CAREER: Algorithms and Software for Computational Algebraic Geometry

在这个项目中，PI将在计算代数几何中开发符号、数值和混合方法，并实现所产生的算法。他希望在数值代数几何中发展启发式和认证方法，目标是求解多项式方程组，并通过近似数值方法从代数变体中提取精确的几何不变量和组合信息。提出的研究包括验证数值同伦延拓的结果，应用于热带几何和Schubert微积分，以及开发Macaulay2的数值代数几何引擎。拟议的多层次教育计划包括一门新的代数计算本科课程、一门研究生专题课程和一项研究生暑期课程。在科学和工程的数学模型中，多项式系统是普遍存在的。随着工程界对代数方法的开放，应用代数几何必然会对许多领域产生影响。本研究项目制作的软件将帮助科学家和工程师解决自然界中的代数问题。同伦跟踪算法的验证将绝对保证计算的正确性，对实践者有价值，并为同伦方法得到的结果提供一种证明状态，对于理论界来说很重要。该项目的教育部分将在佐治亚理工学院数学学院内外推广代数计算，非常适合学生群体多样化的大型工程学校。

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.