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

数学中的一个重要主题是找到对先验包含无限信息量的对象的有限描述。例如，一个无限的数字序列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，但可以分析它们的增长率。数学中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。这个项目将通过有针对性地应用于新类别的例子来发展这样的研究，无论是在环境中还是在技术细节上。PI将进行几个项目，这些项目涉及交换代数和代数几何中不同类型的范畴和经典对称。这项研究的总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。该信息可以采取新的结构性质的形式或关于不变量族的有限结果的形式。这项研究将特别集中于几类数学对象，对于这些对象，这些不变量的性质或计算已经抵抗了更传统的攻击手段。其中包括：序内射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环对多重线性代数上的割簇合子的作用。这是最近研究得很好的“FI模块”的三个变种，它们提供了新的技术挑战，并为“表征稳定性”的新范式打开了大门。

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."