## Research Areas

### Prediction of hysteresis

In phase transformations, hysteresis refers to a transformation temperature on cooling that differs from that on heating. Similar effects occur in ferromagnetism and ferroelectricity, and the understanding of "hysteresis loops" is a longstanding open problem in science. It has an historically close connection to mathematics, particularly functional analysis, but this largely concerns the modeling of hysteresis rather then a fundamental understanding of its origins. Currently, it takes on renewed significance, as the size of the hysteresis loop can often be related to wasted energy in energy conversion materials. One of our goals in this PIRE project is to understand, using theory development, rigorous mathematical analysis and scientific computation, the origins of hysteresis in materials undergoing phase transformations. A emergent theme is the role of compatibility between phases, which finds mathematical expression in the minimization of the energy of transition layers and particularly its sensitive dependence on the lattice parameters of the phases.