Related papers: Lower bounds on the $g$-numbers of spheres without…

We prove that if G is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We also construct a sequence { G_d } of graphs such that this homology is non-trivial in…

This paper follows a previous one in which were introduced deformation invariants $\chi^d_r$, $d \in H_2 (X ; \Z)$, $r \in \N$, of closed real symplectic four-manifolds $(X, \omega, c_X)$, invariants which produced lower bounds in real…

Given a complete Riemannian metric of nonnegative scalar curvature on $\Sigma \times (-\infty, 0 ] $, where $\Sigma$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the…

An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$ points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly $d+1$ points of the set.…

A simplicial sphere $\Delta$ is said to be generically anisotropic over a field $\mathbb{F}$ if, for a certain purely transcendental field extension $\mathbf{k}$ of $\mathbb{F}$, a certain Artinian reduction $A$ of the face ring…

We study the problem of maximizing the minimal value over the sphere $S^{d-1}\subset \mathbb R^d$ of the potential generated by a configuration of $d+1$ points on $S^{d-1}$ (the maximal discrete polarization problem). The points interact…

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced…

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of…

Let $\mu_{\text{2n}}(d,v)$ (respectively, $\mu^{\text{s}}_{\text{2n}}(d,v)$) be the minimal number of facets of a (simplicial) 2-neighborly $d$-polytope with $v$ vertices, $v > d \ge 4$. It is known that $\mu_{\text{2n}}(4,v) = v (v-3)/2$,…

In 2009, Roeglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number $n$ of variables and the maximum density $\phi$ of the semi-random…

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its…

In this article, we introduce and investigate polynomial curvelets on spheres, which form a class of Parseval frames for $L^2(\mathbb{S}^{d-1})$, $d \geq 3$. The proposed construction offers a directionally sensitive multiscale…

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to…

Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure:…

We construct examples of geometrically decomposable aspherical 4-manifolds with non-zero signature. We show that all such 4-manifolds satisfy the inequality (of Bogomolov--Miyaoka--Yau type) $\chi\geq 3|\sigma|$. We also construct examples…

In this article, we study the topological complexity of manifolds with a lower scalar curvature bound. We introduce a small scale index theorem to establish an upper bound for Gromov's simplicial norm of the Poincar\'e dual of the A-hat…

Suppose that $S^n$ is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real…

In this paper, we consider the class $\mathcal{C}^d$ of sphere intersection graphs in $\mathbb{R}^d$ for $d \geq 2$. We show that for each integer $t$, the class of all graphs in $\mathcal{C}^d$ that exclude $K_{t,t}$ as a subgraph has…

A well known Euler's formula consequence's corollary in graph theory states that: For a connected simple planar graph with $n$ vertices and $m$ edges, and girth $g$, we have $m \leq \frac{g}{g-2}(n-2)$. We show that a connected simple plane…

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations…