Numerical real algebraic geometry

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。数值方法是代数几何计算的未来。其原因是，计算能力的增加将是由于大规模并行化和符号算法似乎不并行化，而数值算法很容易并行化。该项目旨在帮助建立这个数字未来的基础设施。 它将通过两个主要的研究项目，并通过学生和博士后研究员的培训来做到这一点。 一个项目是开发和实现一种全新的数值延拓算法，该算法只计算多项式方程组的真实的解， 在 对比 到同伦 延续， 其必然计算所有解，包括真实的和复数。 二是用数值方法研究重要几何问题的微妙几何不变量，即舒伯特问题的伽罗瓦群。 第一个将扩展数值代数几何的工具箱，而第二个将展示其纯数学研究的潜力。 该项目的主要目标是在这一领域培训一名或多名学生，并培训和专业发展博士后研究人员。 两 项目是多年期的 需要软件开发的任务，将涉及团队为基础的研究和合作者在科罗拉多州和格鲁吉亚技术，并将导致出版物，软件包，博士学位。代数几何涉及关于多项式方程组的解的理论问题，但它在应用中具有巨大的潜力，特别是通过实现的算法工具。 数值代数几何是一个应用代数几何中的数值方法并由数学应用所驱动的新领域。 该项目将通过开发新的数值工具来研究真实的解，将其应用于纯数学研究以及学生和博士后研究员的培训来进一步取得进展。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Numerical methods are the future of computation in algebraic geometry. The reason for this is that increases in computing power will be due to massive parallelization and symbolic algorithms do not appear to be parallelizable while numerical algorithms are easily parallelized. This project aims to help build the infrastructure for this numerical future. It will do this through two main research programs and through the training of students and a postdoctoral fellow. One project is to develop and implement a radically new numerical continuation algorithm that computes only the real solutions to a system of polynomial equations, in contrast to homotopy continuation, which necessarily computes all solutions, both real and complex. The other is to use numerical methods to study subtle geometric invariants of important geometric problems, namely the Galois groups of Schubert problems. The first will extend the toolbox of numerical algebraic geometry, while the second will showcase its potential for pure mathematical research. A primary goal of this project is the training of one or more students in this area and the training and professional development of a postdoctoral researcher. Both projects are multi-year tasks requiring software development that will involve team-based research and collaborators at Colorado State and Georgia Tech and will result in publications, software packages, and Ph.D. theses.Algebraic geometry is concerned with theoretical questions about solutions to systems of polynomial equations, but it has great potential in applications, in particular through implemented algorithmic tools. Numerical algebraic geometry is a new field that uses numerical methods in algebraic geometry and has been driven by applications of mathematics. This project will further its progress by developing new numerical tools for studying real solutions, applying it to pure mathematical research, and the training of students and postdoctoral fellows.