IMA Outcomes

Assessment of the success of scientific endeavors, e.g., IMA workshops, is often difficult and subjective, but it is important to make an attempt to document connections between such efforts and their results.

Below, we give a few examples of positive outcomes of IMA activities, and we plan to elaborate on more of these in future issues of Update.

Lax conjecture solved. The Lax conjecture, an open problem of 45 years standing with important applications in optimization, fell in a just few weeks after the right connections were made at the March 2003 IMA workshop on Semidefinite Programming and Robust Optimization. Discussions between Adrian Lewis and Pablo Parrilo led them and coauthor M.~Ramana to pursue connections with work of Helton and Vinnikov. Their joint paper, entitled ``The Lax Conjecture is True'' appeared on the arXiv preprint server on April 8 and has been submitted to the Proceedings of the American Mathematical Society.

Ferromagnetic shape-memory alloys.

FSMA Ni2 MnGa exhibiting

twinned microstructure.

Ferromagnetic shape-memory alloys are one of the hottest subjects in materials science. These materials change shape when a magnetic field is applied, and return to their original shape when the field disappears. They have all sorts of applications in pumps, valves, noise suppressors, sonar transducers, and other devices. Their discovery can be traced directly back to the 1990--91 IMA workshop on Microstructure and Phase Transition, where R. James and D. Kinderlehrer began working on mathematical models for magnetostrictive materials, with the goal

Distortion of an FMSA crystal.

of trying to understand the beautiful domain patterns in Terfenol observed by Donald Lord. They were able to explain the experiments by a new kind of free energy combining symmetry-related ``energy wells'' (an idea which itself has roots in earlier IMA insight immediately suggested how a material could be designed which exhibited much larger field-induced strains (as much as 50 times larger) than Terfenol, and these theoretically derived materials have since been realized as nickel-magnesium-gallium alloys.