Derives an explicit formula for transversal indices on S^1-bundles.
problem Computing transversal indices for S^1-bundles over complex projective spaces.
method Derives an explicit formula using indices of elliptic operators on orbit manifolds.
result Proves Lefschetz formula for complex projective spaces with canonical action.
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
The first purpose of this paper is to generalize the well-known Maslov indices of maps of open Riemann surfaces with boundary lying on Lagrangian submanifolds to maps with boundary lying on coisotropic submanifolds in symplectic manifolds. For this purpose, we first define the notion of {\it Maslov loops} of coisotropi…
We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic…
Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.
problem Restricting Morse indices of saddles in gradient-like flows.
method Analyzing invariant manifolds and their intersections for gradient-like flows.
result Morse indices of saddles are either 1 or n-1, no other indices possible.
New examples of non-simple knots in Lens spaces show rich botany.
problem Understanding the variety of Legendrian and transversal knots in Lens spaces.
method Presented new families of non-simple knots in tight Lens spaces.
result More non-isotopic Legendrians topologically isotopic to the n-twist knot in Lens spaces than in S3. This paper explores equivariant elliptic operators and their invariants.
problem Analyzing invariants of equivariant elliptic operators.
method Equivariant heat asymptotics and group representation multiplicities.
result New integer-valued indices and eta invariant for group actions.
Strict concavity proven for growth indicator function of certain groups.
problem Proving strict concavity of growth indicator function for specific groups.
method Smoothness of Manhattan hypersurface and critical-exponent map.
result Strict concavity of growth indicator function for relatively Anosov groups.
Extends symplectic flow results to foliations.
problem Applying hard Lefschetz theorem to symplectic foliations.
method Generalizes results from symplectic flows to foliations.
result Extends transversal hard Lefschetz theorem to transversely symplectic foliations.
Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Study on transverse Ricci solitons on compact foliated manifolds.
problem Characterizing transverse Ricci solitons on compact foliated manifolds.
method Investigation of self-similar solutions of the transverse Ricci flow, analysis of taut Riemannian foliations.
result Established relations between taut Riemannian foliations and transverse Ricci solitons, found examples of transverse Ricci solitons.
The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
New examples show transverse knots are determined by their branched covers.
problem Transverse knots and their isotopy classes.
method Constructing and analyzing non-isotopic transverse knots with contactomorphic cyclic branched covers.
result Transverse isotopy classes of many transverse knots are determined by the contactomorphism type of their cyclic branched covers.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
Defines transverse symbols for foliated manifolds and proves their K-homology class.
problem Transverse index theory for foliated manifolds.
method Using filtrations of tangent bundles, defining transverse symbols, and constructing equivariant KK-classes.
result Transversally Rockland operators yield a K-homology class and there is a Poincare duality result.
Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
New transverse links solve contact manifold problems.
problem Contact manifold structures.
method Transverse links in 3-sphere contact structures.
result All contact 3-manifolds are contact branched covers over a transverse link.
Smooth maps in o-minimal structures are mostly transverse.
problem Transversality of smooth definable maps in o-minimal structures.
method Definable smooth version of Thom transversality theorem, proving nowhere density of non-transverse maps, and a definable version of Trotman's theorem.
result Non-transverse maps are nowhere dense in the definable smooth topology.
Paper characterizes generic transversality, improving on Mather's result.
problem Understanding and defining generic transversality.
method Characterization of transversality based on Mather's work.
result Improves on Mather's transversality result.
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved
The study classifies transverse spheres in flag manifolds and finds new examples.
problem Classifying transverse spheres in flag manifolds.
method Using topological K-theory and constructions of transverse spheres.
result Classification of transverse spheres in various flag manifolds.
We study the transverse Lusternik-Schnirelmann category of a Riemannian foliation on a compact manifold. We obtain a necessary and sufficient condition when the transverse LS category is finite. We also introduce a variation on the concept of transverse LS category, the essential transverse category, and show that this…
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …
We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.
The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.
Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in R3, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally…
Engel manifolds show transverse tori can be made to have various formal invariants.
problem Understanding transverse tori in Engel manifolds.
method Analogous to transverse knots, classify formal invariants and show their uniqueness.
result Engel manifolds can have infinitely many transverse isotopy classes of tori with specific invariants.
Study on geometry and dynamics of transverse subgroups.
problem Understanding the geometry and dynamics of transverse subgroups.
method Survey of recent research on semi-simple Lie groups.
result Recent findings on transverse subgroups of semi-simple Lie groups.
Study on transverse invariant from Khovanov homology and its properties.
problem Understanding the relationship between knot isotopy and transverse isotopy.
method Analyzing fractional Dehn twist coefficients, quasipositivity, and stability of transverse invariants.
result Stability and triviality of the transverse invariant for specific families of braids and knots.
A transverse knot is a knot that is transverse to the planes of the standard contact structure on real 3-space. In this paper we prove the Markov Theorem for transverse braids, which states that two transverse closed braids that are isotopic as transverse knots are also isotopic as transverse braids. The methods of the…
We classify positive transversal torus knots in tight contact structures up to transversal isotopy.
We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide …
Study on lightlike submanifolds in metallic semi-Riemannian manifolds.
problem Characterizing and investigating properties of lightlike submanifolds in metallic semi-Riemannian manifolds.
method Introduced and analyzed subclasses of screen transversal lightlike submanifolds and investigated their geometric properties.
result Necessary and sufficient condition for an isotropic screen transversal lightlike submanifold to be totally geodesic.
New method computes transverse knot invariant using braids.
problem Compute transverse invariant of knots and links.
method Define new combinatorial complex to compute knot Floer homology.
result Identify BRAID invariant of transverse knots and links.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field
Study transverse and Legendrian invariants of certain link cables using Floer homology.
problem Transverse and Legendrian invariants of specific link cables.
method Combining grid diagrams and Floer homology, using inclusion maps of grid complexes.
result Transverse and Legendrian invariants of certain link cables are zero for large q/p. Study classifies twist knots with maximal self-linking number in S^3.
problem Classifying transverse-universal knots in S^3.
method Classification of transverse twist knots with maximal self-linking number.
result Obtained an infinite family of non-transverse-universal knots.
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.
New conditions found for Riemannian foliations with specific conformal fields.
problem Conditions for Riemannian foliations with transversal conformal fields.
method Analyzes conditions for transversal isometry of foliations.
result Found conditions for F to be isometric to the sphere. Study links in contact manifolds using open books and overtwisted disks.
problem Understanding transverse links in contact manifolds.
method Using open book decompositions and overtwisted disks, the support genus of transverse links is studied.
result The support genus of a transverse knot is zero if there is an overtwisted disk disjoint from it.
Study of cosmetic contact surgeries on knots, excluding some cases.
problem Determining cosmetic contact surgeries on transverse knots.
method Analyzing contact surgeries and their outcomes on transverse knots.
result Exclude non-trivial cosmetic contact surgeries for certain transverse knots.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
The paper proves generic transversality and regularity for minimal submanifolds and area-minimizing currents.
problem Transversality and regularity of minimal submanifolds and area-minimizing currents.
method Proves transversality and regularity for generic metrics using Baire category and minimization properties.
result Generic metrics ensure transversality and regularity for minimal submanifolds and area-minimizing currents.