Asymptotic Commutative Algebra and Multigraded Syzygies

这项研究项目涉及交换代数，它为包括代数几何和代数数论在内的广泛数学奠定了基础。交换代数在科学和工程的许多领域都有应用，包括计算机科学、密码学、编码理论、机器人学、模式识别和理论物理。这个项目的部分工作旨在将Kakeya针问题与交换代数的现代思想联系起来。Kakeya针问题是关于绕一整圈旋转一根针所需空间量的经典分析问题。该项目的另一部分将使用交换代数来设计新的算法，以执行关于一类具有非凡对称性的形状的几何计算，以及其他应用。这项研究将在交换代数和其他领域之间建立新的前沿。第一个项目将交换代数与调和分析联系起来。Kakeya猜想是调和分析中的一个中心问题，它催生了有限域上的平行文献。该项目的目标是将这种并行扩展到p进整数，产生与原始分析问题更紧密的联系。第二个项目是为新的几何环境开发同调代数方法。对于嵌入在射影空间以外的其他东西中的种类，自由分辨率通常不能提供与几何的尖锐联系；这个项目将开发更适合于环面几何的同调机械。这个多方面的项目提供了一系列潜在的应用：环簇上向量丛的分裂定理，Hurwitz空间的合理性证明，以及新的层上同调算法。

This research project concerns commutative algebra, which provides the foundation for a broad range of mathematics, including algebraic geometry and algebraic number theory. Commutative algebra finds application in many fields of science and engineering, including computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. Part of the work in this project aims to connect the Kakeya needle problem, a classical analysis problem about the amount of space needed to turn a needle in a full circle, with modern ideas from commutative algebra. Another part of the project will use commutative algebra to design new algorithms for performing geometric computations about a class of shapes with extraordinary symmetries, among other applications. This research will build new frontiers between commutative algebra and other fields. The first project connects commutative algebra with harmonic analysis. The Kakeya conjecture is a central problem in harmonic analysis that has spawned a parallel literature over finite fields. The project aims to expand this parallel to the p-adic integers, yielding closer connections with the original analytic questions. The second project develops homological algebra methods for new geometric settings. For a variety embedded in something other than projective space, free resolutions often fail to provide sharp connections with geometry; this project will develop homological machinery better suited to toric geometry. This multifaceted project offers an array of potential applications: splitting theorems for vector bundles on toric varieties, rationality proofs for Hurwitz spaces, and new sheaf cohomology algorithms.