TITLE: Applications of information geometry to the natural sciences
QUESTION [25 upvotes]: 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!)

REPLY [8 votes]: The question is an old one, but there is a new-ish book that would make a good answer: Information Theory and its Applications (2016) by Shun-Ichi Amari. (Book information on Springer's website | book on Amazon.com) It assumes pretty much the background you asked for, and half of the book covers applications. (Though it is more oriented to applications in machine learning than the natural sciences, generally speaking.)
Presumably you have long finished your undergrad thesis by now, but I hope this will be useful to future visitors with the same question.