Conformal Welding of Discs and Trees

分形在自然界和科学中普遍存在，其主要特征是结构在不同的尺度上看起来是相同的。它们耐人寻味，既是因为它们的美学吸引力，也是因为尽管它们高度复杂，但它们往往是非常简单的机制的结果。在数学方面，当重复应用简单的规则，如二次多项式时，就会出现这样的结构。在自然界和统计物理模型中，随机性产生了非常相似的图案和形状。首席调查者的研究主要涉及在确定性和随机性环境下具有某种形式的尺度和旋转不变性的二维分形图。使用复杂分析的方法，首席研究员和他的学生将致力于回答基本问题，并理解为什么确定性模式和随机模式如此相似。该项目还支持研究生的培养。共形自相似二维集通常可以通过保形焊接来描述。保形焊接是将两个黎曼曲面沿其边界粘合以形成新曲面的过程。它是Teichmueller理论中的重要组成部分，对几何函数理论具有独立的兴趣，最近在Schramm-Loewner演化(SLE)和Liouville量子引力(LQG)的背景下得到了深入的研究。PI将探索最近发现的Teichmueller理论和Loewner理论之间的联系。他将把他关于焊接性的新条件应用于SLE和LQG的设置，旨在通过粘合量子盘来解析构建SLE。他还将应用这项技术从树的匹配中获得保形球体，无论是在Julia集的保形交配的确定性设置中，还是在布朗映射的概率设置中，作为两个连续统随机树的粘合。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.