Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I would like to see some perspectives on the subject. Maybe some research branches groups may have.

For example: In [1] they seem to work out Mrugala's idea of thethermodynamic phase space on a smooth manifold but then they turn into something more like numerical analysis.

In the GSI reports one finds alot of information but that is over 3 thousand pages of research and there seems to be a lack of motivational texts about the problems they seem to be hunting.

I have found that goemetric control theory looks like a promising area as well but I couldn't find a useful overview of the subject that maybe overlaps with things being done in information geometry.

Finally, in [2] they work out what looks like a non-Riemannian framework for information geometry but the theory seems to be quite general and far from a Msc/PhD thesis level. Can anyone point me out in a direction maybe to narrow down the amount of reading material to be covered? thank you very much.

(I would appreciate maybe something pointing in the topological/geometrical data analysis roadmap as well)

[1] Bravetti, Alessandro, Contact geometry and thermodynamics, Int. J. Geom. Methods Mod. Phys. 16, No. Supl. 01, Article ID 1940003, 51 p. (2019). ZBL1421.80002.

[2] Polysymplectic Geometry of High Order Souriau Lie groups Thermodynamics based on Günther's model, F. Barbaresco.

GSI: Geometric Sicences of Information which are some books result from a conference being done since 2013 every 2 years.


A good place to start learning Information Geometry is the series of blog posts by John Baez, which give an informal survey of some of the central concepts. From there, I think the book by Ay, Jost, Le, and Schwachhöfer [1] is a good reference for learning the subject. What was particularly helpful for me was working through a few examples in detail. In particular, studying the statistical manifolds of multinomial distributions and univariate normal distributions in detail was quite informative.

Due to the page limits on the Proceedings of GSI (which are probably necessary in order to bind the book), it is a bit hard to include much background. As such, the GSI papers might not be a great place to start learning IG. If there is a title in the GSI reports that catches your eye, you might want to see if the authors have an expanded version of the paper somewhere else.

Trying to provide an overview of information geometry might be a bit ambitious for a MathOverflow answer, but one interesting research area is to understand the interaction between optimal transport and information geometry. IG and OT provide two different ways to measure the ``distance" between probability distributions, with the former based off of transport and the latter based off of entropy. Although these notions are genuinely different, there are some interesting relationships between them. Firstly, there are various inequalities that can be proven (for instance the "HWI inequalities" of Otto and Villani). Furthermore, you can interpolate between these two notions of distance, which leads to the Hellinger-Kantorovich distance [2] (which was independently discovered by two other groups in 2015). To give one more example, in stochastic portfolio theory it is worthwhile to study optimal transport on a statistical manifold where the cost is a divergence function (i.e. optimal transport where the individual points already correspond to probability density functions). This is the topic that I'm currently studying, and it seems to be an active area of research.

[1]: Ay, Nihat; Jost, Jürgen; Lê, Hông Vân; Schwachhöfer, Lorenz, Information geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 64. Cham: Springer (ISBN 978-3-319-56477-7/hbk; 978-3-319-56478-4/ebook). xi, 407 p. (2017). ZBL1383.53002.

[2]: Liero, Matthias; Mielke, Alexander; Savaré, Giuseppe, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Invent. Math. 211, No. 3, 969-1117 (2018). ZBL1412.49089.

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Best way to introduce information geometry is to use work of Jean-Louis koszul on geometry of sharp convex cones https://link.springer.com/chapter/10.1007/978-3-030-02520-5_12

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