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尺寸集。在许多感兴趣的情况下，可能无法直接计算这些数字，而是可以分析其增长率。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。对这些结构进行更深入研究的时间是正确的。该项目将通过针对新的示例进行针对新的示例进行新的示例，无论是在设置还是技术细节中。该研究的一般主题是在问题中找到并利用隐藏的对称性，以获取有关它的新信息。这些信息可以采用新结构属性的形式或关于不变的家族的有限结果。这项研究将尤其集中在几类数学对象上，这些数学对象或不变性的计算已经抵制了更传统的攻击手段。其中包括：有序注射对上层矩阵组的同源性的作用，溢流对压缩模量空间的共同体的作用以及Hopf环对由多线性代数产生的分类品种的共氮化物的作用。这是最近研究的“ FI模型”的三个示例，它们提供了新的技术挑战，并为“表示稳定性”的新范式打开了大门。