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: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759107
• 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.

代数几何是数学的核心领域。它不仅在纯数学和应用数学中很重要，而且在自然科学、工程及其他领域（例如经济学）也很重要。代数几何的核心是研究多变量多项式集合的公共零点。该领域的一个普遍主题是将有关这些零集或代数簇的几何问题转化为等效的代数问题。此类几何问题的示例包括“这个零集是否有多个分量？”和“有什么奇点吗？”。拟议的研究将解决具有多种对称性的重要代数簇的这些类型的几何问题。对于此类代数簇，通常可以进一步将某些代数问题转化为组合问题（例如计数问题或有关离散结构的问题）。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，该提议适合数学的特定领域。受多位数学家过去成功经验的启发，我将使用组合交换代数的方法来研究携带对称群的三类代数簇的代数几何性质：Dynkin 箭袋的颤动轨迹、Schubert 簇和相关簇以及某些希尔伯特方案。这些变体在纯数学中很重要，有些已经在其他领域找到了应用。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都有应用；在最近与 Alex Fink 和 Seth Sullivant 的合作中，我们利用舒伯特簇的性质来研究代数统计中的条件独立性。结果将引起数学家的兴趣，并将为有关这些重要代数簇的文献做出贡献。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（例如，沿着我过去连接 A 型颤动基因座和舒伯特簇的联合作品，并使用舒伯特簇来研究条件独立性）。特定方向的结果将为重要的开放问题带来新的见解。最后，拟议的研究包含许多适合各个级别学生的项目，因此该研究计划将通过培养高素质人才产生进一步的影响。