Combinatorial Connections with Algebra, Geometry, Probability and Applications

组合学是数学的一个领域，它在计算机科学、生物学、物理学、化学和工业领域的应用是典型的。在我们的有生之年，基于组合学的优化算法已经彻底改变了商业决策。目前的研究正在揭示数学和科学各个领域之间的联系。这个项目特别关注组合学在代数、几何、概率论和计算机科学中的跨学科应用。组合连接一直是研究者先前工作的核心，并继续激发创新和合作。这个项目描述了三个主要的研究主题。第一部分是关于协不变代数及其表示理论的经典研究。问题是研究其分解为不可约对称群模的渐近性。攻击的方法包括组合学、代数和概率论中的工具。第二部分研究了n维空间中与0-1向量矩阵相关的对称函数族。这个方向的猜想和定理使用了代数几何的工具。这项研究与物理学和经济学有关。第三个主题是组合学、理论计算机科学和离散几何的混合，涉及到平面上圆在包裹条件下的位置。该领域的研究灵感来自于一般离散几何问题，即在图形和优化中出现的平面上寻找适当的区域多边形。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。

Combinatorics is an area of mathematics which is quintessential for applications in computer science, biology, physics, chemistry, and industry. Optimized algorithms based on combinatorics have revolutionized business decisions in our lifetime. Current research is uncovering connections to all areas of mathematics and science. This project focuses specifically on interdisciplinary applications of combinatorics in connection with problems in algebra,geometry, probability, and computer science. Combinatorial connections have been at the core of the investigator's prior work and continue to inspire innovation and collaboration. This project describes three main themes for research. The first relates to the classical study of the coinvariant algebra and its representation theory. The problem is to study the asymptotics of the underlying decomposition into irreducible symmetric group modules. The methods of attack include tools from combinatorics, algebra and probability theory. The second studies a newly proposed family of symmetric functions related to the matroid of 0-1 vectors in n-dimensional space. Conjectures and theorems in this direction use tools from algebraic geometry. This research has connections to physics and economics. The third topic, which is a mixture of combinatorics, theoretical computer science and discrete geometry, pertains to placements of circles in the plane with a wrapping condition. This area of research was inspired by the general discrete geometry problem of finding an appropriate polygonization of a region in the plane which appears in graphics and optimization.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.