An introduction to noncommutative geometry, K-theory of operator algebras,
and their connections to topological insulators will be provided in the first part of this class.


Topological insulators are examples of symmetry protected (topological) phases of matter. That is, they have trivial topological order in the bulk, but in the presence of (unbroken) symmetry there are topologically non-trivial order (defects).  
An alternative (to the K-theory) classification is in terms of gauging finite group symmetries on the trivial topological order, the description of which is cohomological. Higher order TIs may exist that have non-trivial topological order in the bulk, which then has a similar description: symmetry enriched topological phases are modeled by gauging symmetry on non-trivial topological orders.  These underlying topological orders are understood as unitary modular tensor categories.  In the second part of the class, we will develop this description from basic categorical principles.

Prerequisites: Topological Insulators I.