arXiv research

Study numerical invariants under retraction maps between topological spaces.

This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.

New retraction on symplectic Stiefel manifold with closed-form inverse.

The present research work proposes a new fast fixed-point averaging algorithm on the compact Stiefel manifold based on a mixed retraction/lifting pair. Numerical comparisons between fixed-point algorithms based on the proposed non-associated retraction/lifting map pair and two associated retraction/lifting pairs confir…

The paper compares numerical schemes for nonholonomic systems using retraction maps.

Constructs harmonic maps near retractions in hyperbolic spaces.

A new retraction on Stiefel manifold with a closed-form inverse.

NR retraction approximates geodesics on submanifolds efficiently.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

Schmutz Schaller and Thurston's approaches are dual.

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(Z), relative hyperbolic groups and mapping class groups.

Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.

New integrators for mechanical systems on Lie groups simplify based on group properties.

Global homotopies upgrade classical map in differential geometry.

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichmüller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension and construct an intrinsic $\Mod_g$-equivariant deformation retraction from $\T_g$…

Retracted temporarily to fix some erratic things.

Simplicial sets deformation retract onto transverse simplices.

Constructs a moment map flow for isotropic maps on surfaces.

Develops integrators for nonholonomic systems on Lie groups.

Let S be a compact surface, and M be the double of a handlebody. Given a homotopy class of maps from S to M inducing an isomorphism of fundamental groups, we describe a canonical uniformly lipschitz retraction of the sphere graph of M to the arc graph of S. We also show that this retraction is a uniformly bounded dista…

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

We introduce some chain maps between Khovanov complexes. Each of the chain maps commutes with a chain homotopy map and a retraction maps which obtain a Reidemeister invariance of Khovanov homology.

The paper creates a deformation retraction for homeomorphisms of the projective plane.

We construct a one-dimensional deformation retract of the unordered k-point configuration space of a star S. This retract suggests an explicit set of free generators Beta_k for the corresponding braid group of the star B_k and shows that the natural map from B_k-1 to B_k sends Beta_k-1 to Beta_k injectively.

Control data constructed for smooth weak deformation retraction of stratified spaces.

This research solves Hermite interpolation on manifolds using retractions.

Given a hyperbolic domain, the nearest point retraction is a conformally natural homotopy equivalence from the domain to the boundary of the convex core of its complement. Marden and Markovic showed that if the domain is uniformly perfect, then there exists a conformally natural quasiconformal map which admits a bounde…

Selects points from Jordan domains on Riemannian surfaces.

Let f:Σ_1 --> Σ_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in Σ_1\times Σ_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through…

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which ar…

The paper analyzes convergence of Riemannian SA schemes for stochastic optimization.

Contact group retracts to unitary subgroup.

Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.

In this short note, we investigate some features of the space $\Inject{d}{m}$ of linear injective maps from $\bbR^d$ into $\bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this context, as the set of orthonormal systems of $d$ vectors in $\bbR^m$. Finally, we…

A new algorithm avoids retractions to optimize orthogonal matrices efficiently.

Study random walks on sub-Riemannian manifolds using retractions.

The outer automorphism group Out(F_2g) of a free group on 2g generators naturally contains the mapping class group of a punctured surface as a subgroup. We define a subsurface projection of the sphere complex of the connected sum of n copies of S^1 x S^2 into the arc complex of the surface and use this to show that thi…

Construct special Lagrangian fibrations on abelian varieties using retraction techniques.

Embeds complex into higher-dimensional pseudomanifold.

Extends Brouwer fixed point theorem with new conditions for continuous maps.

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.

Modernizes classical theory linking isothermic surfaces to Bonnet pairs.

Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence o…

We characterize metric spaces $X$ whose hyperspaces $2^X$ or $Bd(X)$ of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.

Outer space and Teichmüller space fail well-rounded retract analogy.