Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for the existence of a complete uniform differential structure on a given differenti…
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48 results for “differentiable spaces”
Transforms Aronszajn to Sikorski subcartesian spaces.
problem Mapping between subcategories of differential spaces.
method Constructs a natural transformation.
result Establishes a relationship between Aronszajn and Sikorski subcartesian spaces.
Differentiable spaces derived from Lie group actions have vector fields and forms.
problem Understanding the differential structure of orbit spaces of Lie group actions.
method Analyzing the differential structure of orbit spaces of proper Lie group actions on smooth manifolds.
result Orbit spaces of Lie group actions are differentiable spaces with exterior algebra of differential forms.
Constructs differential forms on -ringed spaces.
problem Developing a theory of differential forms for non-manifold spaces.
method Functorial construction of differential forms on local -ringed spaces.
result Stokes' theorem holds for integrated forms on simplices.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
Study of differential forms and vector fields on orbit spaces.
problem Understanding vector fields and differential forms on orbit spaces.
method Defined differential forms and vector fields as multilinear maps on infinitesimal diffeomorphisms.
result Intrinsic view of vector fields and differential forms on orbit spaces.
Compactifies moduli spaces of abelian differentials with specific zeroes and poles.
problem Constructing a compactification of moduli spaces of abelian differentials.
method Using a blowup of the incidence variety compactification, defining families of projectivized multi-scale differentials, and performing a real oriented blowup.
result The moduli space of multi-scale differentials is a complex orbifold with normal crossing boundary.
Sobolev spaces and -differential operators on manifolds I: basic properties and weighted spacesmath.AP
The paper defines and analyzes -Sobolev spaces and operators on manifolds.
problem Defining and analyzing Sobolev spaces and differential operators on manifolds.
method Coordinate-free approach using connections, proving properties of -Sobolev spaces and operators.
result Equivalent definitions of -Sobolev spaces and operators under certain conditions.
Lectures explore how differential methods improve understanding of algebraic group orbit spaces.
problem Understanding structure of invariants and orbit spaces of algebraic Lie groups.
method Combines algebraic and differential viewpoints to study orbit spaces.
result Differential approach provides deeper insights into invariants and orbit spaces.
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying b…
Survey on moduli spaces of differentials from algebraic geometry perspective.
problem Understanding the topology of moduli spaces of differentials remains limited.
method Algebraic geometry perspective, connections to various fields.
result Many open problems and connections to other fields.
Formula for Euler characteristic of moduli spaces of Abelian differentials.
problem Computing the Euler characteristic of moduli spaces of Abelian differentials.
method Intersection theory on the smooth compactification by multi-scale differentials, Euler sequence for cotangent bundle, and tools in the Chow ring.
result Formula for the full Chern polynomial of the cotangent bundle.
New formalization of curved spaces using pointwise affine spaces.
problem Traditional curved space formalizations like manifolds are complex.
method Introduces pointwise affine spaces and new geometric definitions.
result Simplified and clearer geometric concepts and results.
The distance function (or ) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over , whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjectu…
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Analyzes geometric structures on profinite diffeological spaces.
problem Understanding geometric properties of spaces derived from finite-dimensional manifolds.
method Examines tangent and cotangent spaces, differential forms, metrics, connections, and cohomology.
result Unified geometric constructions across various contexts.
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize distribution spaces in our study of wave and heat equations.
Tensoring -weak differentiable structures preserves their properties.
problem Tensorization of -weak differentiable structures.
method Proving the product of -weak charts is a -weak chart, and showing isometric embeddings.
result Tensorization of -weak differentiable structures is possible under certain conditions.
Develops second order infinitesimal structures on Teichmüller space.
problem Understand the infinitesimal structures of Teichmüller space.
method Formulated second order infinitesimal structures over Teichmüller space.
result Affirmative answers to two folklore problems on Teichmüller space.
Defines vector fields and differential forms on local C-infinity-ringed spaces.
problem No specific problem stated; focuses on mathematical definitions.
method Defines tangent sheaf, contractions, Lie derivatives, and proves Cartan equations.
result Standard Cartan calculus equations hold for local C-infinity-ringed spaces.
Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the actio…
Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with -structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilat…
The paper studies the Poisson transform of differential forms on hyperbolic spaces.
problem Analyzing the Poisson transform of differential forms on real hyperbolic spaces.
method Proving the Poisson transform is a topological isomorphism between boundary forms and eigenforms.
result The Poisson transform is a topological isomorphism for -differential forms on the boundary of hyperbolic spaces.
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
The uniform structure on a differential space defined by a family of generators is considered.
In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just as grad (ient), div (ergence) and rot (ation) are uniquely determined geometric…
The curvature of almost Fuchsian immersions is concave in their Hopf differentials.
problem Understanding the geometry of almost Fuchsian immersions.
method Analyzing the extrinsic curvature and using properties of Hopf differentials.
result The set of Hopf differentials forms a convex subset of holomorphic quadratic differentials.
The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given.
Study shows solutions of differential inclusions are homotopy equivalent in -topology.
problem Homotopy properties of solutions in differential inclusions.
method Analyzes differential inclusion with specific assumptions on corank one distribution.
result Solutions are homotopy equivalent to loop spaces in -topology.
Why Diffeology?math.DG
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
Two types of differentials are shown equivalent for compactifying moduli spaces.
problem Compactifying moduli spaces of curves with prescribed orders of zeros and poles.
method Equivalence of multi-scale and logarithmic differentials, isomorphism of moduli stacks, explicit blowups.
result Multi-scale and logarithmic differentials are equivalent and isomorphic.
New geometric Joyce structures on moduli spaces of quadratic differentials.
problem Constructing Joyce structures on moduli spaces of quadratic differentials.
method Isomonodromic deformations of second-order linear ODEs with rational potential.
result Construction of Joyce structures on moduli spaces of quadratic differentials.
In this paper the notion of an M-th order invariant bilinear differential pairing is introduced and a formal definition is given. If the manifold has an AHS structure, then various first order pairings are constructed. This yields a classification of all first order invariant bilinear differential pairings on homogeneo…
The paper constructs cohomology classes on curve strata.
problem Understanding cohomology classes on curve strata.
method Using geometry of the boundary stratification of moduli space of multi-scale differentials.
result Construction of non-trivial and non-tautological cohomology classes.
Proves even degrees can be realized in Abelian differential strata.
problem Realizing even degrees as stretch factors in Abelian differentials.
method Analyzes Thurston-Veech stretch factors in moduli spaces.
result Every even degree is realized in Abelian differentials.
Study cohomology spaces of sl(2) acting on n-ary differential operators.
problem Computing cohomology spaces for sl(2) action on n-ary differential operators.
method Analyzes polynomial μ-densities as sl(2) modules and computes cohomological spaces H^2.
result Computed cohomological spaces H^2 of sl(2) on n-ary differential operators.
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
Paper solves a class of differential equations with specific solutions.
problem Identifying solutions to a class of nonlinear ODEs.
method Solves using a proposed side condition involving a third-order linear ODE.
result New closed and integral-form solutions for the Tzitzeica curve equation.
On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as …
Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential…
We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the -weak gradient on iter…
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …
Unified approach to stability conditions on surfaces with quadratic differentials.
problem Identifying spaces of stability conditions on triangulated categories.
method Perverse schober and their global sections, mixed-angulations, flips, finite-length hearts, tilts.
result Identification of moduli spaces of quadratic differentials with arbitrary singularity types.
In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left…
Critical graphs of quadratic differentials equidistribute in moduli space.
problem Distribution of critical graphs in moduli space.
method Study of Jenkins-Strebel differentials and their critical graphs.
result Critical graphs equidistribute to the Kontsevich measure.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is -integrable for any .
SageMath package diffstrata calculates intersection theory on abelian differentials.
problem Computing intersection theory on the boundary of strata of abelian differentials.
method Explicit combinatorial description of the boundary, implemented algorithms in SageMath.
result Computes the Euler characteristic of strata using intersection theory.