Combinatorics and commutative algebra of algebraic varieties with group actions

代数几何是数学的一个中心领域。它不仅在纯数学和应用数学中很重要，而且在自然科学、工程等领域也很重要。经济学)。代数几何的核心是研究多变量多项式集合的公共零。该领域的一个普遍主题是将关于这些零集或代数变体的几何问题转化为等价代数问题。这类几何问题的例子包括“这个零集有多个分量吗？”和“有奇点吗？”******提出的研究将解决这些类型的几何问题的重要类代数变异与许多对称性。对于这样的代数变体，通常可以将某些代数问题进一步转化为组合问题(例如：计数问题，或者离散结构的问题)。使用和发展组合工具来研究代数几何问题是组合交换代数的主题，这是该建议适合的特殊数学领域。******在多位数学家过去成功的启发下，我将使用组合交换代数的方法来研究三种承载对称群的代数变量的代数几何性质：Dynkin颤栗的颤振轨迹，Schubert变数及相关变数，以及某些Hilbert格式。这些变量在纯数学中很重要，其中一些在其他领域也有应用。例如，舒伯特变体在代数几何和表示理论中都很重要，在计算机图形学和统计学中也有应用；在最近与Alex Fink和Seth sullivan的合作中，我们利用Schubert变量的性质来研究代数统计中的条件独立性。******结果将是数学家感兴趣的，并将有助于这些重要的代数品种的文献。在某些情况下，结果将连接看似不同的数学对象，或研究不同主题的研究人员群体。沿着我过去的联合工作的路线，连接A型颤振位点和舒伯特变种，并使用舒伯特变种来研究条件独立性)。特定方向的结果将产生对重要开放问题的新见解。最后，拟议的研究包含许多适合各级学生的项目，因此研究计划将通过培养高素质人才产生进一步的影响。

Algebraic geometry is a central area of mathematics. It has importance not just in pure and applied mathematics, but also in the natural sciences, engineering, and beyond (eg. economics). At its core, algebraic geometry is the study of common zeros of collections of polynomials in multiple variables. A general theme in the field is to translate geometric questions about these zero-sets, or algebraic varieties, into equivalent algebraic questions. Examples of such geometric questions include "does this zero-set have multiple components?" and "are there any singularities?". ******The proposed research will address these types of geometric questions for important classes of algebraic varieties with many symmetries. For such algebraic varieties, it is often possible to further translate certain algebraic questions into combinatorics (eg. counting problems, or problems about discrete structures). Using and developing combinatorial tools to study algebro-geometric problems is the subject of combinatorial commutative algebra, the particular area of mathematics in which the proposal fits. ******Motivated by past successes of multiple mathematicians, I will use methods from combinatorial commutative algebra to study algebro-geometric properties of three classes of algebraic varieties carrying groups of symmetries: quiver loci of Dynkin quivers, Schubert varieties and related varieties, and certain Hilbert schemes. These varieties are important in pure mathematics, and some have found applications in other fields. For example, Schubert varieties are significant in both algebraic geometry and representation theory, and have applications in computer graphics and statistics; in recent joint work with Alex Fink and Seth Sullivant, we used properties of Schubert varieties to study conditional independence in algebraic statistics.******Results will be of interest to mathematicians, and will contribute to the literature on these important algebraic varieties. In certain instances, results will connect seemingly different mathematical objects, or communities of researchers studying different topics (eg. along the lines of my past joint works connecting type A quiver loci and Schubert varieties, and using Schubert varieties to study conditional independence). Results in particular directions will yield new insights into important open problems. Finally, the proposed research contains many projects suitable for students at all levels, and so the research program will have further impact through the training of highly qualified personnel.