K-theory Seminar
Lydia Holley
UIC
Higher Scissors Congruence Groups of the Euclidean Plane
Abstract: The study of classical scissors congruence addresses the following problem: if we cut a polytope up into finitely many pieces, and we are allowed to rearrange those pieces by isometries, which polytopes can we build? Higher scissors congruence, or scissors congruence K-theory, expands on the study of scissors congruence by not only searching for equivalent polytopes, but also equivalences between the symmetries themselves. Computations in higher scissors congruence are computations in K-theory, which are notoriously difficult. As such, higher scissors congruence groups have only been computed in dimension 1 for euclidean, spherical, and hyperbolic geometries — dimension 2 and above have been open. My results prove that for 2-dimensional euclidean space, higher scissors congruence groups are uncountable in almost every degree, with the potential exception of degree 1, where the group was conjectured to vanish prior to this work. I will walk the viewer through the strategy of computing approximations of all of the higher scissors congruence groups, and then push uncountably many nontrivial classes forward from these approximations of the groups into the higher scissors congruence groups themselves.
For the zoom link, please contact the speaker or Professor Shipley.
Thursday October 30, 2025 at 4:30 PM in Zoom