CAREER: Complexity, Reality, and Rationality in Large Nonlinear Equation Solving

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

In this Career project, the investigator studies problems that involve an interplay between algebraic geometry, number theory, and complexity theory. Specific projects include: (1) A new generation of fast randomized algorithms for real root counting, allowing analytic as well as algebraic equations, (2) Further explorations into algorithmic fewnomial theory over the real and p-adic numbers, (3) A deeper analysis of numerical conditioning of random sparse polynomial systems, (4) Sharpened quantitative results for rational solutions of straight-line programs, (5) Further exploration of number-theoretic approaches to the P vs. NP question. Solving equations is ubiquitous in applications, cutting across many areas of engineering and science: A brief list includes drug design, faster and more reliable methods for radar imaging and geometric modelling, and more reliable and efficient approaches to robotics and autonomous vehicles. The investigator develops methods for problems related to the solution of polynomial equations, with an eye toward these applications. He also actively recruits students from schools in poorer areas of Texas (with large African-American and Hispanic populations) to help disadvantaged students and channel more bright young people into the computational sciences. He also continues work on related software, so that the broader public can benefit from the discoveries of this project.