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…
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48 results for “stratified Morse theory”
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…
Develops analytic methods for Lefschetz and Morse theories on stratified pseudomanifolds.
problem Analytic framework for Lefschetz and Morse theories on stratified pseudomanifolds.
method Heat kernel and Witten deformation based techniques for global and local Lefschetz numbers and Morse polynomials.
result Formulas for Lefschetz numbers and Morse polynomials as supertraces over cohomology groups of Hilbert complexes.
Holomorphic Witten instanton complexes on stratified pseudomanifolds with Kähler wedge metricsmath.DG
This paper extends Witten's holomorphic Morse inequalities to singular spaces.
problem Applying Witten's holomorphic Morse inequalities to singular spaces.
method Constructing Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds.
result Extends Witten's holomorphic Morse inequalities to singular spaces.
The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…
Given a closed Riemannian manifold of dimension and a Morse-Smale function, there are finitely many -part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of -part broken trajectories is always at least the hyperbolic volume. The proof…
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality…
Link concordance equals homotopy for high-dimensional spheres.
problem Understanding when immersions of high-dimensional spheres are homotopically trivial.
method Developed stratified Morse theory for generic immersions, using gradient-like vector fields and Cerf theory.
result Every link of high-dimensional spheres is homotopically trivial, resolving a long-standing conjecture.
New discrete cobordism category for nested manifolds and relations to algebraic structures.
problem Discrete cobordism category for nested manifolds.
method Stratified Morse theory, Cyl-objects, doubling construction, cylindrical bar construction.
result Relations between Cyl-objects and algebraic structures like Temperley-Lieb algebras.
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 …
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…
Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularitiesmath.GT
New formulae connect topological and geometric properties of singular spaces.
problem Understanding the relationship between singular spaces and their Morse critical points.
method Generalization of Morse theory to non-degenerate locally tame singularities.
result Difference of Brasselet numbers related to Morse critical points of functions.
Let be a smooth closed orientable surface, and let be the space of Morse functions on such that at least critical points of each function of are labeled by different labels (enumerated). Endow the space with -topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\c…
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.
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.
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 describe an algorithm that associates to each positive real number and each finite collection of planar pixels of size a planar piecewise linear set with the following additional property: if is the collection of pixels of size that touch a given compact semialgebraic set , then the …
Adam achieves optimal convergence in deep ReLU networks via novel Kakeya bounds.
problem Training deep ReLU networks using Adam in non-smooth settings.
method Stratified Morse theory and Kakeya bounds to analyze region crossings and convergence.
result First global-optimal convergence for Adam in non-smooth, non-convex ReLU landscapes.
We consider a Morse function and a Morse-Smale gradient-like vector field on a compact connected oriented 3-manifold such that has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of can be isotoped into one so that the trajectory spaces of the new flow pro…
The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation are (1) Looijenga's computation of the rational cohomology ring of M_3, the modul…
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…
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.
Let be a stratum of a compact stratified space . It is equipped with a general adapted metric , which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, has a general type, which is an extension of the type of an adapted metric. A restriction on this …
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 …
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…
We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
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.
Jet spaces on Carnot groupsmath.DG
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 can be embedded in a jet space over a stratified Lie group of step .
New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
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…
Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.
problem Quantum cohomology of symplectic manifolds with -actions.
method Floer theory applied to -actions on symplectic manifolds.
result Constructs a family of filtrations on quantum cohomology for Conical Symplectic Resolutions.
Morse theory extended to noncompact manifolds with complex geometric data.
problem Extending Morse theory to noncompact manifolds with intricate geometric and homotopy data.
method Defining Morse homology for pairs of manifolds and related geometric/homotopy data, constructing a homotopy coherent diagram of linear maps, and showing it computes Morse homology.
result Morse homology can be computed using a chain complex derived from a homotopy coherent diagram.
The paper explores conditions for compactness and finiteness in stratified homotopy theory.
problem Conditions for compactness and finiteness in stratified homotopy theory.
method Analyzes conditions for compactness and finiteness in stratified homotopy theory, providing sufficient conditions and deducing results.
result Conditions for compactness and finiteness in stratified homotopy theory are established.
The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.
problem Understanding symmetries and cohomology in foliated manifolds with stratified boundaries.
method Developed a novel formalism for the Gamma-set and defined an Ihara zeta function to encode symmetries. Investigated the relationship between holonomy and zeta functions, and analyzed how the twist map impacts cohomology.
result Conjectured a duality between holonomy fixed points and the poles of the Ihara zeta function, extending to twisted cohomology classes.
Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regul…
The paper introduces Morse theory for Lie groupoids and proves inequalities.
problem Defining Morse theory for Lie groupoids and studying their properties.
method Introducing Morse Lie groupoid morphisms and proving their Morita invariance.
result Established Morse theory for Lie groupoids and proved Morse inequalities.
This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \cite{K2}, for \emph{traversally generic flows} on -manifolds , we embark on a detailed and somewhat tedious study …
Study compares thimbles to Morse theory on Lie theory models.
problem Exploring thimbles in Landau-Ginzburg models using Morse theory.
method Constructing real Lagrangian thimbles and comparing to gradient flow manifolds.
result Explicit construction and comparison of thimbles to gradient flow manifolds.
New Morse functions on curve moduli space via geodesics.
problem Understanding the moduli space of curves via geometric and combinatorial methods.
method Introducing Morse functions based on geodesic lengths and analyzing their critical points and indices.
result Found new explicit Morse functions on , leading to a combinatorial cell decomposition.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.
problem Computing multiparameter persistence with new tools and methods.
method Adapting Forman's theory to vectorial setting and using combinatorial topological dynamics.
result Established more general result for sublevel sets and found a way to induce Morse decomposition.
Study Morse complexity of manifolds and homology classes, proving bounds and implications.
problem Understanding Morse complexity of manifolds and homology classes.
method Used surgery theory and index theory to prove upper and lower bounds.
result Locally symmetric spaces of Lie groups with discrete series representations do not admit open book decompositions.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.
problem Understanding Morse functions on manifolds with boundaries.
method Constructing a cellular structure and analyzing its algebraic properties.
result Estimation of the number of critical points of a Morse function with boundary conditions.
Study minimax estimation of stratified structure from i.i.d. samples.
problem Estimating stratified structure from i.i.d. samples of stratified mixtures of immersed manifolds.
method Ascending hierarchical co-detection of points belonging to different layers, identifying number of layers and their dimensions, assigning points to layers accurately, estimating tangent spaces optimally.
result Achieves optimal estimation of mixture components at their optimal dimension-specific rates adaptively.