arXiv:1908.10612v6 [math.DG] 8 Jul 2021 Page 1 arXiv:1908.10612v6 [math.DG] 8 Jul 2021
Four Lectures on Scalar Curvature Misha Gromov July 9, 2021 Unlike manifolds with controlled …

Let G be a connected semisimple Lie group such that the associated symmetric space X is
Hermitian and let Gamma be the fundamental group of a compact orientable surface of …

A selection of aspects of the theory of bounded cohomology is presented. The emphasis is
on questions motivating the use of that theory as well as on some concrete issues suggested …

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his
pioneering paper Volume and bounded cohomology. In order to study the main properties of …

The amenable category is a variant of the Lusternik–Schnirelman category, based on covers
by amenable open subsets. We study the monotonicity problem for degree-one maps and …

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 …

A well-known question by Gromov asks whether the vanishing of the simplicial volume of
oriented closed aspherical manifolds implies the vanishing of the Euler characteristic. We …

New constructions in group homology allow us to manufacture high-dimensional manifolds
with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of …

We provide new computations in bounded cohomology: A group is boundedly acyclic if its
bounded cohomology with trivial real coefficients is zero in all positive degrees. We show …

We study PL bordism theories from a quantitative perspective. Such theories include those
of PL manifolds, ordinary homology theory, as well as various more exotic theories such as …