Differentiable groupoids and their inertia spaces are studied with a de Rham theorem.
problem Understanding the de Rham cohomology of inertia spaces.
method Introducing differentiable stratified groupoids and proving a de Rham theorem.
result Inertia groupoids of proper Lie groupoids are locally contractible differentiable stratified groupoids.
Paper proves Whitney stratified spaces can be given a conically smooth structure.
problem Proving Whitney stratified spaces can be given a conically smooth structure.
method Introduced conically smooth structure by Ayala, Francis, and Tanaka. Proved conjecture that any Whitney stratified space admits a canonical conically smooth structure.
result Established a connection between Whitney stratified spaces and conically smooth spaces.
Paper introduces stratified vector bundles and their properties.
problem Understanding singular spaces and their vector bundles.
method Characterization via monoid actions and examples from various fields.
result Functorial properties extended to the stratified case.
The paper studies cohomology on incomplete manifolds and stratified spaces.
problem Analyzing cohomology groups on incomplete Riemannian manifolds and stratified spaces.
method Proves injective/surjective maps between Lp and L2 cohomology groups under certain conditions. result Injective/surjective maps between Lp and L2 cohomology groups are established. Jet spaces on Carnot groups have a canonical Lie group structure.
problem Understanding jet spaces on Carnot groups.
method Constructing jet spaces over stratified Lie groups and showing they are stratified Lie groups.
result Every stratified Lie group of step s+1 can be embedded in a jet space over a stratified Lie group of step s. Study smooth structures and stratifications on orbit spaces of Lie groupoids.
problem Understanding the smooth structure and stratification of orbit spaces of Lie groupoids.
method Utilizing Spallek's framework and extending classical theories of proper Lie group actions.
result Established Morita invariance of smooth structures and stratifications on orbispaces.
Paper investigates pluripotential theory on Teichmüller space using new methods.
problem Understanding pluripotential theory on Teichmüller space.
method Alternative approach to Krushkal formula, natural stratified structure, Levi form description.
result Natural stratified structure and description of Levi form.
Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.
problem Geodesic preservation in infinite-dimensional manifolds.
method Definition of spray-invariant sets and analysis of their properties.
result Different geometric properties of spray-invariant sets based on their regularity.
Maps vector fields between stacks and orbit spaces.
problem Understanding vector fields on stacks and orbit spaces.
method Morita stratifications and geometric vector fields correspondence.
result Derives stacky version of Gauss lemma and extends Palais' theorem.
We study the topology of the inertia space of a smooth G-manifold M where G is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differ…
Analyzes solutions of stratified Lie systems and their geometric structures.
problem Nonautonomous systems of differential equations on manifolds.
method Analyzes particular solutions and properties of stratified Lie systems.
result Generalizes properties of Lie systems to stratified Lie systems.
Symplectic embedding extended to stratified spaces.
problem Symplectic embedding theorem for stratified spaces.
method Defined symplectic structure on stratified spaces, demonstrated embedding in complex projective space.
result Symplectic embedding theorem extended to stratified spaces.
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
The paper introduces knot invariants using stratified homotopy groups.
problem Defining knot invariants in stratified spaces.
method Introducing stratified homotopy groups and proving their properties.
result Stratified Whitehead's theorem holds for these groups.
We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show t…
Survey of Hodge theory on stratified spaces and related compactifications.
problem Analyzing Hodge theory on complex stratified spaces.
method Resolution of Thom-Mather stratified spaces to manifolds with corners, introduction of mezzoperversity, definition of Cheeger spaces.
result Novikov conjecture verified for Cheeger spaces.
We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples…
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…
Computes cohomology classes of strata of meromorphic differentials.
problem Computing cohomology classes of differentials with prescribed poles.
method Defined a space of stable meromorphic differentials, stratified by zero multiplicities, computed Poincaré-dual cohomology classes, proved tautological, and provided an algorithm.
result All cohomology classes are tautological and can be computed.
Optimizes survey design for private mean estimation with reduced variance.
problem Minimizing variance in private mean estimation with privacy constraints.
method Formulates optimal survey design as an optimization problem, determining optimal subsampling sizes to minimize variance.
result Identifies the first privacy-aware stratified sampling scheme that minimizes variance under different privacy mechanisms.
Studied L2−invariants on stratified spaces, proving their stability.
problem Stability of L2−invariants on stratified spaces. method Defined and analyzed L2-Betti numbers and Novikov-Shubin invariants for compact smoothly stratified pseudo-manifolds with a wedge metric, extending results to these pseudo-manifolds. result Invariance of L2-Betti numbers and Novikov-Shubin invariants under smoothly stratified, strongly stratum preserving homotopy equivalence. Bredon's trick helps extend local properties to global topological spaces.
problem Extending local properties to global topological spaces.
method Bredon's trick for local properties to global spaces.
result Bredon's trick allows for natural alternative demonstrations of classic results.
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
We show that conically smooth stratified spaces embed fully faithfully into ∞-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each ∞-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…
The paper proves a rigidity theorem for stratified spaces with positive Ricci curvature.
problem Proving a rigidity theorem for stratified spaces with positive Ricci curvature.
method Analyzing stratified spaces with positive Ricci lower bounds and no cone angles larger than 2π.
result The first non-zero eigenvalue of the Laplacian is equal to the dimension if and only if the stratified space is isometric to a spherical suspension.
A spectral sequence for stratified spaces and configurations helps in understanding cohomology.
problem Understanding cohomology of stratified spaces and configuration spaces of points.
method Sheaf-theoretic construction of a spectral sequence for stratified spaces and configuration spaces of points.
result Representation stability theorem for configuration spaces of points.
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…
The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a c^-iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compac…
This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smoot…
Study of hyperkahler spaces from Lie algebras.
problem Understanding hyperkahler quotients of cotangent bundles.
method Analyzing stratified spaces of cotangent bundles of complex semisimple Lie groups.
result Explicit description of partial order on strata using Lie theoretic data.
We consider the problem of adaptive stratified sampling for Monte Carlo integration of a differentiable function given a finite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where the function oscillates more, while allocating the samples such that they are wel…
The paper extends higher signature invariants to stratified spaces and establishes a new surgery exact sequence.
problem Extending higher signature invariants to stratified spaces.
method Revisiting and extending the surgery exact sequence for stratified spaces.
result Established a natural transformation between the surgery exact sequence and a long exact sequence of K-theory groups for stratified spaces.
Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…
The paper studies control data and equivariant properties of stratified spaces.
problem Equivariant properties of stratified spaces with group actions.
method Equivariant submersion theorem and control data construction.
result Inclusion of G-stratified subspaces is a G-equivariant cofibration. Paper classifies pseudomanifolds over stratified spaces.
problem Classifying pseudomanifolds over stratified spaces.
method Introducing locally standard T-pseudomanifolds and using characteristic data. result Locally standard T-pseudomanifolds over topological stratified pseudomanifolds are classified by their characteristic data. The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally …
New method defines Gysin maps for stratified spaces, preserving signatures.
problem Defining and preserving signatures in stratified spaces.
method Smooth atlas stratified spaces, fiber bundles, Gysin maps, KK-theory.
result Gysin maps preserve analytic signature classes of Witt spaces.
This paper surveys the significant progress over the past couple of decades in the theory of stratified spaces through the application of controlled methods as well as through the application of intersection homology.
This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more general, stratified spaces. This book provides a self--contained treatment of t…
Control data constructed for smooth weak deformation retraction of stratified spaces.
problem Construct control data for smooth weak deformation retraction of stratified spaces.
method Show smooth local triviality with conical fibers, construct control data, use fiber-wise scalar multiplications.
result Obtain neighbourhood smooth weak deformation retraction of stratified spaces.
Gauss-Green theorem proven for vector fields in stratified groups.
problem Establishing the Gauss-Green theorem for vector fields in noncommutative stratified Lie groups.
method Developed a new family of function spaces for divergence-measure fields and proved the Gauss-Green theorem.
result Gauss-Green theorem achieved for vector fields of low regularity on sets of finite perimeter in stratified groups.
New definition of regular points for PL functions on manifolds.
problem Defining regular points for PL functions on combinatorial manifolds.
method Definition based on link of the point, stratification of Jacobi set, Stein factorization of Reeb space.
result Our definition of regularity is distinct from existing definitions.
Smooth contact mappings in a flat (2,3,5)-distribution are shown to be smoother.
problem Characterizing smoothness of contact mappings in a specific geometric setting.
method Study of differential identities and rigidity of stratified Lie groups.
result Smooth contact mappings are actually smoother than initially assumed.
We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf.
Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…
A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…