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、工程学和其他领域也很重要。经济学）。代数几何的核心是研究多元多项式集合的公共零点。该领域的一个普遍主题是将关于这些零集或代数簇的几何问题转化为等价的代数问题。这类几何问题的例子包括“这个零集有多个分量吗？“和“是否存在奇点？". ** 拟议的研究将解决这些类型的几何问题的重要类别的代数簇与许多对称性。对于这样的代数簇，通常可以进一步将某些代数问题转化为组合数学（例如，代数代数问题）。计数问题或关于离散结构的问题）。使用和开发组合工具来研究代数几何问题是组合交换代数的主题，这是该建议适合的数学特定领域。* 受多位数学家过去成功的启发，我将使用组合交换代数的方法来研究三类携带对称群的代数簇的代数几何性质：Dynkin箭图的轨迹，Schubert簇和相关簇，以及某些Hilbert方案。这些变量在纯数学中很重要，有些在其他领域也有应用。例如，舒伯特簇在代数几何和表示论中都很重要，并且在计算机图形学和统计学中有应用;在最近与Alex Fink和Seth Sullivant的联合工作中，我们使用舒伯特簇的性质来研究代数统计中的条件独立性。结果将感兴趣的数学家，并将有助于文献上这些重要的代数品种。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社区（例如。沿着我过去的联合作品的路线，连接A型双位点和舒伯特品种，并使用舒伯特品种研究条件独立性）。特定方向的结果将为重要的开放问题提供新的见解。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.