A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature
We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.
math.DG↗
arXiv subjects
Misha Gromov
Publications and source records attributed to Misha Gromov.
Morse Spectra, Homology Measures and Parametric Packing Problems
We suggest several mathematical counterparts to the idea of "effective degrees of freedom" and formulate specific questions, much of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in balls.
math.MG↗
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.
math.GT↗