New categories for surfaces link to contact geometry.
problem Understanding contact structures on surfaces.
method Associate differential graded categories to surfaces.
result Homotopy category of these categories is triangulated.
New flow category for contact manifolds from Reeb orbits.
problem No direct problem stated; focuses on new construction.
method Adapting Kuranishi charts to contact setting, associating flow category based on Reeb orbits and pseudo-holomorphic buildings.
result Lifts contact homology and associates flow bimodule to exact symplectic cobordisms.
We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The Z2 SFH of these solid tori forms a "categorification of Pascal's triangle", and contact structur…
The paper studies contact categories of disks and their embeddings.
problem Understanding contact structures on disks and their categories.
method Construction of contact categories and their universal covers, embedding into triangulated envelopes.
result The universal cover of the contact category of a disk can be embedded into its triangulated envelope.
Isomorphism found between Floer homology and contact geometry.
problem Connecting Floer homology and contact geometry.
method Demonstrated isomorphism between strand algebra and contact category.
result Direct correspondence between Floer homology and contact geometry concepts.
This paper explores A-infinity structures in contact categories and strand algebras.
problem Understanding A-infinity structures in contact categories and strand algebras.
method Explicit constructions and properties of A-infinity operations are established.
result Conditions for the vanishing and nonvanishing of A-infinity operations are derived.
Constructs non-unital monoidal category of contact manifolds and Legendrian correspondence calculus
problem Constructing a non-unital monoidal category of contact manifolds without contact forms
method Developing contact topology without contact forms and defining the star product
result Proving the associativity of the star product and the pentagon axiom
New algebraic construction of knot contact homology using perverse sheaves.
problem Algebraic construction of knot contact homology.
method Defining a DG category with a braid group action on perverse sheaves.
result The endomorphism algebra of a distinguished object in the category matches fully noncommutative knot DGA.
Functor connects symplectic and contact structures via cutting and blowups.
problem Establishing a functorial relationship between symplectic and contact structures.
method Developed a cutting procedure and its inverse for manifolds with boundary and equivariant transverse maps, then applied it to non-symplectic and non-contact structures.
result Obtained an inverse functor for equivariant radial-squared blowups.
Augmentations and sheaves linked for Legendrian graphs.
problem Understanding categorical Legendrian isotopy invariants.
method Equivalence between augmentation category and DG category of sheaves.
result Proved 'augmentations are sheaves' for Legendrian graphs.
In this paper we construct an A∞-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra A(Λ) and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearised Le…
Develops gluing theory for contact instantons and pseudoholomorphic curves.
problem Constructing contact instanton Floer cohomology and Fukaya-type category.
method Gluing theory of contact instantons and pseudoholomorphic curves in symplectization context.
result Construction of Legendrian contact instanton homology and moduli spaces of holomorphic buildings.
Two rigidity results for Legendrian singularities in complex-analytic category.
problem Understanding singularities of Legendrian subvarieties in contact manifolds.
method Using the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle.
result Normal Legendrian singularities are deformation-rigid.
The paper extends contact geometry concepts to Lie algebroids.
problem Generalizing contact geometry concepts to Lie algebroids.
method Using generalized geometry, the paper constructs an (almost) contact Riemannian Lie algebroid structure.
result An (almost) contact Riemannian Lie algebroid structure is obtained on a vertical Liouville distribution.
Cardinality of augmentation category for Legendrian links determined by ruling polynomial.
problem Counting augmentations of Legendrian links.
method Introducing homotopy cardinality as an invariant, proving it equals ruling polynomial.
result Homotopy cardinality of augmentation category is determined by ruling polynomial.
We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…
We construct families of differential graded algebras R and R \boxtimes R and give an algebraic formulation of the contact category of a disk through the differential graded category DGP(R) generated by some distinguished projective differential graded R-modules. The homology category H^0(DGP(R)) is a triangulated cate…
Study exact Lagrangian cobordisms between Legendrian knots using functorial properties of augmentation categories.
problem Understanding exact Lagrangian cobordisms between Legendrian knots.
method Study the functor between augmentation categories induced by exact Lagrangian cobordisms and establish a long exact sequence.
result Prove the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to exact Lagrangian cobordisms.
Extends Hard Lefschetz Property to isometric flows and shows equivalence.
problem Studying the Hard Lefschetz Property in isometric flows.
method Extending the Hard Lefschetz Property to isometric flows and proving equivalence.
result Equivalence of transverse and non-transverse Hard Lefschetz Properties in isometric flows.
The abstract discusses embedding manifolds in open books and contact structures.
problem Embedding manifolds in open books and contact structures.
method Using open book embeddings and contact structures, the abstract proves embedding theorems for various manifolds.
result Closed manifolds can be embedded in open books and contact structures.
Functor connects sheaf categories of Legendrian submanifolds.
problem Connecting sheaf categories of Legendrian submanifolds.
method Constructs a functor between sheaf categories using Nadler-Shende's work.
result Provides a sheaf theory description of Lagrangian cobordism.
Generalizes surgery techniques for projectively Anosov flows.
problem Creating new projectively Anosov flows from existing ones.
method Introducing a generalized Goodman surgery technique for projectively Anosov flows.
result Generates new examples of projectively Anosov flows on hyperbolic 3-manifolds.
Unified framework for rigidity results on (κ,μ)-manifolds.
problem Rigidity of metrics on (κ,μ)-manifolds. method Study of deviations preserving bi-Legendrian structure, orthogonalizing canonical structure.
result Unified rigidity results in both Riemannian and semi-Riemannian categories.
We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…
Defines a new category structure on Weinstein manifolds using h-principle.
problem Defining a category structure on Weinstein manifolds.
method Defines a co/sheaf of categories on the skeleton of a Weinstein manifold using h-principle.
result Existence and uniqueness of the co/sheaf of categories up to isotopy.
3D foliations can be approximated by contact structures.
problem Approximating taut, transversely oriented foliations in 3-manifolds.
method Showing C0 approximations of taut foliations by contact structures. result Taut foliations not product foliations can be approximated by contact structures.
Classifies contact seaweeds based on their algebraic properties.
problem Determine which seaweed algebras admit a contact structure.
method Analyzed seaweed algebras, showing contact structures for index-one seaweeds.
result Contact seaweeds are quasi-reductive with index one.
Classifies differential operators on symplectic spinors in contact projective geometry.
problem Classifying differential operators on symplectic spinors.
method Classification of homomorphisms of generalized Verma modules and equivariant differential operators.
result Complete classification and construction of differential operators.
A new sheaf theory connects Legendrian knot homologies to generating families.
problem Understanding Legendrian knots and links in contact geometry.
method Associate a constructible sheaf to generating families, linking homology to sheaf theory.
result The generating family homology of a knot is equivalent to its linearized Legendrian contact homology.
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as cons…
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded in the standard contact (2n+1)-space (for n>1) that are contractible in the smo…
Locally symplectic structure found on Kerr space-time.
problem Understanding Kerr space-time using geodesics.
method Identifying locally conformally symplectic structure using characteristic classes and Kerr-Schild coordinates.
result Definition of cobordism category of contact 3-manifolds and locally conformally symplectic cobordisms.
We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product representations V_1^{\otimes n} for n>0. We construct families of DG algebras A, B and R_n, and consider the DG categories DGP(A), DGP(B) and DGP(R_n), which are full DG subcategories of the cat…
Study shows Seifert fibered spaces don't bound rational homology balls.
problem Understanding when Seifert fibered spaces bound rational homology balls.
method Analyzes Seifert fibered spaces with different conditions and orientations.
result Characterizes conditions for Seifert fibered spaces to bound rational homology balls.
New examples of Legendrian links with infinitely many fillings.
problem Understanding the structure of Legendrian links and their fillings.
method New combinatorial formula for Legendrian contact DGAs and Floer-theoretic techniques.
result Construction of the first families of Legendrian links with infinitely many Lagrangian fillings.
We provide an explicit example of a non trivial Legendrian knot Λ such that there exists a Lagrangian concordance from Λ0 to Λ where Λ0 is the trivial Legendrian knot. We then use the map induced in Legendrian contact homology by a concordance and the augmentation category of Λ to show that no Lagrangian co…
Natural metric structures on the tangent bundle and tangent sphere bundles SrM of a Riemannian manifold M with radius function r enclose many important unsolved problems. Admitting metric connections on M with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations o…
Study defines and proves Hard Lefschetz Property for S^3-actions.
problem Formulating Hard Lefschetz Property for S^3-actions.
method Defined and proved two versions of HLP for S^3-actions.
result Agreement of two versions of HLP for 3-Sasakian manifolds.
We analyse the moduli spaces of superconformal field theories (SCFTs). For N=2 we find an enhanced moduli space which in geometrical terms corresponds to tori with two independent complex structures. To explain the precise relation with the moduli space of SCFTs on K3 surfaces as described by Aspinwall and Morrison, we…
Study on L2-cohomology in almost Hermitian manifolds, generalizing previous results.
problem Analyzing L2-cohomology in almost Hermitian manifolds compared to Kähler manifolds. method Generalization of a decomposition theorem by Drăghici, Li, and Zhang to the L2-setting. result The reduced L2 2nd-cohomology group of a four-dimensional almost Hermitian manifold decomposes into invariant and anti-invariant parts. Paper introduces stochastic HJB on Jacobi structures.
problem Stochastic analysis on Jacobi manifolds.
method Global stochastic analysis techniques, extending Bismut and Lázaro-Camí work.
result Proposes a stochastic HJB framework.
The paper explores how configurations of lines can be realized in different geometric settings.
problem Whether configurations of lines can be realized by spheres in various geometric settings.
method Investigates realizability of configurations in topological, symplectic, and smooth categories in the complex projective plane.
result Obstructions to realizability in the topological category for configurations specified by projective planes over finite fields.
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to t…
The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.
problem Characterizing isotopy classes of links in 3-manifolds, especially in contact structures.
method Developed theory of links transverse to a nowhere-zero vector field, constructing examples in both Legendrian and pseudo-Legendrian settings.
result Non-simple isotopy classes of links exist, including Legendrian and pseudo-Legendrian examples.
The paper defines Morse-Bott invariants for critical sets of circles.
problem Homological invariants from Morse-Bott data on unions of circles.
method Axiomatic approach to moduli spaces and evaluation maps, defining homological invariants.
result Construction of a homotopy invariant cascade homology functor.
New examples show contact structures can be isotopic but not contact-isotopic.
problem Identifying contact structures that are isotopic but not contact-isotopic.
method Constructing specific contactomorphisms in all dimensions.
result Infinitely many examples of contactomorphisms that are isotopic but not contact-isotopic.
The study explores embedding closed contact manifolds in higher dimensions.
problem Iso-contact embeddings of closed contact manifolds in co-dimension 2.
method Analyzes conditions for iso-contact embeddings and applies results to specific cases.
result Closed contact 3-manifolds without 2-torsion in their cohomology iso-contact embed in the standard contact 5-sphere.