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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!)

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  • $\begingroup$ This is not a question but rather a follow up. I was wondering if you ended up doing your thesis in this field and if so, what did you research on? Also, even more follow-up, are you still doing research in this field? In your opinion, are there any intersections with ML/AI? Thanks for your answer! $\endgroup$ – user110320 Jan 4 '18 at 19:51
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Personally, I worked in this area producing three papers appeared in conference proceedings (see this paper, this other and this one). I can provide you some book titles:

S. Amari&al., Differential Geometry in Statistical Inference, Institute of Mathematical Statistics (1987). This is currently free online here.

Shun-ichi Amari; Hiroshi Nagaoka, Methods of Information Geometry, AMS (2000).

Khadiga A. Arwini, Christopher T.J. Dodson, Information Geometry, Springer (2008).

You can also read the Wikipedia entry about this matter.

I think these hints should provide you a good starting point.

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  • $\begingroup$ Yes, I have looked at these books, but perhaps I am seeking more of an expert's perspective on the field. Is it interesting/promising research, and has it made you a better physicist/mathematician? $\endgroup$ – user77891 May 16 '13 at 1:46
  • $\begingroup$ @user77891: Well, I have got 3 publications on this matter as you can see and the theorems I proved can have a wealth of applications. The reason is that I have shown a statistical property starting from differential geometry and this is quite general. Generality comes out by the techniques I used. Moreover, the link with Einstein equations makes this result rather interesting. Just imagine to connect gravity with optimal estimators in statistics. I did it in 2 and 3 dimensions. $\endgroup$ – Jon May 16 '13 at 6:34
  • $\begingroup$ I would like to invite you to answer this question if possible, thanks.mathoverflow.net/questions/215984/… $\endgroup$ – Henry.L Aug 30 '15 at 17:46
  • $\begingroup$ @Henry.L Thanks a lot for asking me this. Currently, I am not working in this area of research any more and cannot be of any help. Sorry. $\endgroup$ – Jon Sep 1 '15 at 19:03
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I am an information theorist and I try to give you my personal take. About three years ago, a conference on information geometry was held in Germany. As you can see there, there are lot of applications but most of the applications are deeply tied with Statistics. However, in my opinion, information geometry could not bring results that influence the related research in probability and statistics. And also no fundamentally original results was observed that excite researchers. Differential geometry is itself an obstruction due to the its complexity. Of course these are highly subjective judgments and no one can predict the future but at least currently, researchers are not so excited about the results.

You can have interesting connections with other discoveries of science but still, I think we have not seen some results from information geometry that are being widely used even in statistics.

I believe this is so, partly because the field is still very young and immature. I see a big potential in bring geometrical insights to statistics but I think that is very challenging for a young researcher.

I am looking forward to find a way to apply it in my own research though I have not found any way yet.

The section 2 and 3 of this book discusses the relation with statistics.

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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.

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