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. Local-to-global principle for Morse actions on symmetric spaces.
problem Recognizing Morse actions on symmetric spaces.
method Equivariant Morse quasiisometric embeddings of trees into symmetric spaces.
result Algorithmic recognizability of Morse actions and construction of Morse Schottky subgroups.
The paper studies Morse theory for Lie algebra actions on Riemannian foliations.
problem Morse theory for Lie algebra actions on Riemannian foliations.
method Equivariant Morse-Bott theory on leaf space.
result Established foliated versions of Morse-Bott lemma and handle presentation theorem.
New topology shows Morse boundaries are topologically invariant.
problem Topological invariance of Morse boundaries in CAT(0) cubical groups.
method New hyperbolic topology induced by group actions on hyperbolic spaces.
result Sublinearly Morse boundaries are homeomorphic up to visual topology.
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.
New manifolds help understand group actions on complex chains.
problem Understanding compact Lie group actions on Morse and Floer chains.
method Introduced new manifolds called forest biassociahedra and bimultiplihedra.
result Derived algebraic structures like bialgebras and bimodules.
We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are allowed to "jump" along orbits of the S^1-action for finite time intervalls.
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…
Entropy of critical points generalizes Morse theory.
problem Extending Morse theory to group actions.
method Entropy of homologically detectable critical points.
result Entropy lower bound on critical points.
Let M be a compact oriented simply-connected manifold of dimension at least 8. Assume M is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove M has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Mor…
Proves a theorem for mechanical systems with reflections.
problem Action functionals on paths with reflections.
method Proves a Morse index theorem for action functionals on paths that can reflect.
result Action functionals on paths with reflections have a well-defined Morse index.
We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
Study on Morse homology for reflection actions on manifolds.
problem Finding metrics making equivariant Morse-Smale pairs stable.
method Counting broken trajectories to define equivariant Thom-Smale-Witten complexes.
result Stable Morse-Smale condition is generic for metrics.
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
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…
Analytic realization of Thom-Smale complex for G-manifolds.
problem Realizing Thom-Smale complex for G-manifolds with Lie group action.
method Using G-invariant Witten instanton complex associated with a Morse-Bott function.
result Generalized Thom-Smale complex for G-manifolds including horizontal direction influence.
We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Porti. We show that, if a sequence of Morse representations ρn:Γ→G is (strongly) unbounded in the character variety, the group must have a very particular structure.
Let X be a compact connected CR manifold of dimension 2n−1,n≥2. We assume that there is a transversal CR locally free S1 action on X. Let Lk be the k-th power of a rigid CR line bundle L over X. Without any assumption on the Levi-form of X, we obtain a scaling upper-bound for the partial Szegő …
Study of special subgroups of automorphism groups of Kronrod-Reeb graphs for Morse functions on 2-torus.
problem Characterizing subgroups of automorphism groups of Kronrod-Reeb graphs.
method Analysis of diffeomorphisms preserving Morse functions on 2-torus.
result Full description of special classes of automorphism groups.
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.
Develops Bialynicki-Birula and Morse-Bott theory for complex analytic spaces.
problem Extending classical theories to complex analytic spaces with holomorphic C∗ actions. method Extends Bialynicki-Birula and Morse-Bott theories to non-compact complex manifolds and analytic spaces, proving existence and deriving geometric consequences.
result Existence of Bialynicki-Birula decompositions for C∗-invariant subspaces in complex manifolds. We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…
We introduce Morse-type inequalities for a holomorphic circle action on a holomorphic vector bundle over a compact Kaehler manifold. Our inequalities produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomology in terms of the data of the fixed points and of the symplectic reduction. Th…
Let X be a proper geodesic metric space and let G be a group of isometries of X which acts geometrically. Cordes constructed the Morse boundary of X which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in G by their fixed po…
Study group actions on hyperbolic spaces to find algebraic and geometric properties.
problem Understanding algebraic properties of group actions on hyperbolic spaces.
method Developed an algorithm in the group algebra to show free ideals generated by few elements.
result Derived lower bounds on Morse complexity of closed hyperbolic manifolds.
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.
The paper studies topological and dynamic properties of boundaries in geometric group actions.
problem Understanding the topological and dynamic properties of boundaries in geometric group actions.
method Developed and studied sublinearly Morse and quasi-redirecting boundaries for proper geodesic spaces with geometric group actions.
result Proved that the action of a group on the boundaries is minimal and that the boundaries are topological spaces.
In Garside groups, axes of Morse elements are strongly contracting.
problem Understanding the dynamics of Morse elements in Garside groups.
method Analyzing the Cayley graph of Garside groups modulo their center, using Garside generators.
result Morse elements act loxodromically on the additional length graph of Garside groups.
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.
New spectral estimates for minimal surfaces with boundary conditions.
problem Quantifying the Morse index of free boundary minimal surfaces.
method Adapted Montiel-Ros partitioning methods to compact manifolds with boundary, accounting for mixed and group actions.
result Explicit two-sided linear bounds on the Morse index for minimal surfaces.
Multicurve stabilizers' extensions are hierarchically hyperbolic.
problem Characterizing the structure of multicurve stabilizers' extensions.
method Proving the extensions of multicurve stabilizers are hierarchically hyperbolic groups.
result Extensions of multicurve stabilizers are hierarchically hyperbolic.
This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out in detail. In contrast to standard treatments, gradient flow lines and their str…
The study simplifies complex functions on surfaces using a special transformation.
problem Understanding functions with degenerate singularities on various surfaces.
method Established a 'normal form' for functions using a specific transformation.
result Any function in the class can be simplified to a 'simplest' Morse function through a transformation.
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are q…
The paper proves rigidity for mapping class group actions on metrics of positive scalar curvature.
problem Rigidity of mapping class group actions on metrics of positive scalar curvature.
method Parametrised Morse theory, 2-index theorem, sphere computations.
result Rigidity theorem for mapping class group action on positive scalar curvature metrics.
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.
Cylindrical contact homology computed for links of simple singularities.
problem Computing cylindrical contact homology for links of simple singularities.
method Perturbing degenerate contact form on S3/G with an invariant Morse function, achieving nondegeneracy up to an action threshold. Recovers cylindrical contact homology via direct limit of action filtered homology groups. result Ranks of cylindrical contact homology groups are given in terms of ∣extConj(G)∣, demonstrating a form of the McKay correspondence. New theory for Hamiltonian actions on special geometric structures.
problem Hamiltonian actions on cosymplectic groupoids.
method Developed a moment map theory for 0-shifted cosymplectic structures.
result Established a version of the Kirwan convexity theorem.
The paper studies heat kernel asymptotics for Kohn Laplacians on CR manifolds.
problem Analyzing heat kernel asymptotics for Kohn Laplacians on CR manifolds.
method Establishing asymptotics of heat kernels and equivariant heat kernels on CR manifolds.
result Heat kernel asymptotics for Kohn Laplacians on CR manifolds are derived.
It is well known that the cohomology groups of a closed manifold M can be reconstructed using the gradient dynamical of a Morse-Smale function f:M→R. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of f in term of Betti numbers of $…
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
We study the L2 gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal G-bundle over the sphere S2 from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space ΩG of based loops in the compact Lie group G. An iso…
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic pi…
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
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.
We study the classical action functional $\SMC_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action $\AMC_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M.…
We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existen…