Employing Symmetry in Commutative Algebra

该项目主要涉及交换代数领域，并关注多项式方程组的研究：这些方程应被认为是某些模型中参数集之间的代数关系。通常，自然现象的数学模型配备了大量的对称性，当前项目的主要目标是开发使用对称性的技术，以揭示模型的定性和定量特征。这些问题和方法受到经典代数几何的强烈影响，它从几何的角度分析模型，通过表示论，这是一个代数的对称性研究，并通过D-模理论，这是一个代数的角度对无穷小的研究模型。计算机实验对这个项目的成功很重要。研究者将与学生和合作者一起开发和传播必要的计算机代数软件，并将提供普遍感兴趣的实验结果。本研究项目通过分析对称性对不变量结构的影响，研究与具有大量对称性的代数簇相关的基本不变量的一些开放性问题。主要重点将放在syzygies和当地上同调组的研究，以期了解进一步的不变量，如正则性和投影维数，上同调维数和算术秩，或Lyubeznik数和Bernstein-Sato多项式。该项目将结合联合收割机的方法，从交换代数和经典代数几何，以及技术，从表示论和理论的D-模。例如，研究者将探索与二元形式空间的自然分层相关的局部上同调群的D-模结构。另一方面，研究者将研究矩阵空间上多项式函数环中理想的syzygies之间的联系，以及一般线性李超代数的表示。

The project pertains primarily to the field of commutative algebra and is concerned with the study of systems of polynomial equations: these equations should be thought of as the algebraic relationships between the set of parameters in some model. Oftentimes mathematical models of natural phenomena come equipped with a large group of symmetries, and the main goal of the current project is to develop techniques that employ the symmetry in order to reveal both qualitative and quantitative features of the models. The questions and the methods are strongly influenced by classical algebraic geometry, which analyzes the models from a geometric perspective, by representation theory, which is an algebraic study of symmetry, and by D-module theory, which is an algebraic perspective on the infinitesimal study of the models. Computer experimentation is important for the success of the project. The investigator, together with students and collaborators, will develop and disseminate the necessary computer algebra software and will make available the results of experimentation that are of general interest.This research project investigates a number of open questions concerning fundamental invariants associated to algebraic varieties endowed with a large group of symmetries, by analyzing the implications of the symmetries to the structure of the invariants. The main emphasis will be on the study of syzygies and local cohomology groups, with a view towards understanding further invariants such as regularity and projective dimension, cohomological dimension and arithmetic rank, or Lyubeznik numbers and Bernstein-Sato polynomials. The project will combine methods from commutative algebra and classical algebraic geometry, as well as techniques from representation theory and the theory of D-modules. For instance, the investigator will explore the D-module structure of the local cohomology groups associated to natural stratifications of the spaces of binary forms. On a separate note, the investigator will research the connections between syzygies of ideals in rings of polynomial functions on spaces of matrices, and representations of general linear Lie superalgebras.