Combinatorial K-theory: A Kaleidoscope of Applications

A common strategy employed in mathematics when one wishes to understand an intricate structure is to split it into smaller pieces, study these more manageable pieces first, and then reassemble this piecewise information into data about the original structure. The research area of K-theory was precisely engineered to achieve this, and for this reason it permeates a wide range of mathematical fields, encoding meaningful information about any setting where there is a reasonable notion of “splitting”. A K-theory machinery stores this data in a topological space, which is able to record which objects break up into smaller pieces, as well as how these splittings occur. Unpacking the data present in a K-theory space and reinterpreting it in terms of the original structures of interest, is both the main goal and the main challenge of the field of K-theory. In this project, the PI will carry out two independent research lines in this direction that aim to make strides in our understanding of the K-theory spaces of manifolds (smooth shapes with no sharp edges or corners, central to geometry and physics) and of Lawvere theories (a main character in universal algebra and logic), using the newly-introduced framework of combinatorial K-theory. A third research line, and an underlying theme throughout this project, pushes the state of the art of combinatorial K-theory, developing tools and paving the way for new examples and applications to come. Integrated into this project is the mentoring and training of students, as well as the organization of a collaborative workshop, the creation of a new professional development class for graduate students on mathematical communication skills, and the development of new active learning materials for undergraduate education that will be made broadly available to the mathematical community. The emerging subfield of combinatorial K-theory introduces new techniques to study structures whose splittings have a combinatorial flavor; for instance, by placing a focus on complements, instead of quotients or cofibers. Exciting new developments in the field, such as the introduction of new simplicial models and of robust comparison theorems reminiscent of the best features of classical algebraic and homotopical settings, place combinatorial K-theory in an advantageous spot and make this an ideal time for ambitious explorations. In this project, the PI will exploit these recent developments in order to explore three applications of vastly different mathematical flavors. The PI plans to: (a) Give a first computation of K1 of cut-and-paste K-theory for manifolds; (b) Introduce a K-theory of endomorphisms and a version of the Fundamental theorem of K-theory in the combinatorial setting, with a particular focus on finite sets; and (c) Explore connections between K-theory and logic by constructing localizations for the K-theory of Lawvere theories. 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.