Conformal Welding of Discs and Trees

分形在很大程度上表征为在不同尺度下看起来相同的结构，在自然界和科学中无处不在。它们是有趣的，因为它们的美学吸引力和事实，尽管它们的高度复杂性，但它们往往是非常简单的机制的结果。在数学方面，当重复应用一个简单的规则（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，非常相似的模式和形状都是随机出现的。首席研究员的研究主要涉及二维分形，具有某种形式的尺度和旋转不变性，无论是在确定性和随机设置。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础问题，并理解为什么确定性和随机模式如此相似。共形自相似二维集合通常可以通过共形焊接来描述。保形焊接是将两个黎曼曲面沿着其边界粘合形成一个新曲面的过程。它在Teichmueller理论中有重要作用，在几何函数理论中有独立的兴趣，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。他将把他的新的可焊性条件应用于SLE和LQG的设置，目的是通过粘合量子盘来解析构建SLE。他还将运用这种技术从树木的配对中获得共形球体，无论是在Julia集的共形配对的确定性设置中，还是在布朗映射的概率设置中，作为两个连续随机树的粘合。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.