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

职业：大型非线性方程求解的复杂性、现实性和合理性

• 批准号: 0349309
• 负责人: J Maurice Rojas
• 金额: $ 40万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0349309&HistoricalAwards=false
• 关键词: CAREER; Complexity; Reality; Rationality; Large

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.

在这个职业项目中，研究者研究涉及代数几何、数论和复杂性理论之间相互作用的问题。具体项目包括：(1)新一代的实数根计数快速随机算法，允许解析方程和代数方程；(2)对实数和P进数的算法多项式理论的进一步探索；(3)对随机稀疏多项式系统的数值条件的更深入分析；(4)对直线规划有理解的定量结果的强化；(5)对P与NP问题的数论方法的进一步探索。求解方程在应用中无处不在，跨越了许多工程和科学领域：一个简短的列表包括药物设计，更快更可靠的雷达成像和几何建模方法，以及更可靠更有效的机器人和自动驾驶汽车方法。研究者开发了与多项式方程解有关的问题的方法，着眼于这些应用。他还积极从德州贫困地区（非裔美国人和西班牙裔人口众多）的学校招收学生，帮助贫困学生，引导更多聪明的年轻人进入计算科学领域。他还继续开发相关软件，以便更广泛的公众可以从这个项目的发现中受益。