Conformal Welding of Discs and Trees

圆盘和树木的保形焊接

• 批准号: 1954674
• 负责人: Steffen Rohde
• 金额: $ 24.33万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954674&HistoricalAwards=false
• 关键词: Conformal; Welding; Discs; Trees

Fractals, largely characterized as structures that look the same at different scales, are ubiquitous in nature and the sciences. They are intriguing for both their aesthetic appeal and for the fact that despite their high complexity they often arise as the result of very simple mechanisms. On the mathematical side, such structures arise when repeatedly applying a simple rule such as a quadratic polynomial. In nature and in statistical physics models, very similar patterns and shapes emerge from randomness. The Principal Investigator's research is mostly concerned with two-dimensional fractals that possess some form of scale and rotation invariance, both in the deterministic and in the random setting. Using methods from complex analysis, the principal investigator and his students will work towards answering foundational questions and an understanding of why the deterministic and random patterns are so similar. The project also supports the training of graduate students.Conformally self-similar two-dimensional sets can often be described through conformal welding. Conformal welding is the process of gluing two Riemann surfaces along their boundaries to form a new surface. It is instrumental in Teichmueller theory, of independent interest in geometric function theory, and has recently been intensely studied in the context of the Schramm-Loewner Evolution (SLE) and Liouville Quantum Gravity (LQG). The PI will explore the recently found connections between Teichmueller theory and Loewner theory. He will apply his new conditions on weldability to the setting of SLE and LQG, aiming at an analytic construction of SLE by gluing quantum discs. He will also apply this technique to obtain conformal spheres from matings of trees, both in the deterministic setting of conformal mating of Julia sets and in the probabilistic setting of the Brownian map as the gluing of two continuum random trees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

分形在很大程度上被描述为在不同尺度上看起来相同的结构，本质和科学无处不在。他们的美学吸引力和事实是，尽管它们的复杂性很高，但它们通常是由于非常简单的机制而出现的。在数学方面，当反复应用一个简单的规则（例如二次多项式）时，就会出现这种结构。在自然界和统计物理模型中，随机性出现了非常相似的模式和形状。主要研究者的研究主要与具有某种形式的量表和旋转不变性的二维分形有关，无论是在确定性和随机环境中。使用复杂分析中的方法，主要研究人员及其学生将致力于回答基本问题，并了解为什么确定性和随机模式如此相似。该项目还支持对研究生的培训。通常可以通过共同焊接来描述自相似的二维集合。共形焊接是沿边界粘合两个Riemann表面以形成新表面的过程。它在Teichmueller理论中具有重要意义，对几何函数理论具有独立感兴趣，并且最近在Schramm-Loewner Evolution（SLE）和Liouville量子重力（LQG）的背景下进行了深入研究。 PI将探讨Teichmueller理论与Loewner理论之间最近发现的联系。他将在SLE和LQG的设置上应用他的新条件，以通过粘合量子盘对SLE进行分析结构。他还将应用这项技术从树木的成分中获取保形球，无论是在朱莉娅集合的保形交配的确定性设置和布朗尼地图的概率环境中，作为两个连续树的粘合，这奖奖均反映了NSF的法定任务，并通过评估了基金会的范围和广泛的cr依据，并通过评估了支持。