Laplacian Growth, stochastisity, and selection

提出了不稳定界面生长及其相关问题基础领域的理论研究和培训计划，这是当代研究的一个快速发展的领域。如果一个光滑的初始界面最终发展成分形结构，那么这个界面的生长就是不稳定的。在这个框架下，人们研究重要而独特的现象，如细菌菌落和癌细胞的生长、化学反应的动力学、结晶前沿的传播、量子霍尔液滴的动力学、一维费米子的输移、扩散有限聚集（DLA）、流体动力学流的表面不稳定性、可积模型、二维量子引力和矩阵模型。Laplacian Growth （LG）是这种不稳定增长的最简单的非平凡模型。在LG中，界面的速度与某个标量场的梯度成正比，并附加了不可压缩条件。这个标量场可以表示非常不同的物理量，如静电场、食物浓度或压力场。这是研究不稳定生长的一般现象的一个极好的背景。拟议的项目超越了目前的LG形式主义，考虑到使用对称性和引入随机噪声来解决问题的可能性，使其更加现实和相关。它将允许人们：i)将拉普拉斯生长的任何几何中的选择问题视为对随机噪声的稳定性问题；Ii)为与实验更紧密相关的增长引入新的形式；Iii)探讨了寻找界面（多重）分形性质的问题；iv)将拉普拉斯增长表述为哈密顿问题。更广泛的影响：1；LG和DLA是研究生和本科生学习非线性动力学、可积模型和其他当代数学和物理领域的良好研究领域。它将使学生看到非线性耦合的影响，看到由于随机性而产生的新的基本动力学性质的出现。该学生将获得跨学科研究的经验。2. PI和学生将在同行评议的期刊上发表论文，并在会议上发表演讲。此外，PI将利用一切机会在媒体上露面和公开演讲，向更广泛的公众传播结果。3. 建议的研究将有助于更好地理解上述所有应用中不稳定接口增长的动态。这些领域对技术、医学和环境研究具有重要意义。

A theoretical research and training program is proposed in the fundamental area of growth of unstable interfaces and related problems, a fast developing area in contemporary research. The growth of an interface is said to be unstable if a smooth initial interface eventually develops a fractal structure. Under this umbrella one studies important and distinct phenomena such as growth of bacterial colonies and cancer cells, dynamics of chemical reaction, propagation of crystallization fronts, dynamics of the Quantum Hall droplets, transport of 1D fermions, Diffusion Limited Aggregation (DLA), surface instabilities in hydrodynamical flows, Integrable Models, and 2D Quantum Gravity and Matrix Models. Laplacian Growth (LG) is the simplest nontrivial model of such unstable growth. In LG the velocity of the interface is proportional to the gradient of some scalar field with an additional condition of incompressibility. This scalar field can represent very different physical quantities such as electrostatic field, food concentration, or a pressure field. It is an excellent context in which to study the general phenomena of unstable growth. The proposed project looks beyond current LG formalism to the possibility of using symmetries and the introduction of stochastic noise to the problem, making it more realistic and relevant. It will allow one to: i) consider the selection problem in any geometry of Laplacian Growth as a problem of stability against stochastic noise; ii) introduce new formalisms for the growth more closely connected to experiment; iii) approach the problem of finding the (multi)fractal properties of the interface; iv) formulate the Laplacian Growth as a Hamiltonian problem.Broader impact: 1. LG and DLA are good research area for graduate and undergraduate students to be introduced into nonlinear dynamics, Integrable Models, and other contemporary areas of mathematics and physics. It will allow a student to see the effects of nonlinear couplings, see the emergence of new fundamental dynamical properties which arise due to stochasticity. The student will gain an experience in interdisciplinary research. 2. The PI and the students will publish in peer reviewed journals and give presentations at conferences. In addition the PI will use every opportunity for media appearances and public lectures to communicate the results to a broader public. 3. The proposed research will contribute to the development of a better understanding of the dynamics of growth of unstable interfaces in all the applications mentioned above. These areas are of the great importance for the technology, medicine, and environmental studies.