arXiv research

The study of orthospectrum and simple orthospectrum of hyperbolic surfaces.

New inequality links surface orthospectrum to boundary length.

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd integer n, there is a rational function q_n of degree 2(n-2) so that if M is a c…

We give a short derivation of an identity of Bridgeman concerning orthospectra of hyperbolic surfaces.

In this paper we consider finite volume hyperbolic manifolds X with non-empty totally geodesic boundary. We consider the distribution of the times for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the the first tw…

In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic $n$-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, ou…

We obtain asymptotic counting results with error terms for complex orthospectrum for Schottky groups and orbit counting function for quadratic polynomials. Moreover, we prove equidistribution of holonomy associated to these dynamical systems. Our results are obtained by considering generalized $L$-functions coming from…

Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.

In this paper we will study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment…

Study geodesics on flat tori, focusing on convex bodies.

In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…

Simple lifts of non-simple curves on surfaces.

The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.

3D hyperbolic spaces have endless simple paths.

Most simple braids have positive topological entropy.

The paper characterizes simple closed curves on surfaces using profinite rigidity.

Simple loop conjecture proven for certain discrete representations.

Study geodesics on spherical polyhedra, estimating their number.

We give a new proof of McShane's classification of simple cuspidal geodesics, using simple equivariant methods in the hyperbolic plane.

Study real logarithms of semi-simple matrices, focusing on differential structure.

The paper explores simple and relatively simple transformation groups and their universal coverings.

A version of Dehn's algorithm for simple diagrams on a once punctured surface representing simple diagrams on a closed surface is presented

Rotation systems can't always be drawn in surfaces.

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

The paper finds braid representatives minimizing simple walks for knots.

Geometric models for Lie algebras from simple singularities.

We characterize the kernel of the mixed ray transform on simple $2$-dimensional Riemannian manifolds, that is, on simple surfaces for tensors of any order.

Study finds counterexamples to simple loop conjecture in higher dimensions.

Knowledge graphs contain knowledge about the world and provide a structured representation of this knowledge. Current knowledge graphs contain only a small subset of what is true in the world. Link prediction approaches aim at predicting new links for a knowledge graph given the existing links among the entities. Tenso…

New concept of k-holes in simple drawings and convex drawings.

Proves simple type conjecture for mod 2 Seiberg-Witten invariants.

Simple knots in lens spaces fiber if their order doesn't divide certain Euclidean remainders.

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.

The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The…

All closed geodesics are simple and non-intersecting in dimensions 3 and above.

Mirzakhani obtained the asymptotic growth, when $L\to\infty$, of the number of curves in the mapping class group orbit of some given simple curve and with length at most $L$. Years later she extended this result from simple to arbitrary curves. Here we give a short and relative low-tech argument showing how to derive t…

We characterize absence of arbitrage with simple trading strategies in a discounted market with a constant bond and several risky assets. We show that if there is a simple arbitrage, then there is a 0-admissible one or an obvious one, that is, a simple arbitrage which promises a minimal riskless gain of ε, if the inves…

Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.

Characterizes covers using simple closed curves on surfaces.

We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.

Study on frequencies of non-simple curves in surfaces of large genus.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

The paper classifies geodesic orbit spaces with simple isotropy groups.

Study on functional inequalities on simple edge spaces.

Curvature criteria for A-simple singularities and their parallel curves identified.

Simple geodesics on spherical tetrahedra identified for specific angles.

New examples of manifolds that are homotopy but not simple homotopy equivalent.