arXiv research

Study on moduli spaces of branched projective structures on surfaces.

Study continuous deformations of branched projective structures on surfaces, preserving holonomy and branch points.

Proofs for decomposing branched affine surfaces into triangles and cylinders.

Authors prove a contact structure result using branched covers and overtwisted disks.

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover…

Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.

We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projecti…

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-…

Characterizes representations for complex projective structures with specific branch data.

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary ob…

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that…

StrTransformer recovers sources without labels by optimizing latent matrices and enforcing structural constraints.

New CR structures found for once-punctured torus bundles.

Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$ with boundary admits a Stein structure if and only if $X$ is a simple branched cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$, we constr…

Survey on generalized holomorphic Cartan geometries.

Ideal triangulations of 3-manifolds are shown equivalent up to certain moves.

Study on minimal disks in metric spaces, focusing on branch set structure.

Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khova…

We define sink marks for branched complexes and find conditions for them to determine a branched surface structure. These will be used to construct branched surfaces in knot and tangle complements. We will extend Delman's theorem and prove that a Montesinos knot $K$ of length at least 3 has a persistently laminar branc…

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem …

A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov homology. This provides two advantages: 1) the results hold for integer coefficient…

Simply-connected 4-manifolds are covered by products with a 2-torus.

The abstract discusses how sub-Riemannian manifolds can have branching geodesics.

Let $p:Σ'\toΣ$ be a finite Galois cover, possibly branched, with Galois group $G$. We are interested in the structure of the cohomology of $Σ'$ as a module over $G$. We treat the cases of branched and unbranched covers separately. In the case of branched covers, we give a complete classification of possible module stru…

For a given quasi-Fuchsian representation $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a closed surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $ρ$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the sa…

The study examines the regularity of branched immersions using special coordinate systems.

The paper studies geometric structures of polynomial spaces.

Framework separates chemical and structural contributions to aqueous solubility.

A framework separates chemical and structural contributions to aqueous solubility.

A contamination in a 3-manifold is an object interpolating between the contact structure and the lamination. Contaminations seem to provide a link between 3-dimensional contact geometry and the classical topology of 3-manifolds, as described in a separate paper. In this paper we deal with contaminations carried by bran…

Proves inequality in metric measure spaces with non-branching structure.

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.

In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the categorification of a multiple of the Turaev torsion in the case where the m-fol…

Finite groups can be automorphism groups of translation surfaces with poles.

Automated multi-task learning algorithm that optimizes network topology.

Efficiently models tree-like data with coupled branches using HMMs.

New results on hypersurfaces show no branch points, improving smoothness.

New method identifies causal structure in count data using cumulants and path analysis.

A universal branched 3-manifold $W$ characterizes Sol 3-manifolds.

The paper studies triangulations of surfaces up to branched transit equivalences, proving equivalence conditions and foliations.

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and c…

Minimal surfaces with complex branching structures constructed using various methods.

Analyzes branch points of area-minimizing currents with non-2 planar frequency.

Smooth fractal trees via analytic generators, preserving combinatorial and geometric properties.

This paper proposes a new dimensionality reduction algorithm named branching embedding (BE). It converts a dendrogram to a two-dimensional scatter plot, and visualizes the inherent structures of the original high-dimensional data. Since the conversion part is not computationally demanding, the BE algorithm would be ben…

Isometric embeddings of Teichmüller spaces are derived from branched coverings.

We classify, up to few exceptions, the orbit closures of the $\mathrm{Mod}(Σ)$-action on the affine character variety $χ(\mathrm{Aff}(\mathbb{C}))$. We obtain from this classification that the only obstruction for a non-abelian representation $ρ: π_1 Σ\longrightarrow \mathrm{Aff}(\mathbb{C})$ to be the holonomy of a br…

In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second …