TL;DR
This paper explores the interplay between geometry, information theory, and physics in understanding material models, focusing on symmetry, order, and disorder in packing problems, error-correcting codes, and ground states of particle systems.
Contribution
It provides an interdisciplinary overview connecting packing, coding, and physical ground states, highlighting symmetry phenomena and their underlying reasons.
Findings
Symmetry appears in special optimal structures.
Different perspectives reveal common phenomena.
Techniques from geometry, information theory, and physics are interconnected.
Abstract
These are the lecture notes from my 2014 PCMI graduate summer school lectures. In these lectures, we'll study simple models of materials from several different perspectives: geometry (packing problems), information theory (error-correcting codes), and physics (ground states of interacting particle systems). These perspectives each shed light on some of the same problems and phenomena, while highlighting different techniques and connections. One noteworthy phenomenon is the exceptional symmetry that is found in certain special cases, and we'll examine when and why it occurs. The overall theme of the lectures is thus order vs. disorder. How much symmetry can we expect to see in optimal geometric structures?
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