Materials from Mathematics

Largely due to the extreme forms of nonlinearity encountered in 
the behavior of materials, mathematics and materials science enjoy a 
healthy  interaction.  Like  some  theorems,  the  discovery  of  a 
spectacular new material represents an unmistakable advance, not 
clouded  by  shades  of  meaning.  This  discussion  concerns  the 
mathematical  theory  that  underlies  the  synthesis  of  materials  that 
undergo  phase  transformations,  specifically  in  regard  to  hysteresis 
and reversibility. What elements, in what proportion, and with what 
processing are needed to achieve unprecedented behavior? 
Small bubbles  of  soap froth  disappear while  big ones grow and 
grains of a polycrystalline metal coarsen over time. As in the simplest 
linear elliptic and parabolilc equations, there is a strong tendency to 
simplify  and  smooth.  However,  exactly  the  opposite  happens  in  a 
martensitic phase transition. Here, one begins with a uniform crystal 
of  austentite  and  upon  cooling  one  gets  a  plethora  of  fine 
microstructures  of  martensite.  The  mathematical  origins  of  the 
spontaneous formation of fine structure comprise a fascinating and 
ongoing chapter of nonlinear analysis.