Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Random walk constructs Morse functions on surfaces.
problem Creating Morse functions on surfaces.
method Random walk method to construct Morse functions.
result Small set of Morse functions approximates any other function.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
problem Mapping discrete Morse functions to shellings for topological analysis.
method Inducing Morse shellings on the second barycentric subdivision of a simplicial complex.
result Critical tiles of induced shellings correspond to critical faces of the discrete Morse function.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. Defines concordance of Morse functions on manifolds and presents a condition.
problem Deciding if two Morse functions on the same manifold are concordant.
method Introduces concordance as a stronger equivalence relation than cobordism, and presents a necessary and sufficient condition for concordance.
result A necessary and sufficient condition for two Morse functions to be concordant is presented.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
Study families of Morse functions for manifolds with boundary.
problem Characterize degeneracies in 1-parameter families of Morse functions.
method List all possible degeneracies in generic 1-parameter families.
result Identified all degeneracies in generic 1-parameter families.
In the present paper, we define Morse-Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse-Bott inequalities for these manifolds.
A Morse 2-function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse 2-function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse 2-functions …
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular N-cube chains when the function is constant. We show that the ho…
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…
We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated clo…
Relative cup-length defined for non-Morse functions on manifolds.
problem Defining a lower bound on critical points of non-Morse functions.
method Using local Morse cohomology and cohomology of isolating neighborhoods.
result A lower bound on critical points stronger than absolute cup-length.
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 Mg,n, leading to a combinatorial cell decomposition. The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.
Let f:M→R be a Morse-Bott function on a finite dimensional closed smooth manifold M. Choosing an appropriate Riemannian metric on M and Morse-Smale functions fj:Cj→R on the critical submanifolds Cj, one can construct a Morse chain complex whose boundary operator is…
The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized Heegaard diagram is built. The criteria of Morse function conjugation are given…
The goal of this paper is to classify pairs of Morse functions in general position modulo the action of different groups.In particular, we obtain the classification of generic pairs of Morse functions, with or without target diffeomorphisms, and that of quotients of Morse functions.We will also present a lemma which gi…
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 obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold (M,ω) with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)…
Paper studies gradient fields from discrete Morse functions for watershed-cut computation.
problem Computing watershed-cuts from discrete Morse functions.
method Discrete Morse Theory and simplicial stacks.
result Minimum Spanning Forest of dual graph is induced by gradient vector field.
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.
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.
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…
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graph…
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.
The paper solves conditions for extending circle-valued Morse functions.
problem Conditions for extending circle-valued Morse functions on closed orientable surfaces.
method Provided necessary and sufficient conditions for the existence of a non-singular extension.
result Necessary and sufficient conditions for the existence of a non-singular extension of a circle-valued Morse function.
The Thurston spine's properties are studied in relation to Morse-Smale complexes.
problem Understanding the Thurston spine's local properties and their global implications.
method Analyzes the Thurston spine as a subset of Teichmüller space and studies its local properties in relation to the systole function.
result The Thurston spine satisfies properties analogous to Morse-Smale complexes, demonstrating its topological significance.
We generalize Cohen & Jones & Segal's flow category whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points to an n-category. The n-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to const…
Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.
problem Cohomology of symplectic manifolds under different metrics and Morse functions.
method Symplectic Morse complex with gradient flows and Witten deformation.
result Cohomology of the complex is isomorphic to Tsai, Tseng, and Yau's cohomology and independent of metrics and Morse functions.
We call a Morse function f on a closed manifold k-constrained if neither f nor −f has critical points of indefinite Morse index <k. In this paper we study bordism groups of k-constrained Morse functions, and thus interpolate between the case k=1 of bordism groups of Morse functions (computed by Ikegami…
The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtra…
The systole function has a universal index gap on moduli spaces.
problem Understanding the index gap of systole functions on moduli spaces.
method Analyzing Morse theory properties of systole functions on moduli spaces and their compactifications.
result There exists a universal constant C>0 such that any critical point in Mg,n has Morse index at least Cloglog(g+n). Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper. Also we get an estimation on Arnold's question on a number of critical points of a Morse function with given boundary condition.
Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.
problem Disproving the Borde-Sorkin conjecture on causal continuity of Morse spacetimes.
method Provided a counterexample with low regularity causal structure and causal bubbling.
result Borde-Sorkin conjecture does not hold for Morse spacetimes with large anisotropy.
The study characterizes 3D manifolds using specific Morse-Bott functions.
problem Characterizing 3D manifolds represented as connected sums of Lens spaces, S2imesS1, and torus bundles. method Using Morse-Bott functions to classify the manifolds.
result Explicit characterization of the manifolds via certain Morse-Bott functions.
Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M, and F1 the space of framed Morse functions, both endowed with C∞-topology. The space F0 of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookright…
4-manifolds can be uniquely described as loops of Morse functions.
problem Describing 4-manifolds as loops of Morse functions.
method Proving uniqueness for each of four descriptions of 4-manifolds.
result Two loops of Morse functions yielding diffeomorphic 4-manifolds are related by specific moves.
Study reconstructs Morse-Bott functions with specific preimage conditions on 3D manifolds.
problem Reconstructing Morse-Bott functions with prescribed preimages on 3D manifolds.
method Conditions and approach based on previous work by Sharko and others.
result New result on reconstruction of nice smooth functions with specified preimages.
Regularizers change the geometric properties of loss functions in neural networks.
problem Understanding how different regularizers affect the geometric properties of loss functions in neural networks.
method Examined several regularizers, including weight decay, to determine if the regularized loss function becomes Morse.
result For certain regularizers, the regularized loss function becomes Morse, indicating a change in geometric properties.
Proves unique symplectic Lefschetz fibration from Morse functions.
problem Mapping Morse functions to symplectic Lefschetz fibrations.
method Homotopically unique complex-valued symplectic Lefschetz fibration on cotangent bundles.
result Existence and uniqueness of symplectic Lefschetz fibrations.
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 condition for reconstructing Morse functions on 3D manifolds.
problem Reconstructing Morse functions with specific level sets.
method Studied a necessary and sufficient condition for reconstruction.
result New condition strengthens previous sufficient conditions.