Pathway to Gromov's works My question is really simple and short: I want to understand Gromov's works. Assuming I have background in basic abstract algebra (groups, rings, fields, etc.), general topology and analysis (the last two from Munkres and Rudin respectively), what would be a pathway to study Gromov's works? Kindly mention reference books and/or lecture notes.
 A: What follows is a personal opinion based on trying to read Gromov's writings.  Others will have different ideas about how to approach his work. As a first step, one needs to appreciate that the goal to "understand Gromov's works" is extremely ambitious.
A good place to start understanding Gromov's ideas is his webpage, where he posts nearly everything he has written that is not a book (and some of his book). He appears to be a believer in the unfettered diffusion of ideas. On his webpage he organizes his work thematically in 16 categories. The "Expository" section (and the "recent" subcategory) is a good starting point. His own breakdown of the major themes of his work is as good as any. Among the several major themes he has explored:

*

*distance geometry

*positive scalar curvature

*h-principle

*hyperbolicity (infinite groups)

*symplectic manifolds, pseudoholomorphic curves

*etc...

He has written books or book length manuscripts about each of these and others.
His productivity may be hard to appreciate fully if one has not already tried to develop some ideas and write a few papers explaining them. To appreciate the scope of his work - in my opinion many professional mathematicians could make a happy career out of developing just one of these themes, and many have. (Of course there are other mathematicians with interests and talents as broad as Gromov's, and there are those who understand a lot more of his work than some of us, but I am thinking more from the perspective of ordinary mediocre hacks like me).
Reading Gromov is a wonderful experience because of the flood of ideas in his writings. It is a difficult experience for many reasons. He assumes a lot of knowledge and technique. One should know the best theorems on the existence of solutions to the Beltrami equations, or the results on the size of singular sets of minimal hypersurfaces. The reader is assumed to know little things like these, or the Mostow rigidity theorem, sometimes without any explicit mention. In his writing he doesn't always discriminate well between what I consider elementary and what I consider deep (sometimes this reflects part of his basic strategy, which is to rethink the premises of the basic lines of investigation; he is more interested in general phenomena than in specific concrete examples). So he spends time defining metric spaces, and then states as obvious a characterization of the scalar curvature as the unique function satisfying bla and bla that one can't find in the literature (one is supposed to check it oneself, but sometimes this is not so easy). His papers and books are not always well edited and sometimes contain statements that are false, or not what he meant to write, although he is usually right in his speculations, and is more careful (particularly where it matters) than many seem to think. He does not always clearly distinguish (in an expository sense - in his head surely he does) between what is speculation and what is proved. His exercises for the reader become research papers in someone else's hands. His goal in exposition seems to be to communicate ideas and approaches rather than details, which he assumes the reader competent to supply. This supposition is sometimes unfair to the reader, but this is not his fault.
There are few other mathematicians competent to really assess all of his work (certainly I am not one of them), and there is a lot in his writings that is not fully assimilated still. For example, it took several decades for the h-principle to become a standard tool, with independent textbook expositions.
What is the best starting point outside of his own writings depends on which theme you want to study. For example:

*

*For h-principle there is a book of Eliashberg and Mishachev. For geometric integration theory there is a book of David Spring. (His own book on these topics is very hard.)

*For distance geometry, start with Gromov's own book Metric Structures for    Riemannian and Non-riemannian Spaces, which is probably his most accessible    writing. This area has seen an explosion of work and there are a lot    of other accounts. Which is best depends on specific interests. If    you are interested in nonpositively curved metric spaces, then maybe    something like the book of Bridson and Haefliger. If you are    interested in hyperbolic groups, there are other sources. For metric    measure spaces and concentration of measure, there are others.

*For symplectic manifolds and pseudoholomorphic curves there are the two    books of McDuff and Salamon.

*For Stein manifolds, there is the book of Cieliebak and Eliashberg.

*For positive scalar curvature there is no book form account that I know. One should read, on the one hand, the papers of Gromov and Lawson, and, on the other, the papers of Schoen and Yau (there are 4 or 5 to read in either case). The fundamental thing in this area is to understand how the two    approaches, via Dirac operators, and via minimal hypersurfaces, are    related. There are also surveys by Rosenberg and papers of Stolz.

*His essay on the Sign and Geometric meaning of curvature is an excellent source for any student of differential geometry.

*etc.

The part of his work most accessible to someone with standard undergraduate level preparation is probably the part related to metric geometry and hyperbolicity. This requires somehow less technical machinery than some of the other areas. There are many textbook expositions of related material.
Some of the basic background needed - real analysis, geometric measure theory, elliptic PDE, functional analysis, complex function theory, convex function theory, multivariable complex analysis, probability, differential geometry, harmonic mappings, algebraic topology, differential topology (Morse theory), group theory, Lie theory, dynamical systems. Algebraic geometry, number theory, combinatorics, are not as necessary (nothing hurts); he doesn't use much representation theory explicitly.
A personal sentiment: as for Thurston, it is impossible to appreciate much of his work if one does not know the full sweep of classical (geometric) complex function theory, including things like Nevanlinna theory and quasisymmetric functions. For example, ideas coming from this area permeate his writings about symplectic geometry, Stein manifolds, curvature, and things in between like his recent papers about plateau-stein manifolds.
