CAREER: Categorical and Classical Symmetries in Commutative Algebra and Algebraic Geometry

职业：交换代数和代数几何中的分类和经典对称性

基本信息

批准号：
1849173
负责人：
Steven Sam
金额：
$ 52.97万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1849173&HistoricalAwards=false
关键词：
CAREER Categorical Classical Symmetries Commutative

项目摘要

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 set of 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 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 time is right for a deeper study of these structures. This project will develop such a study through targeted applications to new classes of examples which have been selected for novelty, either in settings or technical details.The PI will pursue several projects that involve different kinds of categorical and classical symmetries in commutative algebra and algebraic geometry. The general theme of the research is to find and exploit hidden symmetries in a problem to gain new information about it. This information can take the form of a new structural property or a finiteness result about a family of invariants. This research will focus in particular on several classes of mathematical objects for which these properties or calculation of invariants have resisted more traditional means of attack. Among these are: the action of ordered injections on the homology of groups of upper-triangular matrices, the action of surjections on the cohomology of compactified moduli spaces, and Hopf ring actions on syzygies of secant varieties of varieties arising from multilinear algebra. These are three examples of variants of the recently well-studied "FI-modules," which offer new technical challenges and open the door to new paradigms of "representation stability."

数学的一个重要主题是为先验包含无限量信息的对象找到有限描述。例如，无限的数字序列可以被紧凑地编码为简单函数的值。与该项目相关的一个这样的事件是一系列空间的尺寸集。在许多有趣的情况下，可能无法直接计算这些数字，但可以分析它们的增长率。数学的一个古老主题是通过在序列上找到一些代数结构来理解此类问题。最近，几个研究小组发现了新的、奇特的代数结构，这些结构适用于拓扑、组合学和代数几何等领域中以前意想不到的此类序列的例子。现在是对这些结构进行更深入研究的时候了。该项目将通过有针对性地应用新类别的示例来开展这样的研究，这些示例在设置或技术细节上都具有新颖性。PI 将开展几个项目，涉及交换代数和代数几何中的不同类型的分类和经典对称性。研究的总体主题是发现并利用问题中隐藏的对称性以获得有关该问题的新信息。该信息可以采用新的结构属性或关于不变量族的有限性结果的形式。这项研究将特别关注几类数学对象，这些对象的这些属性或不变量的计算可以抵御更传统的攻击手段。其中包括：有序注入对上三角矩阵群同调的作用、满射对紧致模空间上同调的作用、以及Hopf环对由多线性代数产生的割簇簇的syzygies的作用。这是最近经过深入研究的“FI 模块”变体的三个示例，它们提出了新的技术挑战，并为“表示稳定性”的新范式打开了大门。