The precipitous loss of Arctic sea ice has outpaced expert predictions. We will explore how mathematical models of key sea ice processes are being developed to improve projections of the future of Earth's sea ice packs and the polar ecosystems they support. Our models are inspired by theories of multiscale composite materials and statistical physics, and are developed in conjunction with field experiments that we have conducted in both the Arctic and Antarctic. The lecture is intended for a wide, interdisciplinary audience, and will conclude with a short video on a recent Antarctic expedition where we measured fluid and electromagnetic transport properties of sea ice.