Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Fall 2016

Speaker:  Miranda Holmes-Cerfon, Courant Institute, NYU
Title: Sphere packings, singularities, and statistical mechanics
Time: 2:30 PM, Monday, November 14, 2016
Place:  Malott 206

Abstract: What are all the ways to arrange $N$ hard spheres to form a rigid cluster? And what can the solution tell us about how materials crystallize? I will explain an algorithm to tackle the first question, and show that it produces many clusters with geometrically unusual properties. Most notably perhaps is the preponderance of ``singular'' clusters, those that are linearly flexible but nonlinearly rigid, so-called because they correspond to singular solutions to a set of algebraic equations. These are also the clusters one sees with unusually high probability in experiments (which consider colloidal particles interacting with a short-range potential), but evaluating this probability is impossible using standard tools from statistical mechanics. I will show that by taking a geometrical approach, we can approximate this probability for all clusters that are ``second-order rigid''. This calculation suggests that the free-energy landscape of a finite collection of hard spheres has a universal shape, and it brings insight into the pathways to crystallization; these observations are empirical and could benefit from a geometric analysis.

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