Hyperuniformity in many-particle systems and its generalizations

Hyperuniform many-particle systems have density fluctuations that are anomalously suppressed at long wavelengths compared to the fluctuations in typical disordered point configurations, such as ideal gases, liquids and glasses. All perfect crystals and perfect quasicrystals are hyperuniform. Disordered hyperuniform many-particle systems can be regarded to be new states of amorphous matter in that they behave more like crystals or quasicrystals in the manner in which they suppress large-scale density fluctuations, and yet are also like liquids and glasses because they are statistically isotropic structures with no Bragg peaks. I will describe a variety of different results and examples of ordered and disordered hyperuniform states of matter that arise in physics, mathematics and biology. I will make contact with minimizers of the Epstein zeta function, nontrivial zeros of the Riemann zeta function, random matrices, fermionic systems, Coulomb gases, classical disordered ground states, and photoreceptor patterns in the eyes of birds. I will describe the generalizations of the hyperuniformity concept to two-phase random media, random scalar fields and random vector fields.