Misha Gromov

Spaces and Questions

Our Euclidean intuition, probably inherited from ancient primates, might have grown out of the first seeds of geometry in the motor control systems of early animals who were brought up to sea and then to land by the Cambrian explosion half a billion years ago. The primates brain had been idling for 30–40 million years. Suddenly, in a flash of one million years, it exploded into growth under the relentless pressure of sexual-social competition and sprouted a massive neocortex (70% neurons in humans) with an inexplicable capability for language, sequential reasoning and generation of mathematical ideas. Then Man came and laid down space on papyrus in a string of axioms, lemmas and theorems around 300 B.C. in Alexandria.

Entropy and isoperimetry for linear and non-linear group actions

We prove inequalities of isoperimetric type for groups acting on linear spaces and discuss related geometric and combinatorial problems, where we use the Boltzmann entropy to keep track of the cardinalities (and/or measures) of sets and of the dimensions of linear spaces.

Cuts in Kähler Groups

We study fundamental groups of Kahler manifolds via their cuts or relative ends. Mathematics Subject Classification (2000). 32Q15, 20F65, 57M07.

Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.

Generalizations of the Kolmogorov-Barzdin embedding estimates

We consider several ways to measure the `geometric complexity of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness, based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.

Crystals, proteins, stability and isoperimetry

We attempt to formulate several mathematical problems suggested by structural patterns present in biomolecular assemblies. Our description of these patterns, by necessity brief and over-concentrated in some places, is self-contained, albeit on a superficial level. An attentive reader is likely to stumble upon a cryptic line here and there; however, things will become more transparent at a second reading and/or at a later point in the article.

Plateau-Stein manifolds

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.

Hilbert volume in metric spaces. Part 1

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov M., Plateau-hedra, Scalar Curvature and Dirac Billiards, in preparation].

Super Stable Kählerian Horseshoe

Following the prints of Smale’s horseshoe, we trace the problems originated from the interface between hyperbolic stability and the Abel-Jacobi-Albanese construction.

A Dozen Problems, Questions and Conjectures About Positive Scalar Curvature

Unlike manifolds with positive sectional and with positive Ricci curvatures which aggregate to modest (roughly) convex islands in the vastness of all Riemannian spaces, the domain ({mathcal{SC}>0}) of manifolds with positive scalar curvatures protrudes in all direction as a gigantic octopus or an enormous multi-branched tree. Yet, there are certain rules to the shape of ({mathcal{SC}>0}) which limit the spread of this domain but most of these rules remain a guesswork. In the present paper we collect a few “guesses” extracted from a longer article, which is still in preparation: 100 Questions, Problems and Conjectures around Scalar Curvature. Some of these “guesses” are presented as questions and some as conjectures. Our formulation of these conjectures is not supposed to be either most general or most plausible, but rather maximally thought provoking.