Applications of Probabilistic Combinatorial Methods

纯数学是基础研究——不能通过它的直接应用来判断它的好坏：历史告诉我们，重要的应用会在若干年后意外出现。按照伟大的大卫·希尔伯特的精神，我们的项目是由问题驱动的。正如他在一百多年前所说：“只要一门科学提供了大量的问题，它就有生命力。”[…研究者正是通过对问题的解决来检验他的钢铁的脾气；他找到了新的方法和新的观点，获得了更广阔、更自由的视野。[…一个数学问题应该是困难的，以便吸引我们，但又不能完全无法理解，以免它嘲笑我们的努力。在这个项目中，要考虑的大多数问题都是60多年前由历史上最伟大的问题提出者保罗·埃尔多斯提出的，这些问题与素数的分布和模算法有关。事实上，这个项目的能量和动力将由一个启动会议来建立，这个会议将汇集杰出的数学家，他们在他的一些最具挑战性的问题上做了动态和有用的工作。几十年来，主要问题似乎遥不可及，但几年前，霍夫和尼尔森证明了一些突破性的成果。在这个项目的范围内，在应用组合学和概率论的方法来解决几个遗留的主要问题时，可以推导出一些新的数学技术来解决这些问题，这些技术可能适用于许多其他问题，包括数论中的问题。该项目有望在该学科内外的许多方面产生影响。例如，整数的因式分解在密码学中具有重要的实际意义，而重建问题可能在生物学中使用（例如，DNA测序）。第二个最短路径问题与提出的精神分裂症诊断测试有关。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。

Pure mathematics is fundamental research -- it cannot be judged by its immediate use: history teaches us that important applications arise unexpectedly, years later. In the spirit of the great David Hilbert, our project is problem-driven. As he said over one hundred years ago, ''As long as a branch of science offers an abundance of problems, so long is it alive. [...] It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon. [...] A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts.'' In the case of this project, most of the problems to be considered were posed over sixty years ago by Paul Erdos, the greatest problem poser in history, and are related to the distribution of primes and modular arithmetic. In fact, the energy and momentum of this project will be established by a kickoff conference which will bring together outstanding mathematicians who have done dynamic and useful work on some of his most challenging problems. For many decades, the main problems seemed to be out of reach, but a few years ago Hough and Nielsen proved some breakthrough results. Within the scope of this project, in applying methods of combinatorics and probability theory to solve several major problems that have remained, some new mathematical techniques for their solutions could be derived that might then be applicable to many other problems, including those in number theory. This project has the promise to be impactful in a number of ways within and beyond the discipline. For example, the factorization of integers is of great practical importance in cryptography, and reconstruction problems are likely to be of use in biology (e.g., DNA sequencing). The second shortest path problem is related to a proposed diagnostic test for schizophrenia.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.