  Persistence homology is a tool used to measure topological features that are
present in data sets and functions. Persistence pairs births and deaths of
these features as we iterate through the sublevel sets of the data or function
of interest. I am concerned with using persistence to characterize the
difference between two functions f, g : M -> R, where M is a topological space.
Furthermore, I formulate a homotopy from g to f by applying the heat equation
to the difference function g-f. By stacking the persistence diagrams associated
with this homotopy, we create a vineyard of curves that connect the points in
the diagram for f with the points in the diagram for g. I look at the diagrams
where M is a square, a sphere, a torus, and a Klein bottle. Looking at these
four topologies, we notice trends (and differences) as the persistence diagrams
change with respect to time.
