I'm a master student and I have read "Monotonicity of the volume of intersection of balls" by Gromov and it was a great experience. When trying to fill the gaps, I often end up finding some very beautiful ideas. I want to keep reading Gromov because he inspires me and was delighted by the ideas that he show on that article, however, I don't know what articles of him are also suitable for a master student. I know some functional analysis, have taken a graduate course on differential geometry, measure theory, complex analysis, commutative algebra, algebraic topology, algebraic geometry, dynamical systems, fourier analysis on number fields and embeddings of finite spaces.

What Gromov's articles do you recommend to read?

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    $\begingroup$ Are you interested in research articles proving a specific theorem, or more sweeping expository accounts? His essay on manifolds (ihes.fr/~gromov/wp-content/uploads/2018/08/…) fits the latter bill. $\endgroup$ Aug 9, 2020 at 17:32
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    $\begingroup$ I'm interested in research articles proving a specific theorem. $\endgroup$
    – HeMan
    Aug 9, 2020 at 18:22
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    $\begingroup$ Gromov's articles are hard reading. You can probably read some parts of his book Metric structures of Riemannian and non-Riemannian spaces. $\endgroup$ Aug 10, 2020 at 14:07

1 Answer 1


I agree with Alexandre Eremenko in that most can be hard to read. But I think you can get a lot out of trying to understand them, as long as you're willing to black-box certain parts which may be inaccessible. Here's a summary of a few major articles. At the least, they all have significant parts which may be understandable, depending on your specific background.

"Curvature, diameter and Betti numbers" is a foundational contribution to Riemannian manifolds of positive sectional curvature, saying essentially that they cannot be arbitrarily topologically complicated. The tool is a Morse theory for the distance function. If you take the Toponogov theorem as given, it should be accessible with some understanding of homology; there is a review of spectral sequences in an appendix. I think it doesn't require too much Riemannian geometry other than the statement of the Toponogov theorem.

"Groups of polynomial growth and expanding maps" has some technical content, but the core idea (in section 6) is purely to do with metric spaces. It is applied by viewing finitely generated groups as examples of metric spaces. I don't have a good understanding of all of the group theory needed to prove the "main theorem", such as the Tits alternative, but the essential ideas of the proof, and many of the technical parts, are interesting (in and of themselves) and accessible. The key idea is to rescale the metric space structure of a finitely generated group to get a new metric space whose group of isometries is a Lie group; this "discrete" to "continuous" passage is somewhat unexpected and fundamental, and is the main point of interest for me.

"A.D. Aleksandrov spaces with curvatures bounded below" (with Yuri Burago and Grigori Perelman) is also largely to do with metric spaces. The idea is to reconstruct some of the theory of sectional curvature in Riemannian geometry by only using a metric space structure. Many parts of it should be accessible, especially with the book "A course in metric geometry" by Burago-Burago-Ivanov as a companion.

"Filling Riemannian manifolds" is very long, and I've never tried to read the whole thing, as it can be read in different parts, and there are already interesting and accessible ideas in the first few sections, to do with "filling radius" and "filling volume" and its relation to systolic inequalities; the only main prerequisite to begin with is simplicial homology. Interestingly a similar filling radius was studied at the same time in a different context in Schoen & Yau's "The existence of a black hole due to condensation of matter"

"The classification of simply connected manifolds of positive scalar curvature" (with Blaine Lawson) proves that certain topological operations on a manifold preserve the existence of Riemannian metrics with positive scalar curvature. Gromov and Lawson prove it by hand by basically elementary means. I've heard that it may have an error in it, but I don't know where it's supposed to be (I've never read it carefully). Schoen & Yau earlier proved the same result in "On the structure of manifolds with positive scalar curvature" by PDE methods that are simpler for me. Taking that theorem as given, Gromov and Lawson give some topological corollaries in cobordism theory that don't appear in Schoen & Yau - this part is too advanced for me.

"Convex symplectic manifolds" (with Yakov Eliashberg) is quite accessible, I recall it not needing too much past standard differential geometry. I think it could definitely be read with any of the standard introductory symplectic geometry textbooks as a companion. The main theorem says that if there is some weak algebraic equivalence between symplectic manifolds, then provided that they are "convex" in Eliashberg and Gromov's sense, one can construct certain symplectomorphisms which realize the equivalence.

The survey article "Spaces and questions" is also interesting to look through. There are also a number of interesting articles on Gromov's website in the non-pure math sections, such as "Mendelian Dynamics and Sturtevant’s Paradigm" and "Crystals, proteins, stability and isoperimetry". Based on your interest in "inspiring and delightful," and distinct from the standard math genre, I would also recommend some of his "ergo" articles like https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/ergo-cut-copyOct29.pdf.


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