Homotopical Algebraic Structures in Algebraic K-theory and Functor Calculus

同伦理论中一个有趣的领域是对代数运算的研究，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法可以形成几何形状。这项研究关注的是更复杂的代数结构，其中我们不仅有运算，还有运算之间的运算，等等。这样的结构有许多应用，但是有许多可能的方法来描述它们，并且该项目的大部分内容涉及开发这样的描述并显示它们本质上彼此等效。在一个相关的项目中，PI将使用这些基础工具在代数k理论和表示理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理论中出现的代数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一种进行更明确比较的途径。第三个项目，是和其他四位女性合作完成的，通过“拓扑学中的女性”项目，是关于将这些方法中的一些应用于类似微积分中出现的泰勒级数的结构。除了支持初级女性研究人员的项目外，本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元化的项目。本研究主要从三个方面探讨了同主题范畴结构模型的发展和应用。在第一篇中，PI将试图给出由简单集的多重简单图和球状图给出的所有高同局部范畴的模型的完整描述。这些模型由Segal条件和离散性条件或完备性条件给出；大多数当前的工作强调完整性，但我们寻求将模型与离散性结合起来，并特别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。这些结构已知是通过Waldhausen s构造产生的，但如何在代数k理论中实际使用它们还有待研究。PI将给出2-Segal空间与Campbell和Zakharevich的cgw范畴之间的明确比较，并发展他们的阿贝尔cgw范畴的类似物。因为2-Segal空间也与霍尔代数结构密切相关，我们试图理解cgw类别如何适应这一图景，更广泛地说，霍尔代数与代数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，其中也开发了模型结构，以及加强齐次函子的分类结果。最后一个项目将作为拓扑学女性研讨会的一部分完成，与其他四位女性合作，其中三位是初级研究人员。该项目包括支持研究生研究相关问题和本科生研究项目的想法，这将有助于学生更多地了解这些领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.