NSF/CBMS Regional Conference in the Mathematical Sciences -- Solving Polynomial Equations

NSF/CBMS 数学科学区域会议——求解多项式方程

• 批准号: 0122220
• 负责人: Paulo Lima-Filho
• 金额: $ 2.75万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2002-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0122220&HistoricalAwards=false
• 关键词: NSF; CBMS; Regional; Conference; Mathematical

One of the most fundamental problems in mathematics is that of solving polynomial equations. Such systems are ubiquitous in applied mathematics, arising in robotics, coding theory, optimization, mathematical biology, computer vision, statistics, and numerous other areas. Of course, the study of polynomialsystems is a beautiful topic in its own right and is the subject of algebraic geometry, which istraditionally regarded as a deep and difficult subject in pure mathematics.In recent years, an explosive development in explicit algorithms and practical software for geometric calculations has revolutionized the area, making many formerly inaccessable problems tractable, and providing a fertile ground for experimentation and conjecture. These algorithms have also generated a surprising interest in algebraic geometry among scientists, engineers, and applied mathematicians. Thetools of the trade employed in this field span the spectrum of mathematics, ranging from numericalmethods and differential equations to algebraic geometry to combinatorics. The purpose of these lectures is toprovide an overview of recent results and the ``state of the art''; and to discuss the wealth of open questions raised by these results and suggest new directions ripe for exploration.

数学中最基本的问题之一是求解多项式方程。这样的系统在应用数学中无处不在，出现在机器人学、编码理论、优化、数学生物学、计算机视觉、统计学和许多其他领域。当然，多项式系统的研究本身就是一个美丽的话题，也是代数几何的主题，代数几何在理论上被认为是纯数学中一个深奥而困难的主题。近年来，几何计算的显式算法和实用软件的爆炸性发展彻底改变了这个领域，使许多以前无法解决的问题变得容易处理，并为实验和猜想提供了肥沃的土壤。这些算法也产生了令人惊讶的兴趣代数几何之间的科学家，工程师和应用数学家。在这一领域中使用的贸易工具跨越了数学的范围，从数值方法和微分方程到代数几何再到组合数学。这些讲座的目的是提供一个最近的结果和“最先进的状态”的概述，并讨论这些结果所提出的开放性问题的财富，并提出新的方向成熟的探索。