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为先验中包含无限信息的对象找到有限描述。例如，无限的数字序列可能被紧凑为简单函数的值。与该项目相关的这种情况之一是一系列空间的维度。在许多感兴趣的情况下，可能不可能直接计算这些数字，而是可以希望分析其增长率。数学中的一个旧主题是通过在顺序上找到一些代数结构来理解此类问题。最近，几个研究人员发现了新的，异国情调的代数结构，这些结构适用于拓扑，组合和代数几何形状等领域中此类序列以前意外的例子。该项目的目的是通过应用和结合工具和组合工具来研究和揭示这些新代数结构的基本属性，并结合工具和代表理论的工具。PI提议在交换代数和代表理论之间的界面上进行多个项目，具体来说是对代数的allaig od ander and commut and commuts and thist twist twist twist od and commut and commut and commut and commut and commutbras mod and的研究。 FI模型）及其应用于经典主题（代数品种的综合性，同源稳定性，算术组的同源性等）。此外，PI将处理Boij-Söderberg理论中的问题，Boij-Söderberg理论是对代数不变的研究，例如分级模块的Betti表和矢量捆绑包的共生表，“到标量倍数”。该主题以间接和微妙的方式与一般主题相关，但仍未得到充分理解。基本主题是确定在这些代数结构之一上具有模块结构的一类示例，并证明它是有限生成的。揭示类似于存在结果的一些信息，但通常这些结果不会带有界限或任何更清晰的信息，因此应将其视为较大程序的第一步。 PI提出的提议从交换代数和同源代数中导入思想，例如希尔伯特功能，投射决议，Castelnuovo-Mumford的规律性和Koszul二元性，以进一步增强对此类示例的理解并揭示新信息。