The study of topological invariants in band theory has led to many insights into the behavior of electrons in insulators and superconductors. In 2013, Kane and Lubensky pointed out that certain "isostatic" mechanical systems can also admit topological boundary modes . This has led to the design and realization of several families of "topological mechanical metamaterials". In my talk I will introduce the "topological polarization" of Kane and Lubensky and then explain how it can be realized in certain mechanical structures, called rigid origami and kirigami, which consist of rigid plates joined by hinges meeting at vertices . Mysteriously, we found in  that triangulated origami structures always seem to be unpolarizable, that is, despite the lack of any apparent symmetry, all of these structures have a vanishing polarization invariant. I will describe recent work with Zeb Rocklin (Georgia Tech), Louis Theran (St. Andrews) and Chris Santangelo (UMass Amherst) which explains this via a "motion to stress" correspondence, that generalizes the 19th-century Maxwell-Cremona correspondence in several directions.
 C.L. Kane and T.C. Lubensky, Nat Phys 10, 39–45 (2014).
 B.G. Chen, B. Liu, A.A. Evans, J. Paulose, I. Cohen, V. Vitelli, and C. Santangelo, Phys Rev Lett 116, 135501 (2016).