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.
