Interactions between Commutative Algebra and Representation Theory

数学中的一个重要主题是为先验包含无限信息的对象找到有限的描述。例如，一个无限序列的数字可能被压缩编码为一个简单函数的值。与这个项目相关的一个这样的事件是空间序列的维度。在许多感兴趣的情况下，可能无法直接计算这些数字，但可以希望分析它们的增长率。数学中的一个古老主题是通过找到序列上的某种代数结构来理解这类问题。最近，几组研究人员发现了新的，奇异的代数结构，适用于以前意想不到的例子，如拓扑学，组合学和代数几何领域的序列。该项目的目的是通过应用和结合交换代数和表示论的工具来研究和揭示这些新代数结构的基本性质。PI建议在交换代数和表示论之间的接口上开展几个项目，特别是研究新的代数对象类（例如，扭交换代数，Delta-模和FI-模）及其在经典课题（代数簇的合势，同调稳定性，算术群的同调等）中的应用。此外，PI将研究Boij-Söderberg理论中的问题，这是代数不变量的研究，例如分次模的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总的主题联系在一起，但仍然没有得到很好的理解。基本的主题是确定一类具有这些代数结构之一的模块的结构的例子，并证明它是随机生成的。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，因此应该被视为更大程序的第一步。PI建议从交换代数和同调代数中引入概念，如希尔伯特函数，投影分解，Castelnuovo-Mumford正则性和Koszul对偶，以进一步加强对这些例子的理解并揭示新的信息。

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.