Related papers: Mogami manifolds, nuclei, and 3D simplicial gravit…
We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed. In the case of no stretching outside the fold, the three-dimensional shape of a single…
We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of…
It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two…
The number of apparent double points of an irreducible projective variety $X$ of dimension $n$ in $\mathbb{P}^{2n+1}$ is the number of secant lines to $X$ passing through a general point of $\mathbb{P}^{2n+1}$. This classical notion dates…
We extend the complete census of orientable cusped hyperbolic $3$-manifolds to $10$ tetrahedra, giving the next $150730$ manifolds and their $496638$ minimal ideal triangulations. As applications, we find the precisely $439898$ exceptional…
In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5,…
We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual…
Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this…
In this paper, we consider a class of Morse-Smale diffeomorphisms defined on a closed 3-manifold (non-necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest…
The main goal of this paper is to show that shellability is NP-hard for triangulated d-balls (this also gives hardness for triangulated d-manifolds/d-pseudomanifolds with boundary) as soon as d is at least 3. This extends our earlier work…
In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend…
Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding we highlight a new area where mock modular forms start to play…
We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has…
The monostatic property of polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in a focus of research ever since Conway and Guy \cite{Conway} published the proof of the existence of the…
We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the…
We determine the combinatorial types of all the 3-dimensional simple convex polytopes in R^3 that can be realized as mean curvature convex (or totally geodesic) Riemannian polyhedra with non-obtuse dihedral angles in Riemannian 3-manifolds…
Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a…
We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed…
We review an approach which aims at studying discrete (pseudo-)manifolds in dimension $d\geq 2$ and called random tensor models. More specifically, we insist on generalizing the two-dimensional notion of $p$-angulations to higher…
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures…