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

在这个职业项目中，研究人员研究涉及代数几何、数论和复杂性理论之间相互作用的问题。具体项目包括：(1)允许解析方程和代数方程的新一代快速随机化算法，(2)进一步探索实数和p-进数上的算法几项式理论，(3)对随机稀疏多项式系统的数值条件进行更深入的分析，(4)加强直线规划有理解的定量结果，(5)进一步探索P与NP问题的数论方法。求解方程在应用中无处不在，涉及工程和科学的许多领域：一个简短的列表包括药物设计，更快和更可靠的雷达成像和几何建模方法，以及更可靠和高效的机器人和自动驾驶车辆方法。研究人员开发了与多项式方程的解有关的问题的方法，着眼于这些应用。他还积极地从德克萨斯州贫困地区(有大量非裔美国人和西班牙裔人口)的学校招收学生，以帮助弱势学生，并引导更多聪明的年轻人进入计算科学。他还继续从事相关软件的工作，以便更广泛的公众可以从这个项目的发现中受益。

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.