Applications of the Geometric Function Theory to Mathematical Physics

几何函数理论在数学物理中的应用

• 批准号: RGPIN-2014-06586
• 负责人: Binder, Ilia
• 金额: $ 1.68万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612315
• 关键词: Applications; Geometric; Function; Theory; Mathematical

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.

几何函数论是分析的一个经典领域，起源于Riemann对他著名的映射定理的证明。最早的两枚菲尔兹奖章中的一枚于1936年授予L.Ahlfors，以表彰他在该领域的研究。1985年，著名的比伯巴赫系数猜想是该理论中的主要猜想之一，自1916年以来一直是公开的，并被L.de Brange成功地证实了。证明中使用的主要技术工具之一是所谓的洛夫纳微分方程。后者是C.Loewner在1923年发明的一种工具，专门用于解决Bieberbach猜想。 洛夫纳演化在几何函数理论之外的统计物理共形不变普适性现象的研究中发现了意想不到的应用。这一现象在60年代末由L.Kadanoff和K.Wilson(后来因其研究而获得诺贝尔奖)首次被观察到，并在80年代由A.Polyakov和他的合作者进一步发展.该原理指出，各种格子模型的长期行为仅取决于“普适类”，并且在共形映射下是局部不变的.直到1998年，O.Schramm发明了Loewner方程的随机版本--Schramm Loewner进化(SLE)，对这一现象的严格数学理解一直遥不可及。该族被猜想为不同普适性类中晶格模型界面的标度极限。G.Lawler、O.Schramm和W.Werner(因这项工作于2006年获得菲尔兹奖)证实了循环擦除随机行走的猜想。S·斯米尔诺夫也获得了菲尔兹奖(2010年)，他证实了临界六角渗流和伊辛模型的猜想。在O.Schramm和S.Shefffield的工作中，对高斯自由场的水平线建立了猜想。 然而，对于大多数晶格模型来说，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。在物理文献中，特别是在我之前的工作中，对光谱的值进行了预测。我们计划严格建立价值观。 另一个重要的物理现象是安德森定域化。非正式地，它指出，在某些条件下，无序可以抑制波。虽然Anderson局域化物理理论是上个世纪凝聚态物理的主要成果之一，但它的数学对应关系仍在积极发展中。我打算研究这种无序的两个数学模型：具有准周期势和随机势的雅可比算子。在第一种情况下，我将主要集中在逆谱问题上，从它的谱来理解势的性质。我计划使用调和测度的估计，这是几何函数论的经典工具之一。在随机设置中，我将致力于改进本地化的定量估计。 几何函数论本身的许多悬而未决的问题可以用平面区域的多重分形谱的最优界来表述。其中之一是著名的布伦南猜想及其推广。我计划使用最近开发的方法来解决这些问题。 我还打算对保角映射及其边值的可计算性有更深入的了解。