arXiv research

Characterizes critical points in convex double and triple bubbles.

Study finds central points of double heptagon surface are not connection points.

Geometric arguments show simplicial arrangements with few double points can't have an irreducible cubic curve dual.

Study on connection points on double regular polygons, providing coordinates and proving non-connection points.

Classifies periodic points on regular and double n-gon surfaces.

Combinatorial proof of grid homology properties.

Study normal operators of double fibration transforms with conjugate points.

Distance, normals, and double normals for real plane curves with singularities

Extends Khovanov homology to surfaces with singularities.

We prove that the "minus" version of Lipshitz's double-point enhanced grid homology is a knot invariant through purely combinatorial means.

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly on…

This paper tabulates prime knot projections up to eight double points.

We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x Z/n on simply connected 4-manifolds with complicated fixed-point sets.

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+…

A new double quasi-Poisson bracket on surface groups.

Let $S$ be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let $S^\circ$ be the same surface equipped with the opposite orientation. We consider the double $S_\mathcal{D}$ obtained by gluing the surfaces $S$ and $S^\circ$ along corresponding boundary components. W…

The study classifies immersed surfaces with knot group Z in simply-connected 4-manifolds.

Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball …

The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic 3-manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with fiber a closed surface of genus 19. We also record arithmeticity of th…

Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.

We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S,θ)=((3,1,1), -id). We also list up all the ca…

In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provi…

New invariant stops certain types of geometric transformations.

Paper proves existence of minimal doublings on surfaces with specific properties.

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around …

Paper addresses underestimation bias in double Q-learning, proposing a method to improve learning performance.

We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from X to X. As an application, we give results restricting the minimal number of do…

Deep learning models can generalize well even when they fit training data perfectly.

We consider (local) parametrizations of Teichmuller space $T_{g,n}$ (of genus $g$ hyperbolic surfaces with $n$ boundary components) by lengths of $6g-6+3n$ geodesics. We find a large family of suitable sets of $6g-6+3n$ geodesics, each set forming a special structure called "admissible double pants decomposition". For …

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar g…

We briefly review our results on the Lie theory underlying vector bundles over Lie groupoids and Lie algebroids, pointing out the role of Poisson geometry in extending these results to double Lie algebroids and LA-groupoids.

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points.

K-stability proven for a specific type of Fano threefold.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

Enhances Hamiltonian systems stability through generalized double bracket vector fields.

Research on refined algebraic domains respecting differential geometry.

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…

Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.

Least squares regression shows unexpected double descent in under-parameterized models.

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish va…

Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and th…

We consider the optimal double stopping time problem defined for each stopping time $S$ by $v(S)=\esssup\{E[ψ(τ_1, τ_2) | \F_S], τ_1, τ_2 \geq S \}$. Following the optimal one stopping time problem, we study the existence of optimal stopping times and give a method to compute them. The key point is the construction of …

The study characterizes slopes for hyperbolic knots and Whitehead doubles.

Study on singularities of frontal surfaces, classifying under equivalence.

Constructs minimal hypersurfaces in S^4(1) by doubling equatorial S^3.

Hexagonal norm double bubble problem solved with minimal configurations.

Given a plane curve $γ: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(γ)$ of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $γ$ is an immersed curve with $D(γ)$ double points and no othe…