Extends Morse structures to partial open books for studying contact 3-manifolds.
problem Analyzing contact 3-manifolds with convex boundaries.
method Develops Morse structures on extendable partial open books.
result New tools for studying contact 3-manifolds with convex boundaries.
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.
The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.
problem Understanding Morse diagrams and their behavior under Murasugi sums.
method Examination of combinatorial Morse structures, open book decompositions, and contact structures.
result Diagrammatic criterion for detecting overtwisted contact structures and classification of Morse diagrams for one-holed torus pages.
This paper equips Morse cochain complexes with A∞-algebra structures.
problem Equipping Morse cochain complexes with A∞-algebra structures. method Analogous to K. Fukaya's definition, this paper provides a detailed treatment of Abouzaid's approach.
result Provides a coherent and detailed treatment of Abouzaid's approach to Morse cochain complexes.
The article details Fukaya's A∞-structure of Morse complexes.
problem Understanding the A∞-structure of Morse complexes.
method Detailed exposition of Fukaya's A∞-structure, emphasizing transversality arguments.
result The A∞-structure is homotopically independent of choices.
Study Morse representations in Lie groups, showing specific group structures.
problem Understanding Morse representations in Lie groups.
method Analyzing sequences of Morse representations and their unboundedness.
result Groups with unbounded Morse representations have specific structures.
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.
Survey on SYZ mirror symmetry using Morse theory.
problem Understanding SYZ mirror symmetry with quantum corrections.
method Using Witten-Morse theory and Fukaya's combinatorial structures.
result Explicit relation between geometric and combinatorial structures.
3D foliation study finds Poisson structure with specific singularities.
problem Characterizing Poisson structures on Bott-Morse foliations in 3D.
method Analyzes Bott-Morse foliations, computes Poisson bivectors and symplectic forms.
result Linear, singular Poisson structure of rank 2 with Bott-Morse singularities found.
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. The Morse complex is shown to be an infinite functor.
problem Understanding the structure of Morse complexes as infinite functors.
method Showed the Morse complex of a compact Lie monoid can be given the structure of an f-bialgebra and defined an ∞-functor.
result Obtained two other ∞-functors mapping manifolds and actions to their Morse complexes.
Study on higher-dimensional quasigeodesics in metric spaces.
problem Understanding asymptotic structure of Morse quasiflats.
method Proving asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity.
result Morse quasiflats exhibit Euclidean volume growth rigidity.
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.
New Coxeter groups have unique boundary structures.
problem Understanding boundaries of Coxeter groups.
method Recursive construction and amalgamation of CAT(0) groups.
result Totally disconnected Morse boundaries for new Coxeter groups.
The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.
problem Analyzing cohomology of complex manifolds using Morse-Novikov theory.
method Establishing invariants, the Leray-Hirsch theorem, and blow-up formula for Dolbeault-Morse-Novikov cohomology.
result Established relations and stabilities of dimensions under complex structure deformations.
Study classifies Morse functions with 4 critical points on immersed 2-spheres.
problem Classifying Morse functions with 4 critical points on immersed 2-spheres.
method Used dual graph of immersion and Reeb graphs to classify functions.
result Found all possible structures of the functions.
We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We gi…
Study describes Morse flows on a torus with up to six singular points.
problem Understanding the structure of Morse flows on a torus with a hole.
method Used separatrix diagrams to describe topological structures and saddle-node bifurcations.
result Identified all possible topological structures of Morse flows with at most six singular points.
The paper explores additional structures on Morse boundaries to distinguish hyperbolic spaces up to quasi-isometry.
problem Distinguishing hyperbolic spaces up to quasi-isometry using additional structures on Morse boundaries.
method Investigates additional structures on Morse boundaries and proves conditions for a homeomorphism to be induced by a quasi-isometry.
result A homeomorphism between Morse boundaries of hyperbolic spaces is induced by a quasi-isometry if and only if it is bihölder, quasi-symmetric, or strongly quasi-conformal.
This paper categorifies Morse theory for manifolds with boundaries.
problem Categorifying Morse theory for manifolds with boundaries.
method Defining a relative Morse complex using handlebody decomposition and constructing an A∞-category structure. result The homology of the relative Morse complex is isomorphic to the relative singular homology.
Classifies Morse flows on 3-sphere with specific saddle connections.
problem Classifying Morse-Smale flows on a 3-sphere with specific saddle connections.
method Used generalized Heegaard diagrams (Pr-diagrams) to classify flows.
result Found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection.
The paper studies Morse flows on 3-manifold boundaries with fixed points.
problem Classifying Morse flows on 3-manifold boundaries.
method Constructing a Pr-diagram as a topological invariant.
result A complete topological invariant of Morse flows on 3-manifold boundaries.
New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.
problem Defining Morse-Bott functions on non-linear Stiefel manifolds.
method Replacing linear height function with a quadratic one, proving it as a Morse-Bott function.
result Critical submanifolds are fibrations of products of Grassmannians, not Grassmannians themselves.
We give elementary constructions of manifold with corner structures and associative gluing maps on compactifications of spaces of infinite, half infinite, and finite Morse flow lines.
Novel Morse theory for mapping cone cohomology.
problem Cohomology of mapping cones varies with closed forms.
method Introduced a Morse complex for mapping cones.
result Cohomology of cone Morse complex is isomorphic to mapping cone cohomology.
New pairing defined from Morse complexes for compact manifolds.
problem Defining a pairing for compact manifolds with Morse functions.
method Constructing Morse complexes and a short exact sequence.
result Induces the intersection product in homology.
This paper proves some results on negative gradient dynamics of Morse functions on Hilbert manifolds. It contains the compactness of flow lines, manifold structures of certain compacti- fied moduli spaces, orientation formulas, and CW structures of the underlying manifolds.
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.
problem Understanding cohomology groups on foliated manifolds.
method Defined and studied Morse-Novikov cohomology relative to a foliation, proving homotopy invariance and extending to more general forms.
result Proved Hodge theorem and Poincaré duality for reduced leafwise Morse-Novikov cohomology groups on Riemannian foliations.
In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular rea…
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.
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.
Minimal Morse functions on Poincaré dodecahedral space are selected via spectral properties.
problem Identifying minimal Morse functions on the Poincaré dodecahedral space.
method Spectral selection property P, obstruction principle, conformal variations, finite dimensional reduction.
result Restoration of minimal Morse selection on the Poincaré dodecahedral space via spectral mechanisms.
A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same …
Solves geometric problems using fully nonlinear equations and Morse theory.
problem Geometric problems, specifically Loewner-Nirenberg and Yamabe problems.
method Investigates structure of fully nonlinear equations and applies Morse theory techniques.
result Constructs admissible metrics under weak conditions and demonstrates topological obstructions.
We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse inde…
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.
Develops graphical calculus for monoidal categories with twisted pivotal structures.
problem Constructing modules for surfaces with Morse functions or foliations.
method Graphical calculus and string nets for monoidal categories with twisted pivotal structures.
result Twisted string net modules assemble in an oriented categorified 2-TQFT.
The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.
problem Understanding the homology of orbifolds.
method Constructing invariant and coinvariant Morse chain complexes for orbifolds.
result The homology of coinvariant Morse complexes computes the singular homology of the underlying space.
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.
New Morse theory for path homology with coefficients.
problem Defining operations on path homology with differential graded coefficients.
method Using tools from Morse theory and string topology.
result Morse-theoretic description of a product on path homology.
Witten uses SUSY to prove classical Morse inequalities.
problem Classical Morse inequalities from a physical perspective.
method SUSY Quantum Mechanics and Schrödinger Operators.
result Witten's proof of Weak Morse Inequalities.
Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.
problem Capturing generic directions in mapping class groups and Teichmüller spaces.
method Develops tools for modeling hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes.
result Sublinear Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph.
Proves immediate transversality for conic singularities.
problem Transversality issues in Morse complexes with conic singularities.
method Proves immediate transversality for conic singularities in Morse complexes.
result Immediate transversality holds for conic singularities.
Given, in the Lagrangian torus fibration R4→R2, a Lagrangian submanifold L, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies associated to the generating function of L, and it is provided with a holomorphic s…
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. We study these invariants using the Morse-Novikov theory and H…
Study LCS structures on Lie algebras of type I, proving trivial Morse-Novikov cohomology and constructing solvmanifolds.
problem Locally conformal symplectic structures on Lie algebras of type I.
method Analyzing Lie algebras of type I, proving trivial Morse-Novikov cohomology, and constructing solvmanifolds.
result LCS structures on Lie algebras of type I are of the first kind and can be used to construct compact solvmanifolds.