Oct 9 - Evan Angelone Β (Northeastern)
Title: Affine Polytopes
Abstract: An abstract π-polytope is a certain graded poset generalizing the incidence structure of the faces of a π-dimensional geometric polytope. The most symmetric are the regular polytopes and are characterized by the string C-groups. The next-most symmetric are the chiral polytopes, admitting maximal rotational symmetry without but not any enantiomorphic symmetry, and are characterized by the string rotation groups. In both cases, the automorphism group of the polytope is a string group. Conversely, the costs of distinguished subgroups of a string group form the faces of a rotary polytope, where incidence between those faces is a canonical relation on the cosets.
In 2025, we demonstrated that the general affine group is remarkably well-suited for the construction of abstract chiral 3-polytopes. The group AGL(1, π), being those maps on π½_π taking π₯ β¦ ππ₯ + π, inspired us to generalize the construction to semidirect products π β πΊ for arbitrary groups πΊ acting as a linear point group along with a finite β€[πΊ]-module π acting as a translation subgroup. This "affine" approach to polytopes immediately opens the floor to entirely new constructions of rotary π-polytopes from Galois groups, affine semilinear groups over both rings and fields, and both the Gaussian and Eisenstein integers. Simultaneously, we're now able to settle a pair of open problems in the classification of minimal chiral polytopes of a fixed order.