Related papers: Counting Frameworks of Bipyramids

In this paper the authors give an infinite series of rigid compact complex manifolds for each dimension $d \geq 2$ which are not infinitesimally rigid, hence giving a complete answer to a problem of Morrow and Kodaira stated in the famous…

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an…

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally…

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very…

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…

In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with…

The goal of this paper is to present a lower bound for the Mahler volume of at least 4-dimensional symmetric convex bodies. We define a computable dimension dependent constant through a 2-dimensional variational (max-min) procedure and…

We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear…

We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group,…

We prove that for $d\geq 3$, the 1-skeleton of any $(d-1)$-dimensional doubly Cohen Macaulay (abbreviated 2-CM) complex is generically $d$-rigid. This implies the following two corollaries (by Kalai and Lee respectively): Barnette's lower…

We propose a quantum version of the quadratic volume simplicity constraint for the EPRL spin foam model. It relies on a formula for the volume of 4-dimensional polyhedra, depending on its bivectors and the knotting class of its boundary…

We give an algebraic characterization of when a $d$-dimensional periodic framework has no non-trivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable, and we provide a simple algorithm that…

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume…

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include…

We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant…

In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed.

Combinatorial characterisations are obtained of symmetric and anti-symmetric infinitesimal rigidity for two-dimensional frameworks with reflectional symmetry in the case of norms where the unit ball is a quadrilateral and where the…

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds…

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…

We prove that every three-dimensional polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting, under two plausible sufficient conditions: (i) the polyhedron has only convex faces and…