  This is a set of notes on some unrelated topics in mathematical physics, at
varying levels of detail. First, I consider certain questions relating to the
decay of correlation functions in matrix product states, in particular those
generated by quantum expanders. This is discussed in relation to recent results
of Brandao and Horodecki on area laws on systems with exponentially decaying
correlation function\cite{areaexp}. Second, I consider some difficulties in
trying to construct a tensor product state (or PEPS) describing a
two-dimensional fermionic system with non-vanishing Hall conductance. Third, I
present some relations between the theory of almost commuting matrices and that
of vector bundles, making the connection between the classifications more
explicit. Fourth, I present an open question about quantum channels, and some
partial results.
