We show that a complete [Formula: see text]-dimensional Riemannian manifold text] with finitely generated first homology has macroscopic dimension if it satisfies the following “macroscopic curvature” assumptions: every ball of radius in volume at most text], and loop is null-homologous concentric text].

Abstract We study the tree scattering amplitudes of Yang–Mills and General Relativity as functions complex momenta, using a homological geometrical approach. This approach uses differential graded Lie algebras, one for YM GR, whose Maurer Cartan equations are classical field equations. The obtained $$L_\infty $$ L</mml...

We study conditions under which a finite simplicial complex K can be mapped to ?d without higher-multiplicity intersections. An almost r-embedding is map f: ? such that the images of any r pairwise disjoint simplices do not have common point. show if prime power and d ? 2r + 1, then there counterexample topological Tverberg conjecture, i.e., an (d +1)(r ? 1)-simplex in ?d. This improves on prev...