Homological Explorations and Symmetry

该奖项支持的研究是关于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模型的几何形状。自然现象通常具有丰富的对称性，这对于识别和将其纳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，我们应该考虑用线性方程定义的线和平面，它们表现出平移对称性，或者用二次方程描述的圆和球，它们表现出旋转对称性。对于更复杂的系统，单靠方程的度数是无法捕捉到几何形状或对称性的。这个项目的目标是系统地分析（更高）定义复杂系统的方程之间的关系，这些方程足够健壮，可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的曲线和构型，多项式及其分解模式，或矩阵空间和高维张量。这些空间不仅对数学的基础研究很重要，而且在整个科学领域的应用中也发挥着核心作用。本研究适合学生参与研究，也适合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。一个重要的重点将放在具体问题的研究上，这些问题表现出丰富的对称性，并触及数学中许多相关领域。PI将继续推进Koszul模的理论和应用，探索它们在群论和曲线的协同、超平面排列或代数变体的同调理论中的应用。PI将研究同调不变量，如合子和Ext模块，正则性和射影维数，局部上同调和吕别兹尼克数，在检查中心对象的背景下，如行列式和单项式理想，Segre和Veronese变体，二进制形式或具有有限多轨道的表示。所采用的方法主要是在交换代数和代数几何领域，但也结合了组合学、d模、颤振和表示理论的工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.