation. The latter is a tool invented by C. Loewner in 1923 specifically for the purpose of solving the Bieberbach conjecture.**The Loewner Evolution found an unexpected application outside of Geometric Function Theory, in the study of conformally invariant Universality Phenomenon of Statistical Physics. The phenomenon was first observed in the late sixties in the works of L. Kadanoff and K. Wilson (who later received a Nobel prize for his research), and further developed in the eighties by A. Polyakov and his collaborators.Intuitively, the principle states that the long-range behavior of the various lattice models depends only on the "universality class" and is locally invariant under conformal maps. The rigorous mathematical understanding of the phenomenon remained out of reach until a Stochastic version of Loewner Equation, the Schramm Loewner Evolution (SLE), was invented by O. Schramm in 1998. The family was conjectured to be a scaling limit of interfaces of lattice models in different universality classes. G. Lawler, O. Schramm, and W. Werner (who received the Fields medal in 2006 for this work) verified the conjecture for the Loop Erased Random Walk. S. Smirnov, who also received the Fields medal (in 2010), verified the conjecture for the Critical Hexagonal Percolation and Ising model. In the works of O. Schramm and S. Sheffield, the conjecture was established for the level lines of the Gaussian Free Fields.*Yet for most of the lattice models, the conjecture is still open. One of the aims of our project is to advance understanding of the conjecture, as well as to establish the relevant rate of convergence.*The fine geometric properties of the SLE curves themselves can be formulated in terms of the multifractal spectrum. The values of the spectrum were predicted in the physics literature, in particular, in my previous work. We plan to rigorously establish the values.**Another important physical phenomenon is Anderson localization. Informally, it states that, under certain conditions, disorder can suppress the waves. While the physical theory of Anderson localization was one of the main achievements of the physics of condensed matter in the last century, its mathematical counterpart is still under active development. I intend to study two of the mathematical models of the disorder: Jacobi operators with quasiperiodic and random potentials. In the first case, I will mainly concentrate on the Inverse Spectral Problems, the understanding of the properties of a potential from its spectrum. I plan to use of estimates on harmonic measure, one of the classical tools of the Geometric Function Theory. In the random setting, I will work on improving the quantitative estimates on localization.**A number of unanswered questions in Geometric Function Theory itself can be stated in terms of the optimal bounds on the multifractal spectrum of planar domains. One of them is the celebrated Brennan's conjecture and its generalizations. I plan to tackle these questions using recently developed methods.*I also intend to acquire deeper understanding of the computability properties of Conformal maps and their boundary values.