Free boundaries and extremal inequalities

自由边界是两种物质（如油和水）之间的界面。另一个例子是勾勒出船只尾迹的曲线。还有一个是聚变反应堆中等离子体和普通物质之间的界面。值得注意的是，描述这些物理现象的自由边界的相同数学可以用于设计最佳形状。例如，当人们想用绝缘材料包裹烤箱或管道时，有一种形状是最佳的，因为隔热成本和热损失成本的总和是最小的。此外，即使远离物理学，人们也可以根据最小化错误（即假阳性或假阴性识别或诊断）的规则，寻求将数据集划分为yes/no区域的最佳方法。这个项目的首要目标是减少寻找这些最佳形状问题的复杂性。我们的期望是，在许多情况下，最优分割线（自由边界）类似于一条适当比例的直线或平面。在这些情况下，人们可以自信地迅速找到接近最佳的形状。除了进行自己的研究外，PI已经并将担任麻省理工学院数十名本科生和高中生研究项目的指导老师。此外，他还在麻省理工学院的开放课件网站上发布了一门广泛观看的单变量微积分课程的视频和课堂笔记。他目前正在制作一门由麻省理工学院发布的在线课程。自由边界是材料之间的界面，在其中材料保留了一些能量。通常情况下，空间被划分为一些数量的水平集，如温度或压力。相比之下，由最小表面表示的界面存在于空的环境空间中。尽管这两种类型的接口之间存在差异，但它们之间存在着深刻的联系。这个项目的主要目标是表明，对于各种各样的问题，最小能量的接口和水平集尽可能简单。PI建议优化器的水平集类似于平行平面，因为这些平面是连接的，并且干净地分开。通常，来自更发达的最小曲面理论的方法指导了自由边界的研究，但在这里，来自自由边界理论的思想将指导最小曲面的研究。给出了一个证明凸对称域上最小能量诺伊曼特征函数的水平集的类似简单行为的途径。这将为劳赫长期存在的“热点”猜想提供一个重要的例子。第二个项目是在凸几何中著名的亚历山德罗夫-芬切尔不等式中确定相等的情况。PI将使用基于建立Brenier（最优运输）映射的新特性的几何方法。第三个项目旨在开发与led设计相关的特征函数和量子隧道定位的高度精确描述。

A free boundary is an interface between two materials like oil and water. Another example is the curve outlining the wake of a boat. Yet another is the interface between plasma and ordinary matter in a fusion reactor. Remarkably, the same mathematics of free boundaries that describes these physical phenomena can be used to design optimal shapes. For example, when one wants to enclose an oven or pipe with insulating material, there is a shape that is optimal in the sense that the sum of the cost of insulation and the cost due to heat loss is minimized. Moreover, even farther from physics, one can seek optimal ways to divide data sets into yes/no regions according to rules that minimize the errors, that is, false positive or false negative identifications or diagnoses. The overarching goal of this project is to reduce the complexity of the problem of searching for these optimal shapes. The expectation is that there are broad classes of situations in which the optimal divider (free boundary) resembles a straight line or plane at an appropriate scale. In those cases, one can be confident of finding a near optimal shape quickly. In addition to conducting his own research, the PI has served and will serve as faculty advisor for research projects by dozens of undergraduates and high school students in programs at MIT. Moreover, he has posted videos and lecture notes of a widely viewed single variable calculus course on MIT's Open Courseware site. He is currently working on an on-line course to be disseminated by MITx.Free boundaries arise as the interface between materials in which the materials retain some energy. Typically, space is divided into level sets of some quantity like temperature or pressure. In contrast, the interface represented by a minimal surface lives in an ambient space that is empty. Despite this difference between these two types of interfaces, there are profound connections between them. The main goal of this project is to show that interfaces and level sets of least energy for a wide variety of problems are as simple as possible. The PI proposes that the level sets of optimizers resemble parallel planes in that these surfaces are connected and cleanly separated. Usually, methods from the more developed theory of minimal surfaces have guided the study of free boundaries, but here ideas from the theory of free boundaries will guide the study of minimal surfaces. The proposal also gives a pathway to proving analogous simple behavior of level sets of the least energy Neumann eigenfunction for a convex symmetric domain. This would yield an important case of the longstanding ``hot spots'' conjecture of J. Rauch. A second project is to identify the cases of equality in the celebrated Alexandrov-Fenchel inequalities in convex geometry. The PI will use a geometric approach based on establishing new properties, of independent interest, of the Brenier (optimal transportation) mapping. A third project is aimed at developing a highly accurate description of localization of eigenfunctions and quantum tunneling, relevant to the design of LEDs.