Hidden Order: The Geometry of Complex Patterns

Complex systems can sometimes display intricate spatial patterns whose underlying order is hidden from simple inspection. Mathematics reveals this organization and, unexpectedly, connects unusual, disordered structures to ordered ones, such as crystals and quasicrystals. These links invite us to ask what order and disorder mean, how they can be measured and why hidden order emerges in matter, abstract spaces and living systems.<br>In this Presidential Lecture, Salvatore Torquato will discuss how the ‘hyperuniformity’ concept sheds light on these fascinating questions. In a hyperuniform point pattern, density fluctuations become small over large distances relative to those in garden-variety random systems. This single idea provides a unified framework for understanding the structure and geometry of periodic and quasiperiodic systems, as well as exotic disordered systems. Examples from physics, mathematics and biology will reveal surprising connections among optimization problems in discrete geometry, disordered ground states, sphere packings, quantum mechanics, number theory, the structure of the universe and avian photoreceptor mosaics. Torquato will also introduce ‘order metrics’ — mathematical measures that place periodic, quasiperiodic and correlated disordered patterns along a spectrum from greater order to greater disorder, for objective structural comparisons.

Torquato is the Lewis Bernard Professor of Natural Sciences and a professor of chemistry, physics, the Princeton Institute for the Science and Technology of Materials and the Program in Applied and Computational Mathematics at Princeton University. Torquato has made fundamental contributions to our understanding of the nature of disorder/order in condensed phases of matter, disordered and ordered jammed states of matter, structural glasses, sphere packings in high dimensions, cancer modeling and more. Among other awards and honors, he is a member of the American Academy of Sciences and Letters, was named a Simons Fellow in Theoretical Physics and was a Guggenheim Fellow. He was the recipient of the ACS Joel Hildebrand Award in Theoretical Chemistry of Liquids, the APS David Adler Lectureship Award in the Field of Materials Physics and the SIAM Ralph E. Kleinman Prize.