Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.
problem Analyzing Calabi-Yau hypersurfaces using non-archimedean geometry.
method Yamamoto's tropical contractions and Li's Fermat degeneration, with toric plurisubharmonic metrics.
result Constant potential along fibers of retraction under discrete symmetry assumption.
This paper constructs a non-Archimedean Teichmüller space using tropical geometry.
problem Constructing a non-Archimedean analogue of Teichmüller space.
method Using techniques from tropical and logarithmic geometry.
result The skeleton of non-Archimedean Teichmüller space is the tropical Teichmüller space.
Defines volume and Monge-Ampère energy on polarized affine varieties.
problem Volume and Monge-Ampère energy on polarized affine varieties.
method Definition of volume using asymptotics of jumping numbers, Monge-Ampère energy using forms and currents on Berkovich spaces.
result Monge-Ampère energy agrees with volume of filtrations and recovers known functionals.
The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
Synthetic approach to pluripotential theory measures finite energy.
problem Global pluripotential theory on compact Kähler manifolds and projective Berkovich spaces.
method Definition and study of measures of finite energy, introduction of twisted and free energy functionals.
result Coercivity of energy functionals is an open condition with respect to polarization.
The study constructs universal invariants for non-Archimedean metrics on projective varieties.
problem Understanding the singularity of non-Archimedean metrics on projective varieties.
method Constructing partial Okounkov bodies and Duistermaat--Heckman measures for non-Archimedean metrics.
result Generalization of Duistermaat--Heckman measures to finite energy metrics on Berkovich analytifications.
Study shows Calabi-Yau metrics converge to a specific form under certain conditions.
problem Degeneration of Calabi-Yau metrics and their limits.
method Optimal transport problem and minimisation of Kontorovich functional.
result Limit data of Calabi-Yau metrics can be encoded into a unique minimiser.
Solves non-Archimedean Calabi-Yau equation on complex log pairs.
problem Non-Archimedean Monge-Ampère equation on Berkovich analytification.
method Solves complex Monge-Ampère equation, then adapts to non-Archimedean setting.
result Non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties.
A new retraction on Stiefel manifold with a closed-form inverse.
problem Efficiency in Riemannian computing applications.
method Introduces a new retraction on the compact Stiefel manifold with a closed-form inverse.
result The retraction is second-order accurate and features a closed-form inverse.
The paper proves a Basmajian identity for non-Archimedean local fields.
problem Proving Basmajian's identity over non-Archimedean local fields.
method Projective Anosov representations and Berkovich hyperbolic geometry.
result A signed finite sum series identity for Basmajian's identity.
This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.
problem Understanding the well-rounded deformation retraction of Teichmüller space.
method Examining the mapping class group-equivariant deformation retraction of Teichmüller space onto a CW complex and comparing it to well-rounded retractions of other spaces.
result The well-rounded deformation retraction of Teichmüller space is analogous to well-rounded retractions of other spaces.
New retraction on symplectic Stiefel manifold with closed-form inverse.
problem Efficient mapping of manifold data to Euclidean domain.
method Introduces a new retraction map with a closed-form inverse.
result The new retraction has a closed-form inverse, unlike previous methods.
In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere C^ using R-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on R-trees: one geometric and one algebraic. The geometric constructio…
We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let (X,B) be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
NR retraction approximates geodesics on submanifolds efficiently.
problem Efficiently approximating geodesics on submanifolds for practical algorithms.
method Introducing Newton retraction (NR) as a class of retractions on submanifolds induced by a foliation of the ambient manifold.
result NR is more stable and computationally cheaper than oblique projection, with superlinear convergence regions.
Contact group retracts to unitary subgroup.
problem Understanding contact structures on 3-sphere.
method Proving deformation retraction to unitary subgroup.
result Group of contactomorphisms retracts to U(2).
Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.
problem Embedding and retraction of topological manifolds in Euclidean spaces.
method Locally flat embedding and retraction of manifolds in high-dimensional Euclidean space.
result Every topological n-manifold can be embedded locally flatly in R2n+1 and is a retract of some neighborhood in R2n+1. 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 $\T_g$…
A new algorithm avoids retractions to optimize orthogonal matrices efficiently.
problem Optimizing functions over the manifold of orthogonal matrices efficiently.
method Landing algorithm that avoids retractions using potential energy.
result The landing algorithm is faster and less prone to numerical errors than retraction-based methods.
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.
problem Existence of special Lagrangian fibrations in Calabi-Yau hypersurfaces.
method Non-Archimedean and tropical Monge-Ampère equations on Berkovich and skeleton spaces, proving uniqueness and deriving solutions.
result Unique solutions to tropical and non-Archimedean Monge-Ampère equations, leading to existence of special Lagrangian fibrations.
The paper describes a stratification of a compactified Hurwitz space using combinatorial trees.
problem Describing the boundary stratification of a compactified Hurwitz space.
method Using decorated trees to describe the boundary strata and their containment relations.
result The boundary strata of the compactified Hurwitz space are in bijection with decorated trees, and containment is given by edge contraction.
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…
Embeds complex into higher-dimensional pseudomanifold.
problem Embedding complex structures into higher-dimensional spaces.
method Deformation retraction and embedding into pseudomanifolds.
result Finite d-dimensional simplicial complex can be embedded as a retract in a closed (2d−1)-dimensional pseudomanifold. Constructs retractions of CAT(1) spaces to convex subsets.
problem Geometric description of an analytic tool.
method Gradient flow of time-dependent locally Lipschitz semiconcave functions.
result Existence of gradient flows proved for independent interest.
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.
problem Classical theory of isothermic surfaces and Bonnet pairs.
method Identifies derivatives of Bonnet pairs with retraction form of isothermic surfaces.
result Modern account and identification of retraction form.
Constructs harmonic maps near retractions in hyperbolic spaces.
problem Finding harmonic maps near retractions in hyperbolic spaces.
method Constructs harmonic maps to the hyperbolic plane from quasidisks, and to convex hulls from sets in the boundary at infinity of pinched Hadamard manifolds.
result Harmonic maps are bounded from nearest-point retractions in hyperbolic spaces.
The paper creates a deformation retraction for homeomorphisms of the projective plane.
problem Deformation retraction of homeomorphisms of the projective plane.
method Equivariant strong deformation retraction from homeomorphism group to special orthogonal group.
result Induces a SO(3)-equivariant strong deformation retraction from projective plane homeomorphisms to SO(3).
The paper compares numerical schemes for nonholonomic systems using retraction maps.
problem Optimal control of nonholonomic systems with numerical approximations.
method Retraction maps used as seed for geometric integrators of Hamilton equations.
result Performance comparison of symplectic and non-symplectic integrators.
We characterize metric spaces X whose hyperspaces 2X or Bd(X) of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.
Simplicial sets deformation retract onto transverse simplices.
problem Deformation retraction of simplicial sets.
method Showed deformation retraction of singular simplicial set onto transverse simplices.
result Singular simplicial set deformation retracts onto transverse simplices.
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.
Outer space and Teichmüller space fail well-rounded retract analogy.
problem Failure of well-rounded retract in Outer space and Teichmüller space.
method Analysed flat tori and metric graphs to show failure.
result Analogue of well-rounded retract does not contain equivariant spine.
Study of non-archimedean μ-entropy and its connection to K-stability.
problem Understanding K-stability in non-archimedean settings.
method Introducing non-archimedean μ-entropy and its properties, connecting it to K-semistability.
result Established a criterion for K-semistability without vector ξ, using the non-archimedean μ-entropy.
This research solves Hermite interpolation on manifolds using retractions.
problem Interpolating data on non-Euclidean spaces with matching derivatives.
method Proposes a novel procedure using retractions for Hermite interpolation on various manifolds.
result Establishes the well-posedness of the method and extends Hermite interpolation results to manifolds.
Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.
Polyhedra collapse to subpolyhedra if they can be continuously shrunk onto them.
problem Characterizing when a polyhedron can be continuously shrunk onto a subpolyhedron.
method Piecewise-linear free deformation retraction and metric considerations.
result A polyhedron collapses to a subpolyhedron if and only if it admits a free deformation retraction onto that subpolyhedron.
We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the…
Being a maximal compact subgroup of SL_nC, SU_n is a deformation retract of the former group. In this note we prove that, for sufficiently large n, there is no retraction of SL_nC to SU_n which preserves commutativity.
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
Paper generalizes K-stability to non-algebraic spaces for Kähler metrics.
problem Existence of constant scalar curvature Kähler metrics on non-algebraic spaces.
method Developed a non-Archimedean theory of complex spaces and applied it to Kähler manifolds.
result Proved K-stability implies existence of unique constant scalar curvature Kähler metric.
Spaces of circle embeddings in curved surfaces indexed by trees.
problem Classifying spaces of braided automorphism groups of trees.
method Indexed connected components with finite rooted trees, constructed strong deformation retract.
result Connected components are classifying spaces of braided automorphism groups.
Schmutz Schaller and Thurston's approaches are dual.
problem Mapping class group-equivariant deformation retractions of Teichmüller space.
method Comparing Schmutz Schaller's and Thurston's methods.
result Schmutz Schaller and Thurston's approaches are dual.
We obtain a classification up to isomorphism of complex-analytic supermanifolds with underlying space CP1 of dimension 1∣3 with retract (k,k,k), where k∈Z. More precisely, we prove that classes of isomorphic complex-analytic supermanifolds of dimension 1∣3 with retract (k,k,k) are in o…