I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has (relatively) deep mathematical applications to biology or physics (I am not planning to pursue a Ph.D. in pure math).
To this extent, I recently stumbled across information geometry. By this I refer to the field of using data to generate a Riemannian manifold with the Fisher information metric. Could you tell me which applications the field has, particularly to (mathematical) biology or (statistical) physics?
Also, are there any good references, at the level of a Ph.D. student well-versed in probability, analysis, and geometry (but not as much so in statistical inference)?
(And, this goes a bit beyond the question, but if you have any other thoughts on what would be interesting subjects given my preferences above, please share!)