The study establishes a lower bound for the number of Reeb chords using stable Morse numbers.
problem Finding a lower bound for the number of Reeb chords on Legendrian submanifolds.
method Using stable Morse numbers and properties of Lagrangian fillings, the study proves a lower bound for the number of Reeb chords.
result The number of Reeb chords is bounded from below by the stable Morse number of the Lagrangian filling.
A symplectic manifold's diffeomorphism has at least its Morse fixed points.
problem Understanding fixed points of symplectic diffeomorphisms.
method Analyzing Morse theory and symplectic asphericity.
result A generic diffeomorphism has at least the stable Morse number of fixed points.
Exponential growth of stable subgroups in Morse geodesics.
problem Growth rates of stable subgroups in complex groups.
method Theory of automatic structures on Morse geodesics.
result Exponential growth of stable subgroups is faster than their infinite index stable subgroups.
Study rational homology of moduli space via Morse functions, proving stability phenomena.
problem Homology of Deligne--Mumford compactification of moduli space of stable curves.
method Using a family of Morse functions, specifically the sys_T functions, and exploiting geometric and Morse properties.
result Homology of Deligne--Mumford compactification is supported entirely on the boundary in low degrees, and rational homology is finite generated and stable across all genera and marked points.
Stable and Morse subgroups coincide in mapping class groups.
problem Understanding subgroup properties in mapping class groups.
method Analyzing stability and Morse properties in mapping class groups.
result Stability and Morse properties coincide for subgroups of infinite index in mapping class groups.
The study bounds Morse indices of Willmore spheres in relation to min-max sweep-outs.
problem Estimating Morse indices of Willmore spheres.
method Analyzing the sum of Morse indices of Willmore spheres in min-max sweep-outs.
result At most one Willmore sphere can have index 1 among those realising min-max sphere eversion.
The number of critical points of a knot's electric potential is at least twice the tunneling number plus two.
problem Understanding the critical points of the electric potential of a knot.
method Morse theory and stable manifold theory.
result The number of critical points of the potential is at least 2t(K) + 2.
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.
Heat equation on projective spaces leads to stable minimal Morse functions.
problem Heat equation behavior on projective spaces.
method Proof of minimal Morse function stability for arbitrary initial conditions.
result Solution of heat equation becomes stable minimal Morse function.
Unified approach to studying hyperbolic groups using stable subspaces and Morse boundaries.
problem Understanding the geometric and algebraic properties of hyperbolic groups.
method Unified approach to viewing geodesic metric spaces as unions of stable subspaces, using quasi-convex subsets and direct limits of Gromov boundaries.
result Unified understanding of stable subgroups and Morse boundaries, leading to new quasi-isometry invariant dimensions.
Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.
problem Proving a method to upgrade Morse-Bott homology to stable homotopy invariants rigorously.
method Rigorous construction of stable normal framings and proof of stable homotopy type recovery.
result The stable homotopy type recovers Σ∞+M and Thom spectra for all reduced KO-theory classes.
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…
We study the structure of the smooth manifold which is defined as the intersection of a stable manifold and an unstable manifold for an invariant Morse-Smale function.
We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associat…
This paper explores when homeomorphisms between Morse boundaries of spaces induce quasi-isometries.
problem When does a homeomorphism between Morse boundaries of spaces imply a quasi-isometry?
method Investigates quasi-mobius homeomorphisms and 2-stability conditions.
result A homeomorphism between Morse boundaries of proper, cocompact spaces is induced by a quasi-isometry if and only if it is quasi-mobius and 2-stable.
CAT(0) spaces' Morse boundaries uniquely identify them.
problem Determining when a homeomorphism of Morse boundaries implies a quasi-isometry.
method Investigates Morse boundaries of cocompact CAT(0) spaces and their homeomorphisms.
result A homeomorphism of Morse boundaries is induced by a quasi-isometry if and only if it is quasi-mobius and 2-stable.
Generalizes Floer homotopy via Morse-Bott theory.
problem Constructing equivariant models in Floer theory.
method Morse-Bott theory, flow categories, stable homotopy types.
result Equivalence of Borel equivariant spectra for certain Lagrangians.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
Let f be a Morse function on a closed manifold M, and v be a Riemannian gradient of f satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function f associates to these data the Morse comple…
Graph products inherit Morse local-to-global property from their components.
problem Generalizing local-to-global property to graph products of infinite groups.
method Generalizing maximization procedure for relatively hierarchically hyperbolic groups and showing stable embeddings.
result Graph products of infinite Morse local-to-global groups have the Morse local-to-global property.
The paper classifies stable hypersurfaces and gives bounds for Morse index.
problem Stability and instability of type-II partitioning problem.
method Complete classification of stable stationary hypersurfaces, topological restrictions, lower bound for Morse index.
result Complete classification of stable type-II stationary hypersurfaces in a ball.
Develops deformation theory for mapping spaces related to Morse theory and Bott-Thom isomorphism.
problem Studying weak homotopy equivalences and decompositions of vector bundles.
method Morse theory on path spaces, deformation theory, Clifford representations, Bott-Thom isomorphism.
result Stable decompositions of vector bundles over sphere bundles derived from Clifford representations.
CW decomposition of manifolds with Morse functions.
problem CW decomposition of manifolds with Morse functions.
method Generic gradientlike vector field and Morse function.
result Stable manifolds provide a CW decomposition.
The local-to-global property is proven for Morse quasi-geodesics in various groups.
problem Proving local-to-global properties for Morse quasi-geodesics in different groups.
method Developing a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces.
result Generalization of combination theorems for stable subgroups of various groups.
Constructs a Morse-Bott function on symplectic Grassmannians.
problem Defines a function on symplectic Grassmannians.
method Uses a compatible linear complex structure to construct a quadratic Morse-Bott function.
result Critical loci consist of subspaces splitting into isotropic and complex parts.
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
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.
Paper examines stability of non-proper smooth functions.
problem Stability of non-proper smooth functions.
method Analyzes properties of Morse and Nash functions to establish stability conditions.
result Shows existence of stable but not infinitesimally stable functions.
In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration usi…
This paper proves properties of convex integrands and their duals.
problem Properties of convex integrands and their duals.
method Analyzing C∞ and stable convex integrands. result Dual convex integrands of stable convex ones are also stable.
Classifies 3D manifold diffeos without heteroclinic curves.
problem Classifying Morse-Smale diffeomorphisms on 3-manifolds.
method Defines equivalence class by embedding heteroclinic laminations.
result Equivalence class determined by laminations' embedding.
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…
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.
We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs (N,L), established in [2,3], as a dynamical thickening of the stable…
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
The Morse-Novikov number MN(L) of an oriented link L in the 3-sphere is the minimum number of critical points of a Morse map from the complement of L in the 3-sphere to the circle representing the class of a Seifert surface for L (e.g., the Morse-Novikov number of L is zero if and only if L is fibered). We develop vari…
Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.
problem Behavior of Morse-Novikov number under knot operations.
method Additivity under connected sum and invariance under cabling.
result Morse-Novikov number is additive under connected sum and unchanged by cabling.
We introduce the \emph{metric spectrum}, which measures the exponential rate of approximation to an isolated invariant set of points starting in its stable set, and relate it to the Lyapunov spectrum. We determine the metric spectrum of each Morse component of the finest Morse decomposition of a linear induced flow on …
The paper characterizes subgroup stability via limit sets on the Morse boundary.
problem Characterizing subgroup stability in various settings.
method Characterization via limit sets on the Morse boundary.
result Stability of a subgroup is equivalent to all limit points being conical or horospherical.
The article shows how to count small eigenvalues without assuming Morse functions.
problem Counting small eigenvalues without assuming Morse functions.
method Using the Witten Laplacian and persistent cohomology.
result The rescaled logarithms of small eigenvalues are determined by bar code lengths.
New result classifies hierarchically hyperbolic groups based on their Morse boundaries.
problem Classifying hierarchically hyperbolic groups using Morse boundaries.
method Generalizing a result on Gromov boundaries to Morse boundaries, showing quasi-isometry if and only if there's a homeomorphism.
result Spaces are quasi-isometric if and only if there's a 2-stable, quasi-möbius homeomorphism between their Morse boundaries.
Uniformly perfect Morse boundaries characterize geometric properties of groups.
problem Characterizing geometric properties of groups using Morse boundaries.
method Introducing and geometrically characterizing uniformly perfect Morse boundaries for proper geodesic metric spaces.
result The Morse boundary of any finitely generated, non-elementary group is uniformly perfect if it is nonempty.
Stability of Morse index for Yang-Mills connections in 4D.
problem Stability of critical points in Yang-Mills energy relaxation.
method Establishing lower semi-continuity of Morse index and upper continuity of Morse index plus nullity.
result Yang-Mills fields are more stable than harmonic maps in 4D.
Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the f…
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. 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-Novikov numbers for frame spun knots and surface-links.
problem Compute Morse-Novikov numbers for frame spun knots and surface-links.
method Apply circle-valued Morse theory to frame spun knots and surface-links.
result Obtain a formula relating the Morse-Novikov numbers of frame spun knots and their complements.