Homological Explorations and Symmetry

同源探索和对称性

基本信息

批准号：
1901886
负责人：
Claudiu Raicu
金额：
$ 28.47万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901886&HistoricalAwards=false
关键词：
Homological Explorations Symmetry

项目摘要

The research supported under this award is concerned with the study of systems of polynomial equations. The equations can be thought of as dependence relations between physical quantities in some model, while the solution set of these equations describes the geometric shape of the model. Natural phenomena often come equipped with a rich symmetry, which is important to identify and incorporate into the modeling process. To imagine how equations, geometry and symmetry naturally fit together, one should think about lines and planes that are defined using linear equations and exhibit translational symmetries, or about circles and spheres that are described by quadratic equations and display rotational symmetries. For more complex systems, the degrees of the equations alone are not able to capture the geometry or the symmetries. The goal of this project is to systematically analyze (higher) relations between the equations defining elaborate systems, that are robust enough to accurately predict shapes and portray symmetries. The techniques developed by the PI will be used to study curves and configurations of linear spaces, polynomials and their factorization pattern, or spaces of matrices and higher dimensional tensors. Such spaces are not only important for the fundamental research in mathematics, but they also play a central role in applications throughout the sciences. The research is suitable for engaging students in research, as well as for computer experimentation and software development.The PI will investigate a number of fundamental invariants in homological commutative algebra and classical algebraic geometry. An important emphasis will be on the study of problems that are concrete, exhibit a rich symmetry, and that touch upon a number of related fields within mathematics. The PI will continue to advance the theory and applications of Koszul modules, exploring their uses to group theory and to syzygies of curves, to hyperplane arrangements or to the homological theory of algebraic varieties. The PI will study homological invariants such as syzygies and Ext modules, regularity and projective dimension, local cohomology and Lyubeznik numbers, in the context of examining central objects such as determinantal and monomial ideals, Segre and Veronese varieties, binary forms, or representations with finitely many orbits. The methods employed fall mainly in the realm of commutative algebra and algebraic geometry, but also incorporate tools from combinatorics, D-module, quiver and representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持的研究与多项式方程式系统的研究有关。可以将方程视为某些模型中物理量之间的依赖关系，而这些方程的解集则描述了模型的几何形状。自然现象通常配备了丰富的对称性，这对于识别和纳入建模过程很重要。为了想象方程，几何和对称性如何自然地融合在一起，应该考虑使用线性方程和展示翻译对称性定义的线和平面，或者围绕二次方程描述的圆和球体，并显示旋转对称性。对于更复杂的系统，仅方程的程度无法捕获几何或对称性。该项目的目的是系统地分析定义精美系统的方程式之间的关系，这些方程足以准确预测形状和描绘对称性。 PI开发的技术将用于研究线性空间，多项式及其分解模式的曲线和构型，或矩阵的空间和更高的维度张量。这样的空间不仅对数学的基本研究很重要，而且在整个科学的应用中也起着核心作用。这项研究适合吸引学生参与研究，以及计算机实验和软件开发。一个重要的重点是研究具体的问题，表现出丰富的对称性，并涉及数学中的许多相关领域。 PI将继续推进Koszul模块的理论和应用，探索它们在群体理论和曲线的共同体中的用途，超平面布置或代数品种的同源理论。 PI将在检查确定性和单一理想的中心物体（例如，Segre和Veronese品种，二进制形式，二进制形式，或具有一定程度的一定程度的orbits的代表）的背景下，研究诸如Syzygies和Ext模块，规律性和投影维度，局部同时性和Lyubeznik数字等同种异体不变式。所采用的方法主要属于交换代数和代数几何的领域，但还结合了Combinatorics，D-Module，Quiver和代表理论中的工具。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子功能和广泛的影响来评估Criteria的智力优点和广泛的CRITERIA。