Related papers: Bounds on Systoles and Homotopy Complexity

Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that…

We provide a reduction in the classification problem for non-compact, homogeneous, Einstein manifolds. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the…

I give a brief informal introduction to the idea and tests of large extra dimensions, focusing on the case in which the space-time manifold has a direct product structure. I then describe some attractive implementations in which the…

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by…

We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable…

We study some new isoperimetric inequalities on graphs. We etablish a relation between the volume entropy (or asymptotic volume), the systole and the first Betti number of weighted graphs. We also find bounds for the volume, associated to…

The paper describes a possible physical characterization for the definition of elementary fluid flow. As consequence, an analytical expansion based on Reynolds number powers for the velocity field is shown in the weakly turbulent case.

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an…

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good…

This expository article discusses some connections between the geometry of a hyperbolic 3-manifold homotopy-equivalent to a surface, and the combinatorial properties of its end invariants. In particular a necessary and sufficient condition…

Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with…

By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space $\mathbb H^{3}$ are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact right-angled polyhedra and a class of…

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable…

The aim of homotopy theory in topology is to simplify, after continuous deformation, continuous maps between topological spaces. What prevents this from happening are homotopy invariants. This raises quantitative questions: $\bullet$ Is the…

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by $-6$. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic $3$-space.

We introduce the notion of topological hyperbolicity to characterize the largeness of the topological fundamental group of a complex variety. Inspired by the Shafarevich conjecture, we propose to study the topological hyperbolicity of…

We develop a general method to calculate entropy numbers of standard Sobolev's classes on an arbitrary compact homogeneous Riemannian manifold. Our method is essentially based on a detailed study of geometric characteristics of norms…

Conjecture F from [VW12] states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the…

We establish the geometric Bogomolov conjecture for semiabelian varieties over function fields. We show a closed subvariety contains Zariski dense sets of small points, if and only if, after modulo its stabilizer, it is a torsion translate…