arXiv research

Study the geometry of nonseparating curves on surfaces with punctures.

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonsepara…

Uniform hyperbolicity proved for nonorientable surface curve graphs.

Study on stable translation lengths of surface homeomorphisms and their approximations.

We construct nontrivial roots of Dehn twists about nonseparating curves.

New bounds on specific torsion lengths for periodic mapping classes.

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

Parabolic mapping class acts on curve graphs of infinite type surfaces.

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $λ: \mathcal{N}(R) \rightarro…

New method produces reflections with nonseparating fixed points.

Conditions for simple closed curves in surface covers.

It is proved that the stable commutator length of a Dehn twist in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating simple closed curve is a product of two commutators. As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given.

Let $R$ be a compact, connected, orientable surface of genus $g$, $Mod_R^*$ be the extended mapping class group of $R$, $\mathcal{C}(R)$ be the complex of curves on $R$, and $\mathcal{N}(R)$ be the complex of nonseparating curves on $R$. We prove that if $g \geq 2$ and $R$ has at most $g-1$ boundary components, then a …

This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves, nonseparating curves, pants, and cut systems are all connected for genus $g \gg…

Proposes NSSNNs to predict nonseparable Hamiltonian systems.

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{\times} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{…

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating cur…

New method solves nonseparable stochastic control problems.

Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the …

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma…

Let $S_g$ be a closed orientable surface of genus $g \geq 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A \textit{fractional power} of $t_C$ of \textit{exponent} $\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that $h^n = t_C^{\ell}$. Unlike a root of a $t…

Bayesian method improves forecasting of nonseparable Hamiltonian systems with noise.

Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C…

Flow IV uses IVs to infer counterfactuals in complex models.

The classical dynamic programming-based optimal stochastic control methods fail to cope with nonseparable dynamic optimization problems as the principle of optimality no longer applies in such situations. Among these notorious nonseparable problems, the dynamic mean-variance portfolio selection formulation had posted a…

This paper restricts efficient geodesics to non-separating curves.

Characterizes minor-minimal separating projective planar graphs and their generalizations.

D. Margalit and S. Schleimer found examples of roots of the Dehn twist about a nonseparating curve in a closed orientable surface, that is, homeomorphisms whose nth power is isotopic to the Dehn twist. Our main theorem gives elementary number-theoretic conditions that describe the values of n for which an nth root exis…

On a compact oriented surface of genus $g$ with $n\geq 1$ boundary components, $δ_1, δ_2,\ldots, δ_n$, we consider positive factorizations of the boundary multitwist $t_{δ_1} t_{δ_2} \cdots t_{δ_n}$, where $t_{δ_i}$ is the positive Dehn twist about the boundary $δ_i$. We prove that for $g\geq 3$, the boundary multitwis…

Bayesian Complementary Kernelized Learning models complex spatiotemporal data.

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist t_c about a nonseparating circle c in the mapping class group M(N_g) of a nonorientable surface N_g of genus g. We explore the existence of roots and, following the work of …

Deep neural networks enforce non-crossing quantile regression curves.

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books…

Recent years have witnessed the rapid development of block coordinate update (BCU) methods, which are particularly suitable for problems involving large-sized data and/or variables. In optimization, BCU first appears as the coordinate descent method that works well for smooth problems or those with separable nonsmooth …

This paper classifies knots with simple curves in genus 2 handlebodies.

We prove the $3$-manifold $\RP^3 \# \RP^3$ is of $\Z_{2}$-coefficient homology $(1, 2)$-systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define $\Z_{2}$-coefficient homology $1$-systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{1}(\RP^3\#\…

Computes extendable mapping classes for knotted surfaces in $S^4$.

The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the min…

This paper analyzes popular time-nonseparable utility functions that describe "habit formation" consumer preferences comparing current consumption with the time averaged past consumption of the same individual and "catching up with the Joneses" (CuJ) models comparing individual consumption with a cross-sectional averag…

Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the empirical risk is, however, highly nonconvex. This alone presents a challenge to appl…

Unique arcs create slopes in genus two handlebodies.

We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a higher-dimensional setting. (2) They do not arise as nonseparating weak contact-type hypersurf…

New mapping classes of knotted surfaces are computed via surgery.

This manuscript provides optimization guarantees, generalization bounds, and statistical consistency results for AdaBoost variants which replace the exponential loss with the logistic and similar losses (specifically, twice differentiable convex losses which are Lipschitz and tend to zero on one side). The heart of the…

Study describes frequencies of geodesics on hyperbolic surfaces as genus grows.

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of $S$ develops into a torus that splits. If the geodesic is nonseparating then the Jacobian torus of $…

Study of codimension-1 embeddings in 3-manifolds using twist maps and push maps.

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. This system is known to admit closed orb…