Geometry and the simplex: results, questions and ideas
type: Item
instance of: scholarly article
title: Geometry and the simplex: results, questions and ideas
zbMATH Open document ID: 1466.51010
DOI: 10.4171/NEWS/118/7
author: Karim Adiprasito
published in: European Mathematical Society Newsletter
publication date: 2021-02-23T00:00:00Z
review text: In \(\mathbb{R}^2\), Euler and Descartes showed: a planar graph without double edges or loops can have at most three times as many edges \(E\) (one-dimensional simplices) as it has vertices \(V\), i.e. \(E\le 3V\). The paper under review is in fact a survey telling about simplices, graphs, embedding problems, counting specific spaces, and the like. It gives a perfect illustration of the kind of things that are done in our days. Among others, the author mentions his theorem Theorem 1.1 Given a simplicial complex \(\Delta\) that piecewise-linearly embeds in \(\mathbb{R}^4\), then \(T\le 4E\), i.e. the number of triangles \(T\) (two-dimensional simplices) exceeds the number of edges \(E\) (one-dimensional faces) by a factor at most 4. The paper informs the reader about several results by the author and others. In addition, six problems and one conjecture are printed in clear sight. Let us give those! \begin{itemize} \item[1)] Do the results of Theorem 1.1 extend to topological embeddings? Do they extend to cell complexes that are strongly regular; that is, all complexes whose partially ordered set of faces is an atomistic lattice? \item[2)] Are there infinitely many combinatorially distinct types of polytopes of dimension 4 whose moduli space is a point? \item[3)] Is the number of contractible complexes with a given number of triangles \(n\) and a uniform bound on the vertex degree exponential, or superexponential? What happens if I only restrict to complexes with vanishing reduced homology? \item[4)] Given a triangulation of a connected \(d\)-manifold on \(n\) vertices, how many steps does one need from any facet to any other facet? \item[5)] Is it true that for every \(k=k(\ell)\) such that every \(d\)-polytope (or polyhedral sphere), \(d\ge k\), has a Boolean interval of length \(\ell\) in its face lattice? \item [6)] Is it true that for every \(p\ge 2\) there exists a \(k=k(p)\), such that every Riemannian metric \(R\) on \(S^d\), \(d\ge k\), has a point \(x\) and …
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51-02
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 7314412
zbMATH Keywords: simplicial complex
zbMATH Keywords: planar graphs
zbMATH Keywords: topological embeddings
zbMATH Keywords: face numbers
zbMATH Keywords: Poincaré duality
zbMATH Keywords: cell complexes
zbMATH Keywords: homology polyhedra
zbMATH Keywords: curvature
reviewed by: Robert W. van der Waall
describes a project that uses: FLAGTOOL
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7314412