Free Boundaries, Level Surfaces, and Stochastic Growth

这个项目涉及自由边界表面，界面，最大限度地减少某种能源或成本。两个例子是最小面积的表面将固体分成两半等体积和第一节点集，最低能量本征函数改变符号的表面。该项目同时考虑了从一个阶段或材料到另一个阶段或材料的突然过渡以及更渐进的过渡。一个主要的目标是研究过渡的所有层次的相互作用，并建立过渡发生在最简单的方式。该提案提出了大规模的几何机制，导致优化水平表面类似于平行平面在较小的尺度。该项目的第二个方面是研究称为内部扩散限制聚集（内部DLA）的晶格生长模型的波动。在极限情况下，随着晶格尺寸趋于零，这个过程接近连续自由边界演化，称为Hele-Shaw流。在第一个关键步骤中，首席研究员和他的合著者莱昂内尔莱文和斯科特谢菲尔德已经表明，波动与高斯自由场密切相关，高斯自由场是一种二维或多维的随机分布，类似于一维的布朗运动。在这个提议中处理的界面出现为尾流和射流的轮廓，作为油和水或冰和水之间的边界面，以及物理学中的其他相变。 在每一种情况下，最佳形状都由物理原理决定。此外，还有一些受经济原则支配的界面的例子，例如绝缘材料与外部空气相遇的表面，这些界面被优化以最小化绝缘材料的总成本和维持所需温度所需的能量。上面提到的概率格模型是由化学家Paul Meakin和John M. Deutch描述了工业化学过程中的分子运动，如电解抛光，蚀刻和腐蚀。在这个项目中研究的波动是化学家关注的问题，他们希望在分子尺度上估计金属表面的粗糙度。除了进行自己的研究外，主要研究员还组织和监督本科生和高中生的研究生导师参加两个麻省理工学院暑期课程，其中一个是国际公认的研究科学研究所（RSI）。

This project concerns free boundary surfaces, interfaces that minimize some kind of energy or cost. Two examples are the surface of least area dividing a solid into two halves of equal volume and the first nodal set, the surface where the lowest energy eigenfunction changes sign. The project considers at the same time abrupt transitions from one phase or material to another and more gradual transitions. A main goal is to study the interaction of all levels of the transition and to establish that the transition happens in the simplest possible way. The proposal suggests large-scale geometric mechanisms that cause optimizing level surfaces to resemble parallel planes at smaller scales. A second aspect of the project is to study the fluctuations of a lattice growth model known as internal diffusion-limited aggregation (internal DLA). In the limit, as the lattice size tends to zero, this process approaches a continuum free boundary evolution known as Hele-Shaw flow. In a first, key step, the principal investigator and his coauthors, Lionel Levine and Scott Sheffield, have shown that the fluctuations are closely related to the Gaussian Free Field, a random distribution in two or more dimensions that is analogous to Brownian motion in one dimension.The interfaces treated in this proposal arise as profiles of wakes and jets, as the boundary surface between oil and water or between ice and water, and in other phase transitions in physics. In each of those cases, the optimal shape is dictated by a physical principle. There are, moreover, examples of interfaces governed by economic principles, such as the surface where an insulating material meets the outside air, optimized to minimize the total cost of the insulating material and the energy needed to maintain a desired temperature. The probabilistic lattice model mentioned above was proposed in the 1980s by chemists Paul Meakin and John M. Deutch to describe the movement of molecules in industrial chemical processes such as electropolishing, etching, and corrosion. The fluctuations studied in this project are the very issue of concern to chemists, who want to estimate the roughness of metal surfaces at the molecular scale. In addition to conducting his own research, the principal investigator organizes and supervises graduate student mentors of undergraduates and high school students in two MIT summer programs, one of which is the internationally recognized Research Science Institute (RSI).