CAREER: Algorithms and Software for Computational Algebraic Geometry

在这个项目中，PI将开发计算代数几何中的符号、数值和混合方法，并实现结果算法。他期望在数值代数几何中推进启发式和证明方法，旨在解决多项式方程系统，并通过近似数值方法从代数变量中提取精确的几何不变量和组合信息。拟进行的研究包括数值同伦延拓结果的证明，在热带几何和舒伯特微积分中的应用，以及Macaulay2的数值代数几何引擎的开发。提议的多层次教育计划包括代数计算的新本科课程，研究生主题课程和研究生暑期课程。多项式系统在科学和工程数学模型中无处不在。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。在这个研究项目中产生的软件将帮助科学家和工程师面对本质上的代数问题。对同伦跟踪算法的证明将对计算的正确性带来绝对的保证，这对实践者来说是有价值的，同时也为同伦方法得到的结果提供了一种证明的状态，这对理论家来说是很重要的。该项目的教育部分将促进佐治亚理工学院数学系内外的代数计算，非常适合拥有多样化学生群体的大型工程学院。

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.