UBC Math Department Colloquium: Jacob Shapiro

Topological insulators are novel materials which are on the one hand insulators in their bulk, yet on the other hand excellent conductors along their edge. This distinguished property, and many more, are explained via the rich topological structure of the space of quantum-mechanical Hamiltonians. This structure can be measured experimentally, and gives rise to an entirely new field in condensed matter physics as well as mathematical physics. After presenting the general setting, we will survey two recent results: (1) connecting PDE models of continuum systems with discrete models, and the magnetic double-well problem, and (2) a new analytic framework for calculating the path-components of 1D insulators in various symmetry classes, which goes beyond K-theory.