Interactions between Commutative Algebra and Representation Theory

交换代数与表示论之间的相互作用

• 批准号: 1848744
• 负责人: Steven Sam
• 金额: $ 4.54万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1848744&HistoricalAwards=false
• 关键词: Interactions; between; Commutative; Algebra; Representation

An important theme in mathematics is to find finite descriptions for objects that a priori contain an infinite amount of information. For example, an infinite sequence of numbers might be compactly encoded as the values of a simple function. One such occurrence relevant to this project is the dimensions of a sequence of spaces. In many cases of interest, it may not be possible to directly compute these numbers, but one can instead hope to analyze their rate of growth. An old theme in mathematics has been to understand such problems by finding some algebraic structure on the sequence. Recently, several groups of researchers have found new, exotic algebraic structures that apply to previously unexpected examples of such sequences in areas such as topology, combinatorics, and algebraic geometry. The purpose of this project is to study and uncover the fundamental properties of these new algebraic structures by applying and combining tools from commutative algebra and representation theory.The PI proposes to work on several projects at the interface between commutative algebra and representation theory, specifically the study of new classes of algebraic objects (e.g., twisted commutative algebras, Delta-modules, and FI-modules) and their applications to classical topics (syzygies of algebraic varieties, homological stability, homology of arithmetic groups, etc.). Additionally, the PI will work on problems in Boij-Söderberg theory, which is the study of algebraic invariants, such as Betti tables of graded modules and cohomology tables of vector bundles, "up to scalar multiple." This topic is connected to the general theme in indirect and subtle ways that are still not well understood. The basic theme is to identify a class of examples that have the structure of a module over one of these algebraic structures and to prove that it is finitely generated. The finite generation reveals some information akin to an existence result, but often the results do not come with bounds or any sharper information, so should be seen as a first step in a larger program. The PI proposes to import ideas from commutative algebra and homological algebra, such as Hilbert functions, projective resolutions, Castelnuovo-Mumford regularity, and Koszul duality, to further enhance the understanding of such examples and to reveal new information.

数学中的一个重要主题是找到先验包含无限量信息的对象的有限描述，例如，无限的数字序列可能被紧凑地编码为一个简单函数的值。项目是一系列空间的维度。在许多有趣的情况下，可能无法直接计算这些数字，但人们可以希望分析它们的增长率。数学的一个古老主题是理解此类问题。最近通过在序列上找到一些代数结构。几个研究小组发现了新的、奇异的代数结构，这些结构适用于拓扑学、组合学和代数几何等领域中以前意想不到的序列示例。该项目的目的是研究和揭示这些新代数结构的基本属性。通过应用和结合交换代数和表示论的工具。PI建议在交换代数和表示论之间的接口上开展几个项目，特别是新类别代数对象的研究（例如，扭曲交换代数、Delta 模和 FI 模）及其在经典主题中的应用（代数簇的合性、同调稳定性、算术群的同源性等）。此外，PI 将研究 Boij 中的问题。 -Söderberg 理论，这是对代数不变量的研究，例如分级模的 Betti 表和向量丛的上同调表，“直到标量倍数。”这个主题以间接和微妙的方式与一般主题联系在一起，但目前还没有很好地理解。基本主题是识别一类具有这些代数结构上的模块结构的示例，并证明它是有限生成的。有限生成揭示了一些类似于存在结果的信息，但结果通常没有边界或任何更清晰的信息，因此应该被视为大型程序中的第一步。来自交换律的想法代数和同调代数，例如希尔伯特函数、射影解析、Castelnuovo-Mumford 正则性和 Koszul 对偶性，以进一步增强对此类示例的理解并揭示新信息。