CAREER: Learning, testing, and hardness via extremal geometric problems

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。如果P不同于NP，就会有许多重要的计算问题无法有效解决。对于应用程序更重要的是（因为在实践中通常不需要精确的解决方案），甚至在计算上也很难近似地解决其中一些问题。研究这一主题的领域，被称为“近似硬度”，在过去的二十年里取得了突飞猛进的进展。该领域的开创性成就之一是在计算复杂性与几何和概率论中的等周型问题之间建立了深刻的联系。平面的等周问题——人们已经知道并研究了2000多年——问的是给定区域的哪种形状的周长最小（答案是：圆）。如果对这个问题的某些概率的、高维的变体有了更好的理解，它将在近似的硬度上解决几个开放的问题。更好地理解有效近似计算的极限将反过来导致更好的算法来解决现实世界的计算问题。这个项目是关于加强硬度的近似，几何和概率之间的联系。通过解决几何和概率中的新的最优划分问题，研究者将开发算法并证明新的算法硬度结果。这些划分问题的困难之一是存在组合的许多鞍点或局部最小值，但研究者最近（与E. Milman）对高斯双泡猜想的解决包括一种新的方法来绕过这个困难。这些最优划分问题的算法结果包括(i)改进了测试和学习几何概念类的边界；（ii）改进的非交互式相关蒸馏算法（应用于随机信标和信息协调的密码学中的一个问题）；（iii）在经典和量子通信复杂性之间有更强的分离。该奖项将允许研究生和本科生参与相关研究项目，它将资助用于数值计算的开源软件的开发，并将支持K-12学生的推广活动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).If P differs from NP, there are many important computational problems that cannot be solved efficiently. Even more importantly for applications (because in practice exact solutions are often not needed), it is 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. One of the seminal achievements of the field was the forging of a deep connection between computational complexity and isoperimetric-type problems in geometry and probability. The isoperimetric problem in the plane -- which has been known and studied for more than 2 millenia -- asks which shape of a given area has a minimal perimeter (the answer: a circle). If there were a better understanding of certain probabilistic, high-dimensional variants of this problem, it would close several open problems in hardness of approximation. A better understanding of the limits of efficient approximate computation will in turn lead to better algorithms for real-world computational problems.This project is about strengthening the link between hardness of approximation, geometry and probability. By solving new optimal partitioning problems in geometry and probability, the investigator will develop algorithms and prove new algorithmic hardness results. 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. Algorithmic consequences of these optimal partitioning problems include (i) improved bounds for testing and learning geometric concept classes; (ii) improved algorithms for non-interactive correlation distillation (a problem in cryptography with applications to random beacons and information reconciliation); and (iii) a stronger separation between classical and quantum communication complexity. This award will allow graduate and undergraduate students to participate in related research projects, it will fund 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.