Gromov’s macroscopic dimension conjecture

Macroscopic Dimension and Essential Manifolds

M. Gromov asked whether the macroscopic dimension of rationally essential n-dimensional manifolds equals n. We show that the answer depends only on the corresponding group homology class and give an affirmative answer for certain classes. In particular, the answer is positive for manifolds with amenable fundamental groups.

A short proof of the maximum conjecture in CR dimension one

In this paper and by means of the extant results in the Tanaka theory, we present a very short proof in the specific case of CR dimension one for Beloshapka's maximum conjecture. Accordingly, we prove that each totally nondegenerate model of CR dimension one and length >= 3 has rigidity. As a result, we observe that the group of CR automorphisms associated with each of such models contains onl...

The Manin Conjecture in Dimension 2

Scaled Dimension and the Berman-Hartmanis Conjecture

In 1977, L. Berman and J. Hartmanis [BH77] conjectured that all polynomialtime many-one complete sets for NP are are pairwise polynomially isomorphic. It was stated as an open problem in [LM99] to resolve this conjecture under the measure hypothesis from quantitative complexity theory. In this paper we study the polynomial-time isomorphism degrees within degm(SAT ) in the context of polynomial ...

On Keller's Conjecture in Dimension Seven

A cube tiling of R is a family of pairwise disjoint cubes [0, 1) + T = {[0, 1) + t : t ∈ T} such that ⋃ t∈T ([0, 1) d + t) = R. Two cubes [0, 1) + t, [0, 1) + s are called a twin pair if |tj − sj | = 1 for some j ∈ [d] = {1, . . . , d} and ti = si for every i ∈ [d] \ {j}. In 1930, Keller conjectured that in every cube tiling of R there is a twin pair. Keller’s conjecture is true for dimensions ...