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)提供资金的。数值方法是代数几何计算的未来。这是因为计算能力的提高将是由于大规模并行化，而符号算法似乎不是可并行化的，而数值算法很容易并行化。这个项目旨在帮助为这个数字未来建立基础设施。它将通过两个主要的研究项目以及通过培训学生和一名博士后研究员来实现这一点。一个项目是开发和实现一种全新的数值连续算法，该算法只计算多项式方程组的实数解，而不是同伦连续，同伦连续必须计算所有的解，包括实数和复数。二是用数值方法研究重要几何问题的精细几何不变量，即Schubert问题的Galois群。第一个将扩展数值代数几何的工具箱，而第二个将展示其在纯数学研究方面的潜力。该项目的一个主要目标是培训该领域的一名或多名学生，并培训和发展一名博士后研究人员。这两个项目都是多年的任务，需要软件开发，将涉及科罗拉多州立大学和佐治亚理工学院的团队研究和合作者，并将产生出版物、软件包和博士学位。代数几何关注多项式方程系统的解的理论问题，但它在应用方面有巨大的潜力，特别是通过实现的算法工具。数值代数几何是在数学应用的推动下，将数值方法应用于代数几何的一个新领域。这个项目将通过开发研究真实解的新的数值工具，将其应用于纯数学研究，以及对学生和博士后研究员的培训来进一步推进其进展。

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.