Collaborative Research: K-theory, manifolds, and polyhedra

This research combines ideas and techniques from several of the most exciting directions of contemporary mathematical thought, developing powerful connections between seemingly distant fields. These include homotopy theory (the study of shapes up to continuous deformation), algebraic K-theory (a broadly applicable framework that captures how mathematical objects can be decomposed into pieces and reassembled), smooth manifolds (shapes with no edge or boundary, of fundamental importance to geometry and physics), polyhedra (the higher-dimensional version of polygons), and knots (frictionless loops of string in three-dimensional space). The PIs will pursue several new directions, building on the longstanding and successful program of combining these theories to classify important geometric objects, along with some striking new results relating K-theory to polyhedra and knots. The project also includes the training of junior mathematicians, the development of a much-needed textbook in homotopy theory, and the expansion of college-level mathematics in prison education, which was shown to curb recidivism and save taxpayer money. In this project the PIs will develop algebraic K-theory and its applications to geometric objects such as smooth manifolds, polyhedra, and knots. They plan to bring the connection between equivariant algebraic K-theory and h-cobordisms to fruition, in order to further our understanding of the homotopy type of the moduli space of G-manifolds. They will study the higher versions of Hilbert’s 3rd problem for polyhedra through scissors congruence K-theory. They will integrate recent insights from the setting of polytopes into the scissors congruence of manifolds, in order to perform computations, construct new higher invariants of smooth manifolds up to cut-and-paste relations, and investigate the tantalizing relation to cobordism categories. Lastly, they will collaborate with low dimensional topologists in order to apply stable homotopy theory and algebraic K-theory to prove new properties of Khovanov homology. 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.