Homotopical Algebraic Structures in Algebraic K-theory and Functor Calculus

同伦理论领域的一个感兴趣的领域是对代数运算的研究，例如，行为类似于乘法的运算，但其中将不同元素相乘的许多可能方法可以形成几何形状。这项研究涉及更复杂的代数结构，在这种结构中，我们不仅有运算，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涉及开发这样的描述，并表明它们本质上是彼此等价的。在一个相关的项目中，PI将使用这些基础工具在代数K-理论和表示理论的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-理论中出现的那些代数结构有几个相似之处，但确切的关系仍然未知。一些新的例子为进行更明确的比较提供了一条途径。第三个项目是通过女性拓扑学项目与其他四名女性合作完成的，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。除了这项支持初级女性研究人员的计划外，这项计划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化的计划。本研究着眼于从三个主要方向开发和应用不同的同伦范畴结构模型。在第一章中，PI将寻求通过单纯集的多单纯图和球形图给出更高同伦范类的所有模型的完整描述。这类模型是由西格尔条件和离散性或完备性条件给出的；目前的大多数工作都强调完备性，但我们试图将离散性模型结合起来，特别是考虑在哪些情况下可以使用这两种条件的组合。第二个方向是考察2-西格尔空间在代数K-理论中的应用。众所周知，这些结构是通过Waldhausen S构造而产生的，但它们如何实际用于代数K-理论还有待研究。PI将给出2-西格尔空间与Campbell和Zakharevich的CGW-范畴的明确比较，并发展它们的交换CGW-范畴的类似。因为2-西格尔空间也与霍尔代数构造有很深的联系，所以我们试图理解CGW-范畴如何适合这幅图景，更广泛地说，仅仅是霍尔代数与代数K-理论有什么关系。最后，PI将考察离散和阿贝尔函子演算中的模型范畴结构，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。最后一个项目将作为拓扑学中的妇女讲习班的一部分，与另外四名妇女合作完成，其中三名是初级研究人员。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NSF的法定使命，并通过使用基金会的智力优势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.