Isoperimetric Clusters and Related Extremal Problems with Applications in Probability

等周簇和相关极值问题及其在概率中的应用

• 批准号: 2204449
• 负责人: Joseph Neeman
• 金额: $ 6.24万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204449&HistoricalAwards=false
• 关键词: Isoperimetric; Clusters; Related; Extremal; Problems

The isoperimetric problem in the plane -- which has been known and studied for more than 2 millennia -- asks which shape of a given area has a minimal perimeter (the answer: a circle). The last hundred years have seen significant advances and generalizations in two directions: on one hand, we now how how to study related problems on other spaces including on the sphere, in hyperbolic space, and in Gaussian space. On the other hand, in some situations we can describe what happens when we optimize multiple sets at the same time. In particular, we can describe what happens to soap bubbles when they touch, which also gives insight into the structure of foams. Surprisingly, these geometric problems have a close link to computational complexity: is it widely believed that many important computational problems cannot be solved efficiently. More importantly for applications (because in practice we don't often need exact solutions), it's computationally hard even to approximately solve some of these problems. The field that studies this topic, known as "hardness of approximation," has progressed in leaps and bounds over the last two decades, and one of its seminal achievements was the forging of a deep connection between computational complexity and isoperimetric-type problems in geometry and probability. If we had a better understanding of certain probabilistic, high-dimensional, multi-part isoperimetric problems, it would close several open problems in hardness of approximation. The project will also develop software for numerical computation and support the advising and mentoring of students.This project is about introducing and exploiting new techniques for multi-part isoperimetric problems, with an emphasis on both problems that are natural in geometry (such as the double-bubble conjecture on the sphere) and problems coming from computer science. One of the difficulties with these partitioning problems is the presence of combinatorially many saddle points or local minima, but the investigator's recent resolution (with E. Milman) of the Gaussian double-bubble conjecture included a new method to circumvent this difficulty; the PI will build on this success by extending the method to related settings. This project will allow graduate and undergraduate students to participate in related research projects, it will aid the development of open-source software for numerical computation, and it will support outreach activities for K-12 students.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.

飞机中的等等问题（已知并研究了2千年以上）询问给定区域的哪个形状的周长最小（答案：一个圆圈）。在过去的百年中，两个方向都取得了重大进步和概括：一方面，我们现在如何在包括球体，双曲线空间和高斯空间在内的其他空间上研究相关问题。另一方面，在某些情况下，我们可以描述当我们同时优化多个集合时会发生什么。特别是，我们可以描述肥皂泡接触时会发生什么，这也可以深入了解泡沫的结构。令人惊讶的是，这些几何问题与计算复杂性有着密切的联系：是否广泛认为无法有效地解决许多重要的计算问题。更重要的是，对于应用程序（实际上我们通常不需要确切的解决方案），即使大约解决了其中一些问题，在计算上也很难。在过去的二十年中，研究该主题的领域（称为“近似硬度”）在突飞猛进中取得了进步，其开创性成就之一是在几何学和概率和概率中的计算复杂性与等速度型问题之间建立了深厚的联系。如果我们对某些概率，高维，多部分的等速度问题有了更好的了解，它将解决近似硬度的几个开放问题。 该项目还将开发用于数值计算的软件，并支持学生的建议和指导。该项目旨在引入和利用新技术，以解决多部分的等速度问题，重点是几何学中的两个问题（例如领域的双重掩盖），以及来自计算机科学的问题。这些分区问题的困难之一是组合上存在许多马鞍点或局部最小值，但是调查员最近对高斯双重气泡的猜想（与E. Milman）的解决方案包括一种新方法来解决这一困难。 PI将通过将方法扩展到相关设置，以此为基础。该项目将允许研究生和本科生参与相关研究项目，它将有助于开发用于数值计算的开源软件，并将支持针对K-12学生的外展活动。该奖项反映了NSF的法定任务，并认为通过使用该基金会的智力和更广泛影响的评估来审查Criteria，并被认为是值得通过评估的支持。