Homological Explorations and Symmetry

该奖项支持的研究涉及多项式方程组的研究。这些方程可以看作是某个模型中物理量之间的依赖关系，而这些方程的解集描述了该模型的几何形状。自然现象往往带有丰富的对称性，这对于识别和融入建模过程非常重要。要想象方程、几何和对称是如何自然地结合在一起的，人们应该考虑使用线性方程定义的线和平面并显示平移对称，或者考虑由二次方程描述并显示旋转对称的圆和球体。对于更复杂的系统，仅靠方程的阶数不能反映几何或对称性。这个项目的目标是系统地分析定义精细系统的方程之间的(更高的)关系，这些系统足够健壮，足以准确地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、多项式及其因式分解模式、或矩阵和高维张量空间的曲线和构形。这样的空间不仅对数学基础研究很重要，而且在整个科学的应用中也发挥着核心作用。这项研究适合学生从事研究，也适合于计算机实验和软件开发。PI将研究同调交换代数和经典代数几何中的一些基本不变量。一个重要的重点将是研究具体的、表现出丰富对称性的、涉及数学中许多相关领域的问题。PI将继续推进Koszul模的理论和应用，探索它们在群论和曲线合集、超平面排列或代数簇的同调理论中的应用。PI将研究同调不变量，如合子和Ext模，正则性和射影维度，局部上同调和Lyubeznik数，在检查中心对象的背景下，如行列式和单项理想，Segre和Veronese簇，二元形式，或具有有限多个轨道的表示。所采用的方法主要属于交换代数和代数几何领域，但也融合了组合学、D-模、箭图和表示理论的工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.