US-China Collaboration: Problems in Computational Algebraic Geometry

该项目促进了路易斯安那州立大学和中国三个研究所在计算代数几何领域的新合作。研究计划在三个不同的领域进行：算术代数几何问题。具体地说：(1)具有特殊雅可比自同态的亏格3曲线的模。(2)第(1)部分中曲线产生的伽罗瓦表示的模性问题；(3)L-函数的特殊值。利用Pommaret基等对合基，开发求解多项式系统的高效算法。将已有的解0维系统的工作推广到更高维的广义超几何方程。研究附着在有趣多面体上的GKZ超几何系统，例如那些来自根系统的超几何系统。这些产生了有趣的变种家族，如Calabi-Yau变种。所有这三个领域都与目前代数几何中正在进行的活跃研究有关，项目将涉及理论和计算工具。代数几何-关注多项式方程组的解。在纯数学和应用数学中，多项式是普遍存在的。它们是发生在几乎每一个科学领域的基本对象，它们的研究是当前数学研究的许多领域的中心。该项目的一部分是研究由多项式定义的空间。另一部分是研究多项式方程的有效解的算法。这个项目还探索了与数学的其他部分的联系，例如数论。这项建议还支持研究生和年轻的博士后研究员各访问一个月。本研究的研究对象是中国，尤其是北京和天津。国际数学家协会将与中国数学家合作，并教授相关主题的课程。这个项目允许两位年轻的美国数学家参与到中国令人兴奋的数学世界中来。建立和扩大这种国际合作不仅可以对研究项目的狭隘目标产生重大积极影响，而且可以对美国研究和教学的长期发展产生重大积极影响。

The proposed project catalyzes a new collaboration between the Louisiana State University and three institutes in China, in the area of computational algebraic geometry.Research is planned in 3 distinct domains:Problems in Arithmetic Algebraic Geometry. Specifically: (1) Moduli of genus 3 curves with special endomorphisms of their Jacobians. (2) Modularity questions for the Galois representations that arise from curves in part (1); (3) Special values of L-functions for Galois representations of the shape ρf ⊗ φ where f is a modular form and φ is a nonabelian Artin representation; (4) Dynamical systems arising from integer matrices.Polynomial systems. Develop efficient algorithms for solving systems of polynomials via in- volutive bases such as Pommaret bases. Extend work already done for solving 0-dimensional systems to higher dimensions.Generalized hypergeometric equations. Study the GKZ hypergeometric systems attached to interesting polytopes, for instance those coming from root systems. These give rise to interesting families of varieties, such as Calabi-Yau varieties.All three areas are related to currently active research going on in algebraic geometry, and projects will involve both theoretical and computational tools.Algebraic Geometry - is concerned with solutions to systems of polynomial equations. Polynomials are ubiquitous in pure and applied mathematics. They are fundamental objects occurring in practically every domain of science, and their study is central to many areas of current mathematical research. Part of the project is a study of spaces defined by polynomials. Another part is to study algorithms for the efficient solution to polynomial equations. This project also explores connections to other parts of mathematics, for instance number theory.This proposal also supports one month visits each for a graduate student and a young post- doctoral researcher. The locus of this research is China, especially Beijing and Tianjin. The PI will be collaborating with Chinese mathematicians, and also teaching courses on related topics. This project allows two young U.S. mathematicians to participate in the exciting mathematical world developing in China. Establishing and extending such international collaborations can have a major positive impact not only on the narrow goals of a research project, but also on the long-term development of research and teaching in the US.