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

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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.