arXiv research

The paper explores rational functions with 3 branching points on the Riemann sphere.

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible …

We consider surface branch data with base surface the sphere, odd degree d, three branching points, and two partitions of d of the form (2,...,2,1) and (2,...,2,2h+1). If the third partition has length L, this datum satisfies the Riemann-Hurwitz necessary condition for realizability if h-L is odd and at least -1. For s…

Paper solves the Hurwitz existence problem using fiber products.

In this work we study the decomposability property of branched coverings of degree $d$ odd, over the projective plane, where the covering surface has Euler characteristic $\leq 0$. The latter condition is equivalent to say that the defect of the covering is greater than $d$. We show that, given a datum $\mathscr{D}=\{D…

Rational maps structure theorem with geometric decomposition and realizability proof.

\noindent Given a Riemann surface $M$, the \emph{complexity} of a branched cover of $M$ to the Riemann sphere $S^2$, of degree $d$ and with branching set of cardinality $n \geq 3$, is defined as $d$ times the hyperbolic area of the complement of its branching set in $S^2$. A branched cover $p \colon M \to S^2$ of degre…

Solves weighted bi-colored plane tree enumeration and applies to geometric problems.

This paper studies three aspects around dimension datum: (1), a generalization of the dimension datum, which we call the tau-dimension datum; (2), dimension data of disconnected subgroups; (3), compactness of isospectral sets of normal homogeneous spaces.

In this paper we investigate the following existence problem for rational functions: for a given collection $Π$ of partitions of a number $n$ to define whether there exists a rational function $f$ of degree $n$ for which $Π$ is the branch datum. An important particular case when the answer to this problem is known is t…

Given a smooth bounded domain $Ø\subseteq \R^2$, we consider the equation $\D v = 2 v_x \wedge v_y$ in $Ø$, where $v: Ø\to \R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple…

Study of tori of revolution under Willmore flow converges to Clifford Torus.

We assume that a high-dimensional datum, like an image, is a compositional expression of a set of properties, with a complicated non-linear relationship between the datum and its properties. This paper proposes a factorial mixture prior for capturing latent properties, thereby adding structured compositionality to deep…

Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.

We prove that the Calabi-Yau equation on the Kodaira-Thurston manifold has a unique solution for every $S^1$-invariant initial datum.

Ancient solutions found for a specific flow on symplectic half-flat structures.

Proves existence of multi-phase flows from arbitrary initial data.

A Bayesian treatment of latent directed graph structure for non-iid data is provided where each child datum is sampled with a directed conditional dependence on a single unknown parent datum. The latent graph structure is assumed to lie in the family of directed out-tree graphs which leads to efficient Bayesian inferen…

Global existence and convergence of heat flow for p-harmonic maps.

The paper addresses errors in online selective conformal prediction and proposes new strategies to ensure valid inference.

Proves existence and uniqueness of curvature motion for regular networks.

Study shows diffused interface flows to single diffused balls over time.

We introduce a first order flow of $G_{2}$-structures and construct its explicit solution in case of a cone over $S^3\!\times\! S^3$. Also we prove for this situation that starting from certain initial datum the flow deforms corresponding to $G_2$-structure metric to a conic metric up to homotheties.

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^\infty$ minimal graphs with prescribed small boundary datum…

We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-M…

The paper studies non-Kähler LVMB manifolds and their metrics.

Study on moduli spaces of branched projective structures on surfaces.

We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are…

New criterion for branched covers between 2-spheres.

Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This …

Uniformly branching trees are equivalent to certain metric spaces.

Unique solution found for quaternionic Monge-Ampère equation on specific HKT manifolds.

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

We consider 3-dimensional pseudo-manifolds M with a given set of marked point V such that M-V is the interior of a compact 3-manifold with boundary. An ideal triangulation T of (M, V ) has V as its set of vertices. A branching (T, b) enhances T to a Delta-complex. Branched triangulations of (M, V ) are considered up to…

The paper details folding of branched covers of the 3-sphere over knots.

In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.

Course on knots using branched coverings.

A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched cover…

Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…

Noise stabilizes solutions to transport equations, preventing blow-up.

Formula compares metrics on branched coverings of line bundles.

A branched covering surface-knot is a surface-knot in the form of a branched covering over a surface-knot. For a branched covering surface-knot, we have a numerical invariant called the simplifying number. We show that branched covering surface-knots with degree three have the simplifying numbers less than three.

New examples show transverse knots are determined by their branched covers.

Study traveling waves in hyperbolic space for Fisher-KPP equations.

We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.

Quantized Coulomb branches linked to skein algebras.

Characterizes groups of branched twist-spun knots.