Homotopical Algebraic Structures in Algebraic K-theory and Functor Calculus

同伦理论领域的一个有趣领域是代数运算的研究，例如行为类似于乘法的运算，但其中不同元素相乘的多种可能方式可以形成几何形状。 这项研究涉及更复杂的代数结构，其中我们不仅有运算，还有运算之间的运算，等等。这样的结构有很多应用，但是有很多可能的方法来描述它们，并且项目的大部分内容都涉及开发这样的描述并表明它们本质上是彼此等效的。 在相关项目中，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。 一些新的例子提出了进行更明确比较的途径。 第三个项目是通过“拓扑学女性”计划与其他四位女性合作完成的，该项目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，该提案的活动还包括支持研究生、与本科生一起开展研究以及参与促进数学界进一步多样性的计划。这项研究涉及在三个主要方向上开发和应用同伦分类结构的不同模型。首先，PI 将寻求对由单纯集的多重单纯形图和球状图给出的更高同伦类别的所有模型进行完整描述。这些模型由 Segal 条件和离散性或完整性条件给出；当前的大多数工作强调完整性，但我们寻求将模型与离散性结合起来，特别是考虑在哪些情况下我们可以使用两种条件的组合。 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 Waldhausen S 结构产生的，但它们如何实际用于代数 K 理论仍有待研究。 PI 将给出 2-Segal 空间与 Campbell 和 Zakharevich 的 CGW 范畴之间的明确比较，并开发其阿贝尔 CGW 范畴的类似物。由于 2-Segal 空间也与霍尔代数结构密切相关，因此我们试图了解 CGW 范畴如何适应这幅图景，以及更广泛地了解霍尔代数与代数 K 理论的关系。 最后，PI 将研究离散和阿贝尔函子微积分中的模型类别结构，目的是与已开发模型结构的其他类型函子微积分进行比较，并加强齐次函子的分类结果。 最后一个项目将作为拓扑学中的女性研讨会的一部分，与其他四名女性合作完成，其中三名是初级研究人员。 该项目包括为研究生研究本科研究项目的相关问题和想法提供支持，这将有助于学生更多地了解这些领域。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。

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.