Related papers: Extremal examples of collapsible complexes and ran…

The number of faces of the convex hull of $n$ independent and identically distributed random points chosen on the boundary of a smooth convex body in $\mathbb{R}^d$ is investigated. In dimensions two and three the number of $k$-faces is…

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs.…

We show, among other things, that for each integer $n \ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic…

We consider a finite simplicial complex $K$ together with its successive barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower bounds for…

Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. We compute an explicit…

We introduce the \emph{Morse ensemble polynomial} $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$,…

The \emph{strong collapse} of a simplicial complex, proposed by Barmak and Minian (\emph{Disc. Comp. Geom. 2012}), is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational…

We construct $d$-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with $n$ vertices whose combinatorial diameter grows as $c_d n^{d-1}$ for a constant $c_d$ depending only on $d$, which is the maximum possible…

We study bootstrap percolation processes on random simplicial complexes of some fixed dimension $d \geq 3$. Starting from a single simplex of dimension $d$, we build our complex dynamically in the following fashion. We introduce new…

In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous…

We investigate simplicial complexes deterministically growing from a single vertex. In the first step, a vertex and an edge connecting it to the primordial vertex are added. The resulting simplicial complex has a 1-dimensional simplex and…

Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as…

In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete…

The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…

We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group…

We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$…

Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this…

We give a counterexample to a conjecture of S.E. Morris by showing that there is a compact plane set X such that R(X) has no non-zero, bounded point derivations but such that R(X) is not weakly amenable. We also give an example of a…

In contrast with the 3-dimensional case (cf. \cite{RaMo}), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally…

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex…