Combinatorics and commutative algebra of algebraic varieties with group actions

具有群作用的代数簇的组合学和交换代数

• 批准号: RGPIN-2017-05732
• 负责人: Rajchgot, Jenna
• 金额: $ 1.53万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=662933
• 关键词: Combinatorics; commutative; algebra; algebraic; varieties

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.

代数几何形状是数学的中心区域。它不仅具有纯粹和应用数学的重要性，而且在自然科学，工程及其他（例如经济学）中也具有重要意义。代数的几何形状是对多个变量中多项式集合的共同零的研究。该领域的一个一般主题是将有关这些零元或代数品种的几何问题转化为同等的代数问题。此类几何问题的示例包括“此零集有多个组件？”和“有奇点吗？”。 ******拟议的研究将解决与许多对称性的重要类别的代数品种类型的几何问题。对于此类代数品种，通常可以将某些代数问题进一步转化为组合学（例如计数问题或离散结构问题）。使用和开发组合工具研究代数几何问题是组合交换代数的主题，该代数是该提案适合的数学领域。 ******由多位数学家过去的成功动机，我将使用组合交换代数的方法来研究三种代数品种的代数几何特性，这些代数品种携带的对称性组：Quiver of Symorties：Quiver of dynkin Quiver of Dynkin Quiver of Schubert Quivers，Schubert Varieties and Resiendles和某些Hilbert Schemes。这些品种在纯数学中很重要，有些品种在其他领域发现了应用。例如，舒伯特品种在代数几何学和表示理论中都很重要，并且在计算机图形和统计中都有应用。在与Alex Fink和Seth Sullivant的最新联合合作中，我们使用Schubert品种的特性来研究代数统计中的有条件独立性。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（例如，按照我过去的联合作品连接A Quiver loci和Schubert品种的界限，并使用Schubert品种来研究有条件的独立性）。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训高素质的人员而进一步影响。