I apologize in advance if this question is too general, that is, not a request for a specific reference, but more of a request for a road map, perhaps from someone that knows the material and, in retrospect, can see what knowledge was necessary for them in their development. I also apologize for the lack of brevity.
I am asking from the perspective of someone with only an undergraduate mathematics degree and a small amount of slightly more advanced self-study in differential geometry and topology that would like to work toward an understanding of the second half of Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio, Gigli and Savare. I am aware that this is an advanced area that relies on many others and thus the present question.
I have read the post on mathoverflow: https://mathoverflow.net/questions/43083/textbooks-or-notes-on-gradient-flows-in-metric-spaces as well as the post: Gradient flows in metric spaces on stackexchange. Additionally, I have taken taken a look at the notes linked to in the above, some of which is accessible to me and some of which is not. I feel that I am sufficiently distant skill-wise from what is needed to read the material that textbooks are probably the first place that I want to head. It has been pointed out, for example, that Optimal Transport, old and new by Cédric Villani is useful and have begun to take a look at that. This is a good example however as, optimal transport itself touches a diversity of mathematical areas.
Investigating the references (on Arxiv), I finds a wide array of core mathematics that are used by the authors in referenced material: analysis and ODEs, differential/Riemannian geometry, metric geometry, analysis of PDEs, probability... Related to these are other areas, e.g., geometric measure theory, which is itself a good example of the interdependance of material (differential geometry, measure theory, calculus of variations, analysis). This interdependence, while exciting to the uninitiated, does not allow for a quick understanding of what, exactly, the prerequisites are and what the "prerequisites to the prerequisites" may be in some limited cases.
Finally, in related/referenced work, one also sees technical material accessible mainly to experts in say geometry, e.g., Gromov's Metric Structures for Riemannian and Non-Riemannian Spaces which I bring up as an example because in this case, it would be great for someone to provide a pointer to A Course in Metric Geometry by Burago^2 and Ivanov. Ideally, that is the kind of information I am looking for.
For example responses along the lines of, "Well, the book and related research rely heavily on results from metric geometry, some of which are available only in advanced monographs. A good introduction to the field is A Course in Metric Geometry." Or perhaps responses like, "For the PDE-related material, a thorough understanding of Evan's book is sufficient." Or, "Check out Stroock's books on probability, analysis and PDE."
I have found the notes, blibiography and even the preliminary remarks and program from the course at https://www.math.leidenuniv.nl/~vangaans/topicsinanalysis2011.html to be helpful in beginning to situate myself. It seems there is material from analysis, measure theory, probability, optimal transport, ODE, PDE, stochastic processes and other fields that must be mastered in order to really engage with this material.
As I am beginning this process as a self-study to hopefully be continued in graduate school, I am really just looking for a list of prerequisite math, good references to learning those areas, and tips on the knowledge that one must accumulate in order to understand the above book (especially the second half) at the level of applying it and reading related papers in the field. It would be very helpful if this advice came from those with direct experience learning these ideas.
Thank you in advance for any help!