A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Paper proves Whitney stratified spaces can be given a conically smooth structure.
We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct `fundamental Whitney…
Paper introduces stratified vector bundles and their properties.
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
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…
In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a -stratified space carries a system of -equivariant control data. As an application, we show that if $A \su…
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…
Control data constructed for smooth weak deformation retraction of stratified spaces.
We study the topology of the inertia space of a smooth -manifold where 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…
Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…
Geometric model explains music perception combining neuroscience and acoustics.
The paper extends structures to generic closed two-forms on manifolds.
We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a L…
The paper explores conditions for compactness and finiteness in stratified homotopy theory.
SGD avoids critical points on weakly convex functions.
Global stability bounds for matrix frames in phase retrieval problems.
Mathematical foundation for phylogenetic tree uncertainty quantification.
Symplectic embedding extended to stratified spaces.
Extends functions on symmetric spaces to analytic functions.
Extends h-principle to stratified spaces using sheaf and jet theories.
The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.
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.
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.
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…
Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.
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…
We compute the group of link homotopy classes of link maps of two 2-spheres into 4-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn-Rolfsen link map and detected by two self-intersection invariants introduced by Paul Kirk in this setting. As a corollary, we show that any …
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…
Studied invariants on stratified spaces, proving their stability.
Proves Verdier duality for sheaves on stratified spaces.
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…
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 -iterated edge metric on its regular part -parabolic. Moreover, besides stratified pseudomanifolds, the -parabolicity of other classes of singular spaces, such as compac…
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 …
Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stie…
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…
Paper classifies pseudomanifolds over stratified spaces.
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 …
3D analog of Whitney's planarity criterion for 2-complexes.
The paper studies cohomology on incomplete manifolds and stratified 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.
New method defines Gysin maps for stratified spaces, preserving signatures.
We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.
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.
New definition of regular points for PL functions on manifolds.
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…
We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with th…