arXiv research

Paper proves Whitney stratified spaces can be given a conically smooth structure.

Symplectic embedding extended to 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…

Jet spaces on Carnot groups have a canonical Lie group structure.

The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.

Paper proves collapsing result for orbifolds without curvature bounds.

Study minimax estimation of stratified structure from i.i.d. samples.

Proves Verdier duality for sheaves on stratified spaces.

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are us…

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…

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…

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth st…

The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.

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

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{α=1}^rg_αX_α$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification …

The study lists low-dimensional stratified groups and their properties.

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of …

ISP improves GNN expressivity by stratifying nodes based on graph invariants.

The paper explores conditions for compactness and finiteness in stratified homotopy theory.

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with c…

By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…

Develops deep generative models for stratified learning.

Study reveals RL game's embedding space is stratified, not a manifold.

New homotopy theory reveals the structure of stable curves.

The paper extends $L_\infty$ structures to generic closed two-forms on manifolds.

R package stagedtrees learns staged tree structures from data.

We study stratified G-structures in ${\cal N}=2$ compactifications of M-theory on eight-manifolds $M$ using the uplift to the auxiliary nine-manifold ${\hat M}=M\times S^1$. We show that the cosmooth generalized distribution ${\hat {\cal D}}$ on ${\hat M}$ which arises in this formalism may have pointwise transverse or…

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…

Paper introduces stratified vector bundles and their properties.

We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between …

Extends h-principle to stratified spaces using sheaf and jet theories.

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.

Stratified models depend in an arbitrary way on a selected categorical feature that takes $K$ values, and depend linearly on the other $n$ features. Laplacian regularization with respect to a graph on the feature values can greatly improve the performance of a stratified model, especially in the low-data regime. A sign…

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…

Optimizes survey design for private mean estimation with reduced variance.

The paper proves symplectic neighbourhood theorems for stratified subspaces.

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…

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…

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…

Characterizes hypergenerated stratified groups with flat boundaries.

Studied $L^2-$invariants on stratified spaces, proving their stability.

Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and Kaehler reduction with reference to the adjoint action yields a stratified Kaehler structure on the resulting adjoint quotient. …

This work describes compactifications of metric spaces and vector spaces using asymmetric norms.

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…

We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Paw…

Paper classifies pseudomanifolds over stratified spaces.

Study of quantum decorated character stacks and their quantizations.

In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…