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.
