Homotopical Algebraic Structures in Algebraic K-theory and Functor Calculus

同伦理论领域的一个有趣的领域是研究代数运算，例如行为类似乘法的运算，但其中不同元素相乘的许多可能方式可以形成几何形状。 这项研究关注的是更复杂的代数结构，其中不仅有运算，还有运算之间的运算等等。这种结构有许多应用，但有许多可能的方法来描述它们，该项目的大部分内容都是关于发展这种描述，并表明它们本质上是相互等效的。 在一个相关的项目中，PI将使用这些基础工具在代数K理论和表示论领域之间建立新的联系。 在后者中，某些被称为霍尔代数的代数结构与K理论中出现的结构有一些相似之处，但精确的关系仍然未知。 一些新的例子为进行更明确的比较提供了一条途径。 第三个项目是与其他四名妇女通过妇女拓扑方案合作进行的，涉及将其中一些方法应用于类似于微积分中出现的泰勒级数的结构。 除了支持年轻女性研究人员的计划外，该计划的活动还包括支持研究生，与本科生一起开展研究，并参与促进数学界进一步多样性的计划。本研究涉及在三个主要方向开发和应用同伦范畴结构的不同模型。在第一，PI将寻求给一个完整的描述所有模型的更高的同伦范畴给出的多单和球形图的单纯集。这种模型是由西格尔条件和离散性或完整性条件;目前大多数工作强调完整性，但我们试图将模型与离散性，特别是考虑在哪些情况下，我们可以使用这两种条件的组合。 第二个方向是研究2-Segal空间在代数K-理论中的应用。已知这些结构是通过瓦尔德豪森S-构造产生的，但它们如何实际用于代数K-理论还有待研究。PI将给出2-Segal空间与坎贝尔和Zakharevich的CGW-范畴之间的一个明确的比较，并发展它们的交换CGW-范畴的类似物。由于2-Segal空间也与Hall代数结构有着密切的联系，我们试图理解CGW范畴如何适应这幅图景，以及更广泛地理解Hall代数与代数K理论有什么关系。 最后，PI将研究离散和阿贝尔函子演算中的模型类别结构，目的是与其他类型的函子演算进行比较，这些函子演算的模型结构也已经开发出来，以及加强齐次函子的分类结果。 这最后一个项目将作为妇女在拓扑研讨会的一部分，与其他四名妇女，其中三人是初级研究人员合作。 该项目包括支持研究生从事相关问题和本科生研究项目的想法，这将有助于学生了解更多关于这些领域。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

One area of interest in the field of homotopy theory is the study of algebraic operations, for example operations that behave like multiplication, but in which the many possible ways of multiplying different elements can form a geometric shape. This research is concerned with more complicated algebraic structures, in which we not only have operations, but operations between operations, and so forth. Such structures have a number of applications, but there are many possible ways to describe them, and much of the project is concerned with developing such descriptions and showing that they are essentially equivalent to one another. In a related project, The PI will use these kinds of foundational tools to make new connections between the fields of algebraic K-theory and representation theory. In the latter, certain algebraic structures called Hall algebras bear several similarities to those that appear in K-theory, yet a precise relationship is still unknown. Some new examples suggest a path for making a more explicit comparison. A third project, which is being done in collaboration with four other women through the Women in Topology program, is concerned with applying some of these methods to structures which resemble the Taylor series which appear in calculus. In addition to this program for supporting junior women researchers, the activities of this proposal also include supporting graduate students, developing research with undergraduate students, and participating in programs to promote further diversity in the mathematics community.This research is concerned with developing and applying different models for homotopical categorical structures in three main directions. In the first, the PI will seek to give a full description of all models for higher homotopical categories given by multisimplicial and globular diagrams of simplicial sets. Such models are given by Segal conditions and either discreteness or completeness conditions; most current work emphasizes completeness but we seek to incorporate models with discreteness, and in particular consider in which cases we can use a combination of the two kinds of conditions. The second direction is to look at applications of 2-Segal spaces in algebraic K-theory. These structures are known to arise via the Waldhausen S-construction, but how they can actually be used in algebraic K-theory is yet to be investigated. The PI will give an explicit comparison between 2-Segal spaces and the CGW-categories of Campbell and Zakharevich, and to develop the analogues of their abelian CGW-categories. Because 2-Segal spaces are also deeply connected to Hall algebra constructions, we seek to understand how CGW-categories fit into this picture, and more broadly just what Hall algebras have to do with algebraic K-theory. Finally, the PI will look at model category structures in discrete and abelian functor calculus, with the goal of comparison to other kinds of functor calculus for which model structures have also been developed, as well as of strengthening classification results for homogeneous functors. This last project will be done as part of the Women in Topology workshop, in collaboration with four other women, three of whom are junior researchers. This project includes support for graduate students working on related problems and ideas for undergraduate research projects which would facilitate students learning more about these areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.