US-China Collaboration: Problems in Computational Algebraic Geometry

拟议的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：算术代数几何问题。具体来说：(1)具有雅可比矩阵特殊自同态的3属曲线的模量。(2)第(1)部分中曲线伽罗瓦表示的模块化问题；(3)形状&amp；#961的伽罗瓦表示的l函数的特殊值；f和# 8855;, # 966;其中f是模形式&amp；#966；是一个非abel Artin表示；(4)由整数矩阵产生的动力系统。多项式系统。开发有效的算法来解决系统的多项式通过在进化基，如波玛雷特基。将已经完成的求解0维系统的工作扩展到更高的维度。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，例如来自根系的多面体。这就产生了有趣的品种家族，比如卡拉比-丘品种。这三个领域都与当前代数几何领域的活跃研究有关，项目将涉及理论和计算工具。代数几何——研究多项式方程组的解。多项式在纯数学和应用数学中无处不在。它们是几乎每个科学领域都存在的基本对象，对它们的研究是当前许多数学研究领域的核心。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算法。这个项目还探索了与数学其他部分的联系，例如数论。该提案还支持一名研究生和一名年轻的博士后研究员分别进行为期一个月的访问。本研究的地点是中国，特别是北京和天津。PI将与中国数学家合作，并教授相关主题的课程。该项目为两位年轻的美国数学家提供了参与中国蓬勃发展的数学世界的机会。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的积极影响，而且可以对美国研究和教学的长期发展产生重大的积极影响。

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.