Continuous metrics on isometry classes of 2-dimensional lattices.

12^th November 2021, 14:00
Vitaly Kurlin
MIF

Abstract

A periodic lattice is an infinite set of all integer linear combinations of basis vectors in a Euclidean space. A natural equivalence on lattices is rigid motion (a composition of translations and rotations) or isometry (also including reflections). All 2-dimensional lattices can be parametrised by three real parameters, but there was no easy metric on isometry classes of lattices, which is also continuous under basis perturbations. Using a recent parametrisation of 2-dimensional lattices by root forms, we introduce metrics that are invariant up to isometry or up to rigid motion, which preserves orientation. The latter case also quantifies chirality by measuring a distance from a chiral lattice to its closest mirror-symmetric neighbour. The talk is based on the updated dimension 2 paper joint with Matt Bright and Andy Cooper.