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

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In this Career project, the investigator studies problemsthat involve an interplay between algebraic geometry, numbertheory, 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, cuttingacross many areas of engineering and science: A brief listincludes drug design, faster and more reliable methods for radarimaging and geometric modelling, and more reliable and efficientapproaches to robotics and autonomous vehicles. The investigatordevelops methods for problems related to the solution ofpolynomial equations, with an eye toward these applications. Healso actively recruits students from schools in poorer areas ofTexas (with large African-American and Hispanic populations) tohelp disadvantaged students and channel more bright young peopleinto the computational sciences. He also continues work onrelated software, so that the broader public can benefit from thediscoveries of this project.