Surprising geometries and novel materials

From disordered to ordered spatial structures, complex geometries can define new states of matter, rigorously formulated in mathematics and experimentally observed in physics. Surprising probabilistic and geometric phenomena can lead to both fundamental insights and innovative material designs. By presenting recent examples of intriguing geometries, this talk spans a bridge from an anomalous suppression of large-scale density fluctuations to stable matchings and from a hidden order in tessellations to photonic networks, which shape the flow of light.