Applications of Probabilistic Combinatorial Methods

University Of Memphis, Memphis TN

Investigators

Abstract

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.