Algebraic and Geometric Combinatorics

这项工作针对代数和几何组合学中的几个问题。这些问题在纯数学和应用数学和理论物理的许多领域都有应用，如凸几何、拟阵理论、表示论、代数几何、交换代数、逆边界问题、全正性、簇代数、超平面排列理论、热带几何、支化聚合物、孤子理论、李理论、Schubert微积分、量子上同调、判别式等。许多问题涉及特殊类型的凸多面体的研究、它们的组合结构、赋值和三角剖分，以及相关组合对象的研究。组合学在科学中发挥着越来越重要的作用。该项目的重点是组合方法在代数和几何问题中的应用。其结果将对纯数学和应用数学和物理学的几个领域产生影响。这项工作将导致新的组合技术和结构的发展。这个项目中的一些主题适合本科生研究。这个项目将提高公众对组合学和数学的总体认识。

This work targets several problems in algebraic and geometric combinatorics. These problems have applications to many areas of pure and applied mathematics and theoretical physics, e.g., convex geometry, matroid theory, representation theory, algebraic geometry, commutative algebra, inverse boundary problems, total positivity, cluster algebras, theory of hyperplane arrangements, tropical geometry, branched polymers, theory of solitons, Lie theory, Schubert calculus, quantum cohomology, discriminants, and other areas. Many problems involve the study of special classes of convex polytopes, their combinatorial structure, valuations, and triangulations, and the study of related combinatorial objects. Combinatorics plays an increasingly important role in science. The project focuses on applications of combinatorial methods to algebraic and geometric problems. Its results will have impact in several areas of pure and applied mathematics and physics. This work will lead to development of new combinatorial techniques and constructions. Some topics from this project are suitable for undergraduate research. This project will increase public awareness of combinatorics and mathematics in general.