Over the past several years, our understanding
of topological electronic phases of matter has advanced
dramatically. A paradigm that has emerged is that insulating
electronic states with an energy gap fall into distinct topological classes.
Interfaces between different topological phases exhibit gapless conducting
states that are protected topologically and are impossible to get rid
of. In this talk we will discuss the application of this idea to
the quantum Hall effect, topological insulators, topological
superconductors and the quest for Majorana fermions in condensed
matter. We will then show that similar ideas arise in a completely
different class of problems. Isostatic lattices are arrays of
masses and springs that are at the verge of mechanical instability.
They play an important role in our understanding of granular matter, glasses
and other 'soft' systems. Depending on their geometry, they can exhibit
zero-frequency 'floppy' modes localized on their boundaries that are
insensitive to local perturbations. The mathematical relation
between this classical system and quantum electronic systems reveals an
unexpected connection between theories of hard and soft matter.
Watch a video of the colloquium.