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.
Proves Weinstein's and Arnold's conjectures using contact instantons.
problem Proving Weinstein's and Arnold's conjectures in contact geometry.
method Existence of fundamental class in Legendrian contact instanton cohomology, evaluation transversality, and geometric construction of contactomorphisms.
result Proves Weinstein's and Arnold's conjectures in full generality.
Study contact instantons and Legendrian links, proving energy inequalities.
problem Estimating Reeb-untangling energy of Legendrian submanifolds.
method Develop contact Hamiltonian geometry, introduce tame contact manifolds, construct moduli spaces, prove convergence results.
result Self Reeb-untangling energy of compact Legendrian submanifolds is greater than period gap.
Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established C∞ convergence of perturbed contact instantons under finite energy hypothesis. 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
The main result of this paper is that, off of a `fundamental class' in degree 1, the linearized Legendrian contact homology obeys a version of Poincare duality between homology groups in degrees k and -k. Not only does the result itself simplify calculations, but its proof also establishes a framework for analyzing coh…
We describe a semi-local canonical form for Legendrian foliations on contact manifolds in the neighbourhood of a Legendrian submanifold. This result generalizes local results by Libermann and Pang on Legendrian foliations on contact manifolds, and is analogeous to a semi-local result by Weinstein in the symplectic case…
We study a natural contact instanton (CI) equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (tHYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreduci…
One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces J1M, equipped with the standard contact structure, is through the Morse theoretic technique of generating families. This paper extends the invariant of generating family cohomology by giving it a product μ2. To define the product, moduli…
We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …
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…
Study confoliations' symplectic fillability, finding obstructions.
problem Understanding confoliations' symplectic fillability in higher dimensions.
method Generalized Massot-Niederkrüger-Wendl's bordered Legendrian open book for confoliations.
result Obstructions to weak symplectic fillability for confoliations.
This research studies the smoothness of moduli spaces of self-dual contact instantons on Sasakian manifolds.
problem Understanding the smooth structure of moduli spaces of self-dual contact instantons on Sasakian 7-manifolds.
method Computing a Weitzenböck formula and using a Bochner-type method to obtain a vanishing theorem.
result Conditions for the smoothness of moduli spaces of self-dual contact instantons, particularly when the Sasakian manifold is transversely Ricci positive and the curvature operator is positive.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
Study confirms contact cosmetic surgery for most knots, with exceptions.
problem Determining contact cosmetic surgeries for Legendrian knots.
method Analyzing Legendrian knots, including unknots, and using contact surgery theory.
result Some Legendrian unknots have unique contact surgeries without a cosmetic pair.
Proposes a contact dynamics framework using generalized geometries.
problem Contact dynamics and related geometries.
method Generalizes symplectic and Morse families to contact framework.
result Establishes contact Hamiltonian and Lagrangian Dynamics as Legendrian submanifolds.
The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R3 are canonically isomorphic. Recently we constructed the first examples where Vassiliev in…
Persistent Legendrian contact homology distinguishes knots using height functional.
problem Distinguishing Legendrian knots in R3. method Persistent homology applied to Chekanov-Eliashberg DGA, with height functional.
result Strong Morse inequalities for persistent Legendrian contact homology.
New algebra structure for Legendrian knots preserves contact homology invariants.
problem Constructing an L∞ algebra for Legendrian knots. method Combining rational Symplectic Field Theory and combinatorial methods.
result Invariant Poisson algebra of Legendrian links under isotopy.
In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsváth-Szabó invariant for contact (+1)-surgery along certain Legendrian two-component links. The main tool is a link surgery formula for Heeg…
Study cone structures on contact manifolds to understand their geometric properties.
problem Characterize cone structures on holomorphic contact manifolds.
method Characterize subadjoint varieties among Legendrian submanifolds in terms of contact prolongations.
result Holomorphic horizontal splitting of the canonical distribution on contact G-structures.
Extends LOSS invariant naturality to positive contact surgeries.
problem Naturality of LOSS invariant under positive contact surgeries.
method Proves naturality of LOSS invariant for Legendrian knots under positive integer contact surgeries.
result Contact invariant of +n surgery depends only on classical invariants of S.
We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres −Σ(2,3,6n−1).
Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.
problem Contact cosmetic surgeries for Legendrian knots in L-spaces.
method Adapting techniques from S3 to L-spaces, incorporating Heegaard Floer theory constraints.
result Contact cosmetic surgery conjecture holds for non-trivial Legendrian knots, except for Lagrangian slice knots.
Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
problem Classifying convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
method Contact isotopy classification, h-principle, fundamental groups, contact mapping class group.
result Establishes an h-principle for convex disks with Legendrian boundary in overtwisted contact 3-manifolds.
New surgeries on knots preserve contact structures.
problem Understanding how surgeries on Legendrian knots affect their contact structures.
method Analyzing surgeries on specific types of knots (twist and two-bridge knots) and proving distinct contact structures for certain surgeries.
result Negative rational surgeries on certain Legendrian knots yield distinct contact 3-manifolds.
In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on (S3,ξstd) along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an inf…
The study constructs and shows isotopy of high-dimensional Legendrian spheres.
problem Understanding Legendrian spheres in contact manifolds of any dimension.
method Three Legendrian sphere constructions using open books and a doubling procedure.
result These constructions are isotopic to the Legendrian unknot.
Upper bounds for Legendrian links in tight contact 3-manifolds.
problem Bounding the complexity of Legendrian links in tight contact 3-manifolds.
method Constructing exact Lagrangian cobordisms and defining minimal Lagrangian genus.
result Established upper bounds for Legendrian links with a common rotation number.
Contact round surgeries on (S3,ξst) help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in (S3,ξst). Torsion found in knot homology, challenging augmentation theories.
problem Torsion in linearized contact homology for Legendrian knots.
method Examples of Legendrian knots with non-trivial homology over Z.
result Augmentations not induced by exact fillings, even mod 2.
Study the spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.
problem Homotopy types of spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.
method Recursive formula and contractibility proofs for specific cases.
result Homotopy equivalence and contractibility results for spaces of Legendrian embeddings.
Contact surgeries yield algebraically overtwisted manifolds.
problem Understanding algebraically overtwisted contact manifolds through surgeries.
method Contact (+1)-surgeries on Legendrian spheres in flexibly fillable contact manifolds. result Yielding algebraically overtwisted manifolds when the Legendrian's homology class is not annihilated.
Contact homology for Legendrian submanifolds in standard contact (2n+1)-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to b…
We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.
Study contact instantons on Sasakian 5-manifolds with Calabi-Yau structures.
problem Anti-self-dual contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures.
method Singularity data of leaf spaces and computation of moduli spaces.
result Explicit computation of complex dimensions of moduli spaces for 95 orbifold K3 surfaces.
Expanded Legendrian knot atlas for 10-arc index knots.
problem Lack of Legendrian knot data for knots with high arc index.
method Created an atlas of Legendrian knots up to arc index 10.
result Legendrian knots of arc index 10 have been cataloged.
We give a procedure to ``average'' canonically C1-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby.
Combinatorial method computes Legendrian knot invariant.
problem Computing the Heegaard Floer contact invariant for Legendrian knots.
method Combining Plamenevskaya's combinatorial description with Heegaard Floer theory.
result Hat version of LOSS invariant can be computed combinatorially.
Proves h-principle for loose Legendrian embeddings in contact topology.
problem Existence of non-loose Legendrian embeddings.
method h-principle from microlocal sheaf theory.
result Existence of non-loose Legendrian embeddings.
Study contact structures on lens spaces, classifying rational knots.
problem Classify rational knots in lens spaces.
method One-parametric convex surface theory to classify Legendrian and transverse rational unknots.
result Determine the contact mapping class group of lens spaces.
To a Legendrian knot, one can associate an A∞ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomolog…
Algorithm converts curves on ribbon surfaces to contact surgery diagrams.
problem Legendrian realization of curves on ribbon surfaces.
method Explicit algorithm to Legendrian realize homologically nontrivial curves.
result Any two Legendrian realizations of the same curve are Legendrian isotopic.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legen…
A rigidity theorem for smooth Legendrian self-shrinkers is proven.
problem Understanding the structure of Legendrian self-shrinkers.
method Estimating weighted volume to prove optimal volume growth.
result Rigidity theorem for entire smooth Legendrian self-shrinkers.
New knots are found to be non-simple in Legendrian contact geometry.
problem Identifying non-simple Legendrian knots in contact geometry.
method Utilized knot Floer homology and the distinguished surgery triangle.
result Whitehead doubles of the trefoil are Legendrian non-simple.
Study Legendrian surfaces using N-graphs and flag moduli.
problem Characterize and apply Legendrian surfaces in contact geometry.
method Develop diagrammatic calculus and algebraic-geometric characterization.
result Show applications in Lagrangian concordance, exact fillings, and rational point counts.