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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地包含无限信息的对象找到有限的描述。例如，一个无限数列可能被紧凑地编码为一个简单函数的值。与这个项目相关的一个事件是一系列空间的维度集。在许多感兴趣的情况下，可能无法直接计算这些数字，但可以分析它们的增长率。数学中一个古老的主题是通过在数列上找到一些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，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这种序列的例子。现在正是深入研究这些结构的时候。本项目将通过有针对性的应用来开展这样一项研究，这些应用被选择为新颖性的新类别的例子，无论是在设置上还是在技术细节上。PI将进行几个项目，涉及交换代数和代数几何中不同种类的范畴对称和经典对称。该研究的总体主题是发现和利用问题中隐藏的对称性，以获得有关该问题的新信息。这个信息可以是一个新的结构性质，也可以是关于一组不变量的有限结果。本研究将特别关注几类数学对象，这些对象的性质或不变量的计算抵制了更传统的攻击手段。其中有：有序注入对上三角矩阵群的同调的作用，抛射对紧化模空间的上同调的作用，以及Hopf环对由多线性代数产生的割变的变的合的作用。这是最近得到充分研究的“fi模块”变体的三个例子，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。