CAREER: Algorithms and Software for Computational Algebraic Geometry

点击翻译按钮获取中文摘要

In this project the PI will develop symbolic, numerical, and hybrid methods in computational algebraic geometry and implement the resulting algorithms. He expects to advance both heuristic and certified methods in numerical algebraic geometry aimed at solving systems of polynomial equations and extracting exact geometric invariants and combinatorial information from algebraic varieties via approximate numerical methods. The proposed research includes certification of results obtained with numerical homotopy continuation, applications to tropical geometry and Schubert calculus, as well as the development of the numerical algebraic geometry engine of Macaulay2. The proposed multi-level educational program includes a new undergraduate course in algebraic computation, a graduate topics course, and a graduate summer program. Polynomial systems are ubiquitous in mathematical models in science and engineering. As engineering community opens up to algebraic methods, applied algebraic geometry is bound to make impact on a multitude of areas. The software produced in this research project will help scientists and engineers confronted with problems algebraic in nature. Certification in homotopy tracking algorithms will bring absolute assurance of the correctness of computation valuable to practitioners and provide a status of proof to the results obtained with homotopy methods important to theoreticians. The educational part of the project will promote algebraic computation in Georgia Tech within and outside the School of Mathematics and is perfectly suited for a large engineering school with a diverse student body.