Estimates in computational complexity, fluid mechanics, additive combinatorics and analysis

可以粗略地描述为“硬分析”的方法可以应用于许多领域。这些方法涉及不等式，即使用估计而不是精确恒等式。为了得到这些，我们可以使用分而治之的方法，就像那些在谐波分析中特别出现的方法，比如卡尔德隆-齐格蒙引理和停止时间。人们可以使用变换方法，特别是傅里叶变换来确定频率在问题中的作用。通常最重要的是对问题进行建模，这在数学中意味着找到简单的模型问题，表现出原始问题的困难。在本提案中，我们计划使用这些硬分析方法来研究各个领域的一些感兴趣的问题。通过分析接近极值的二部图不存在完全二乘二子图的性质，寻找一类复杂性问题的下界。根据Beale、Kato和Majda的研究，我们计划在接近极值的两个维度研究流体的行为。我们计划研究算术组合学中与不包含长度为3的等差数列的集合的结构和大小有关的问题。该项目更广泛的影响将包括围绕所讨论的一些要素组织会议，并培训不同层次的学生，博士后、研究生和本科生，使他们能够参与和理解数学中的重要工作。PI是IPAM一个为期3个月的项目的组织者之一，该项目以离散极值组合中的代数方法的使用为中心，他在与Guth一起研究关节问题和Erdos离散距离问题时帮助开创了这一实践。这次会议将汇集数十名数学家：高级教员，初级教员，尤其是博士后和研究生，使他们能够富有成果地会面并讨论这个令人兴奋和不断发展的领域。这样的会议对于培养年轻数学家队伍是必不可少的。这笔经费将帮助PI培养研究生和博士后，在某些情况下提供一些支持。高级研究人员通过解决问题对初级研究人员进行有针对性的培训，这对于创造我们在纯数学领域拥有的充满活力的劳动力至关重要。他在加州理工学院开始了一份新工作，对此他非常兴奋。他将带来的一个广泛影响是教授一门名为数学1a的课程，这门课程是加州理工学院所有新生的必修课。它本质上是一种针对科学家和工程师的加速基本分析。这样的课程将由研究分析师教授，通过分析中的基本普遍思想对前沿问题产生影响，这将丰富课程，让学生对数学的真正含义有一个很好的感觉。

The methods of what can loosely be described as "hard analysis" can be applied in many areas. These methods involve inequalities, that is the use of estimates rather than exact identities. To obtain these, one might use divide and conquer methods like those which specifically occur in harmonic analysis, such as Calderon-Zygmund lemmas and stopping times. One might use transform methods, especially the Fourier transform to determine the role of frequency in the problem. Often most important, is the role of modeling the problem, which in mathematics means finding simple model problems exhibiting the difficulties of the original problem. In this proposal, we plan to use these methods of hard analysis to work on a number of problems of interest in various areas.We plan to look for lower bounds in a certain kind of complexity problem by analyzing the complexity of bipartite graphs which are close to extremal for the property of having no complete two-by-two subgraphs. We plan to study the behavior of fluids in two dimensions which are close to extremal for estimates coming from the work of Beale, Kato, and Majda. We plan to study problems in arithmetic combinatorics related to structure in and size of sets having no arithmetic progressions of length 3.The broader impact of the project will consist in organizing meetings around some of the elements discussed and in the training of students at various level, postdocs, graduate students, and undergraduates in ways that allow them to participate in and understand important work in mathematics. The PI is one of the organizers of a 3 month program at IPAM centered on uses of the algebraic method in discrete extremal combinatorics, a practice he helped pioneer in his work with Guth on the Joints problem and on the Erdos discrete distance problem. This meeting will bring together scores of mathematicians: senior faculty, junior faculty, but especially postdocs and graduate students, allowing them to fruitfully meet and discuss this exciting and growing field. Such meetings are essential in the development of a workforce of young mathematicians. This grant will help the PI in his training of graduate students and postdocs, in some cases providing a bit of support. Focused training of junior researchers by more senior one, which is carried on through the solution of problems, is crucial to the creation of the dynamic work force we have in the pure mathematics community. The PI is starting a new job at Caltech which he is very excited about. One broad impact that he will have is teaching a course called Math 1a, required of all Caltech freshmen. It is essentially an accelerated elementary analysis aimed at scientists and engineers. That such a course will be taught by a research analyst, making an impact on cutting edge problems through essentially universal ideas in analysis, will enrich the course, giving the students a good feel of what mathematics is really about.