  This paper introduces the key concepts and problems of the new research area
of Periodic Geometry and Topology for applications in Materials Science.
Periodic structures such as solid crystalline materials or textiles were
previously studied as isolated structures without taking into account the
continuity of their configuration spaces. The key new problem in Periodic
Geometry is an isometry classification of periodic point sets. A required
complete invariant should continuously change under point perturbations,
because atoms always vibrate in real crystals. The main objects of Periodic
Topology are embeddings of curves in a thickened plane that are invariant under
lattice translations. Such periodic knots were classified in the past up to
continuous deformations (isotopies) that keep a fixed lattice structure, hence
are realized in a fixed thickened torus. The more practical equivalence is a
periodic isotopy in a thickened plane without fixing a lattice basis. The paper
states the first results in the new area and proposes further problems and
directions.
