New directions arising from a special diffusion process on the integer lattice

所谓的阿贝尔沙堆是一个从简单的设定中产生惊人数学的显著例子。沙堆的规则很简单：给定无限棋盘（即整数格）上的“筹码”配置，一个人可以“推翻”任何至少有4个筹码的方块，通过从这个方块向它的四个相邻方块中的每个方块发送一个筹码。当从一个大的筹码堆栈开始，并继续，直到没有正方形有超过3筹码，结果是一个惊人的分形配置。研究这一过程，发现了看似不同的数学领域之间令人惊讶的联系，这个项目旨在利用这些新的联系，既为了更深入地了解沙堆过程，也丰富我们对其他领域的理解。这个数学领域与物理中的相变和共形场论也有联系。更具体地说，这个项目涉及Bak， Tang和Wiesenfeld所谓的Abelian沙堆模型。通过识别整数超调和函数、阿波罗圆填料和欧几里得平面上的某些规则瓦片之间的新联系，我们现在能够表征沙堆的尺度极限并分析其局部分形结构。这个项目有两个主要目标。第一个是扩展我们对沙堆的知识：例如，我们希望加强我们可以证明沙堆过程允许的收敛性，表征长期研究的沙堆Dirichlet问题实例的解，并将我们对沙堆过程的标度极限的知识扩展到正方形晶格的范围之外。另一方面，我们希望将沙堆分析的新观点应用到其他领域，从而有可能证实关于阿波罗圆填料的数论猜想，或者描述具有某些对称性质的平面瓷砖。

The so-called Abelian sandpile represents a striking example of surprising mathematics arising from a simple setting. The rules of the sandpile are simple: given a configuration of "chips" on an infinite chessboard (i.e., the integer lattice), one can "topple" any square with at least 4 chips, by sending one chip from this square to each of its four neighboring squares. When begun from a single large stack of chips and continued until no square has more than 3 chips, the result is a striking fractal configuration. Studying this process has lead to the discovery of surprising connections between seemingly disparate areas of mathematics, and this project aims to leverage these new connections both for the sake of a deeper understanding of the sandpile process, and to enrich our understandings of these other areas as well. This area of mathematics has connections with phase transitions and conformal field theory in physics as well.More specifically, this project concerns the so-called Abelian sandpile model of Bak, Tang, and Wiesenfeld. By identifying new connections between integer superharmonic functions, Apollonian circle packings, and certain regular tilings of the Euclidean plane, we are now able to characterize the scaling limit of the sandpile and analyze its local fractal structure. This project has two primary goals. The first is to extend our knowledge of the sandpile: for example, we would like to strengthen the kind of convergence we can prove the sandpile process admits, characterize solutions to long-studied instances of the Dirichlet problem for the sandpile, and extend our knowledge of the scaling limit of the sandpile process beyond the universe of the square lattice. On the other hand, we would like to bring new perspectives developed for our analysis of the sandpile to bear on other areas, with the possibility of confirming number-theoretic conjectures regarding Apollonian circle packings, or characterizing tilings of the plane with certain symmetry properties.