arXiv research

Simple lifts of non-simple curves on surfaces.

In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia, and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised s…

This paper extends braid lifting to coloured braid groupoids for all simple disc covers.

Characterizes covers using simple closed curves on surfaces.

Proves conjecture about surface cover homology.

Suppose $S$ is a closed orientable surface and $\tilde{S}$ is a finite sheeted regular cover of $S$. The following question was posed by Julién Marché in Mathoverflow: Do the lifts of simple curves from $S$ generate $H_{1}(\tilde{S},\mathbb{Z})$? A family of examples is given for which the answer is "no".

Lifts isometries in orbit spaces for compact groups.

Researchers lift knot coloring polynomial to Habiro ring.

Unified complexity bound for sampling logconcave distributions

We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps $φ:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}…

The paper studies which branched covers can be lifted to braided embeddings.

We prove that for each sufficiently complicated orientable surface $S$, there exists an infinite image linear representation $ρ$ of $π_1(S)$ such that if $γ\inπ_1(S)$ is freely homotopic to a simple closed curve on $S$, then $ρ(γ)$ has finite order. Furthermore, we prove that given a sufficiently complicated orientable…

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

We define compactifications of vector spaces which are functorial with respect to certain linear maps. These "many-body" compactifications are manifolds with corners, and the linear maps lift to b-maps in the sense of Melrose. We derive a simple criterion under which the lifted maps are in fact b-fibrations, and identi…

Let $γ$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $ρ$ for which $γ$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreov…

Transports along path in fibre bundles are axiomatically introduced. Their general functional form and some their simple properties are investigated. The relationships of the transports along paths and lifting of paths are studied.

We provide linear lower bounds for $f_ρ(L)$, the smallest integer so that every curve on a fixed hyperbolic surface $(S,ρ)$ of length at most $L$ lifts to a simple curve on a cover of degree at most $f_ρ(L)$. This bound is independent of hyperbolic structure $ρ$, and improves on a recent bound of Gupta-Kapovich. When $…

Mining association rules is an important technique for discovering meaningful patterns in transaction databases. Many different measures of interestingness have been proposed for association rules. However, these measures fail to take the probabilistic properties of the mined data into account. In this paper, we start …

New method finds unbranched covers with non-kernel homology.

New constructions and examples from moduli spaces.

Random curves on surfaces have predictable properties as they grow.

For a generic degree d smooth map f: N^n -> M^n we introduce its "transverse fundamental group" π(f), which reduces to π_1(M) in the case where f is a covering, and in general admits a monodromy homomorphism π(f) -> S_{|d|}; nevertheless, we show that π(f) can be non-trivial already for rather simple degree 1 maps S^n …

Existence and rigidity results for lifts in Carnot groups.

Study lift metrics and connections on tangent bundles of Riemannian manifolds.

Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. Thi…

Mathematical analysis of Prytz planimeter using sub-Riemannian geometry.

We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such a Higgs bundle is equivalent to a certain class of solutions to the affine Toda equations. We further explain thi…

In this paper, we define a complete lift for semisprays. If $S$ is a semispray on a manifold $M$, its complete lift is a new semispray $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond to the Jacobi fields for $S$, and second, this complete lift generalizes and unifies previ…

Unified approach to constructing integrable systems using Stäckel lifts.

Geometric structures are lifted to higher tangent bundles preserving statistical properties.

The article classifies liftings of connections on differential manifolds for geodesic modeling.

Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.

In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^…

We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…

The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.

In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff($M$)-equivariant pencil liftings for first order operators,…

Study covers of sphere with homeomorphisms lifting property.

In this paper we study the homeomorphisms of the disk that are liftable with respect to a simple branched covering. Since any such homeomorphism maps the branch set of the covering onto itself and liftability is invariant up to isotopy fixing the branch set, we are dealing in fact with liftable braids. We prove that th…

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $ζ$-invariants using lifted …

Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.

The paper constructs Sasakian lifts from Kähler manifolds and studies their properties.

New quantum code lacks sparse lift.

Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.

We construct some lift of an almost complex structure to the cotangent bundle, using a connection on the base manifold. This generalizes the complete lift defined by I.Sato and the horizontal lift introduced by K.Yano and S.Ishihara. We study some geometric properties of this lift and its compatibility with symplectic …

We give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is…

Proves harmonicity equivalence on manifold metrics.

In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…

The Eisenhart lift connects Hamiltonian systems to geodesics in pp-wave spacetimes.