Isoperimetric Clusters and Related Extremal Problems with Applications in Probability

平面上的等周问题--人们已经知道并研究了2000多年--询问给定区域的哪个形状的周长最小(答案是：一个圆)。在过去的一百年里，我们在两个方向上取得了重大的进展和概括：一方面，我们现在如何研究其他空间上的相关问题，包括球面空间、双曲空间和高斯空间。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，这也让我们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：人们普遍认为，许多重要的计算问题无法有效地解决。对于应用程序来说，更重要的是(因为在实践中，我们通常不需要精确的解)，甚至在计算上也很难近似地解决其中的一些问题。在过去的二十年里，研究这一主题的领域--被称为“逼近的困难”--取得了突飞猛进的进展，其开创性的成就之一是在计算复杂性与几何和概率中的等周型问题之间建立了深刻的联系。如果我们更好地理解某些概率的、高维的、多部分等周问题，就会结束几个在逼近难度方面的公开问题。该项目还将开发数值计算软件，并支持对学生的建议和指导。该项目旨在介绍和开发多部分等周题的新技术，重点是几何中的自然问题(如球面上的双泡猜想)和来自计算机科学的问题。这些划分问题的困难之一是存在组合的许多鞍点或局部极小值，但研究人员最近与E.Milman一起对高斯双泡猜想的解决方案包括一种新的方法来规避这一困难；PI将通过将该方法扩展到相关设置来建立这一成功。这个项目将允许研究生和本科生参与相关的研究项目，它将帮助开发用于数值计算的开源软件，它将支持K-12学生的外展活动。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.