Combinatorial and Algebraic Aspects of Varieties

代数组合学的核心是这样一种哲学，即通过识别关键的组合结构，可以使数学的各个方面更加精确，更加具体，并且在计算上更加可行。 这个建议解决了几个具体的问题，跨越了广泛的数学，包括拓扑，代数几何，表示论，统计，计算机科学和组合学。 中心主题是促进这些领域的计算和理解。 这四个研究领域包括组合结构、k-Schur函数、支化聚合物和用于分析有序数据的组合/统计算法。所提出的工作将在纯数学、理论物理、计算机科学和统计学的几个领域产生广泛的影响。 所有拟议的工作将有一个计算的重点，这将进一步发展算法和证明技术。 所有拟议的工作都将通过对本科生、研究生和从事研究的博士后的教学和指导，产生对人类的影响。 在教育方面，PI发起了一个服务学习课程，通过数学解决我们社区的非营利组织和小企业的问题。

At the heart of algebraic combinatorics is the philosophy that every aspect of mathematics can be made more precise, more concrete and more computationally feasible by identifying key combinatorial structures. This proposal addresses several specific problems that span a wide range of mathematics including topology, algebraic geometry, representation theory, statistics, computer science and combinatorics. The central theme is to facilitate computation and understanding in these areas. The four areas of research include varieties with combinatorial structures, k-Schur functions, branched polymers, and combinatorial/statistical algorithms for analyzing ordered data.The work proposed will have a broad impact in several areas of pure math, theoretical physics, computer science and statistics. All of the proposed work will have a computational focus which will further develop algorithms and proof techniques. All of the proposed work will have a human impact component through teaching and mentoring of undergraduates, graduate students and postdocs doing research. In terms of education, the PI has initiated a service learning course to address problems in non-profit organizations and small businesses in our community through mathematics.