Related papers: A Low-Dimensional Counterexample to the HK-Conject…

We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound.…

Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group $H_k$ for all $k\geq 3$, either it is flat or it has Gauss-Bonnet…

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky-Kimelfeld theorem does not hold. The…

We prove a sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group. Namely, let $G$ be a compact connected Lie group of dimension $d_G$, we show that for for all measurable subsets $A$, we have…

The article is devoted to the question whether the orbit space of a compact linear group is a topological manifold and a homological manifold. In the paper, the case of a simple three-dimensional group is considered. An upper bound is…

For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$…

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with…

We provide counterexamples to the stable equivalence problem in every dimension $d\geq2$. That means that we construct hypersurfaces $H_1, H_2\subset\mathbb{C}^{d+1}$ whose cylinders $H_1\times\mathbb{C}$ and $H_2\times\mathbb{C}$ are…

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact…

A riemannian manifold is secure if the geodesics between any pair of points in the manifold can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure,…

In 2017, Geordie Williamson proved the existence of counterexamples to James's conjecture on the decomposition matrices of symmetric groups and their Hecke algebras. The smallest counterexample detectable by Williamson's method occurs in…

In this paper we consider the class of graphs which are redundantly $d$-rigid and $(d+1)$-connected but not globally $d$-rigid, where $d$ is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It…

In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to…

In a Note added in proof to a 1984 paper, Daniel A. Marcus claimed to have a counterexample to his conjecture that a minimal positively k-spanning vector configuration in R^m has size at most 2km. However, the counterexample was never…

In this paper, we develop new ideas regarding the finitistic dimension conjecture, or the findim conjecture for short. Specifically, we improve upon the delooping level by introducing three new invariants called the effective delooping…

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $\Phi$ does not preserve a proper affine subspace, we show that one of the following holds: (1) the dimension is equal to the…

In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized…

In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces…

In this note we construct a closed 4-manifold having torsion-free fundamental group and whose universal covering is of macroscopic dimension 3. This yields a counterexample to Gromov's conjecture about the falling of macroscopic dimension.

Let $\Delta$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $\Delta$ is that $g_2$ of $\Delta$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $\Delta$ has…