US-China Collaboration: Problems in Computational Algebraic Geometry

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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.