Paper establishes an isomorphism between Fukaya category and bordered knot Floer homology.
problem Connecting Fukaya category and bordered knot Floer homology.
method Using A-infinity deformations and Hochschild cohomology calculations.
result Established an isomorphism between endomorphism algebras and star algebras.
Geometrically generates Fukaya categories of Weinstein manifolds.
problem Generating Fukaya categories of Weinstein manifolds.
method Geometric approach using surgery formula and Floer cohomology.
result Wrapped Fukaya category generated by index n critical points.
Researchers create a Fukaya category for knot complements using specific metrics.
problem Constructing a Fukaya category for knot complements.
method Using cylindrical adjustments and wrapping, they construct the Fukaya category and its wrapped Floer complex.
result The quasi-equivalence and quasi-isomorphism classes are independent of the choice of metrics.
We define a torus algebra for Heegaard Floer homology.
problem Developing algebraic structures for 3-manifold homology.
method Combinatorial and abstract algebraic constructions.
result Established connection to wrapped Fukaya category.
New constraints found for Lagrangian embeddings in symplectic fillings.
problem Understanding Lagrangian embeddings in symplectic fillings with semisimple cohomology.
method Deriving constraints through Lagrangian embeddings and symplectic cohomology analysis.
result Existence of many non-toric monotone symplectic manifolds with proper wrapped Fukaya categories.
The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshit…
Study knot Floer cohomology using hyperbolic metrics and wrapped Fukaya categories.
problem Computing knot Floer cohomology for hyperbolic knots.
method Formalizes Floer complex on ideal boundary using hyperbolic metrics and wrapped Fukaya categories.
result Knot Floer cohomology is isomorphic to wrapped Floer cohomology for hyperbolic knots.
Homological mirror symmetry proved for symmetric squares of punctured spheres.
problem Proving homological mirror symmetry for symmetric squares of punctured spheres.
method Constructed quasi-equivalences between wrapped Fukaya categories and derived categories of coherent sheaves, using categorical resolutions and localisation.
result Wrapped Fukaya category of symmetric square quasi-equivalent to coherent sheaves on a singular surface.
New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
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.
Study of Fukaya-Seidel categories for surface Lefschetz fibrations.
problem Understanding Fukaya-Seidel categories for surface Lefschetz fibrations.
method Proving exact Lefschetz fibrations and deriving Fukaya-Seidel categories.
result Derived Fukaya-Seidel categories are independent of symplectic structure and contain more information than Milnor lattices.
Study of Fukaya category on surfaces with mapping class group action.
problem Understanding the Fukaya category of surfaces with topological modifications.
method Construction of Fukaya category, disregarding area form, and studying mapping class group action.
result Established faithfulness of mapping class group action on Fukaya category.
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
Given a J-holomorphic Morse function on a symplectic manifold, a new construction of the Fukaya-Seidel category is outlined. Applying this construction in an infinite dimensional case, a Fukaya-Seidel-type category is associated to a smooth three-manifold. In this case the construction is based on a five-dimensional ga…
In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as perturbation of the Lagrangian skeleton of the exact symplectic manifold.
Survey on stability conditions and Fukaya type categories, focusing on their topological structure.
problem Understanding the topological structure of stability conditions and Fukaya type categories.
method Reviewing and analyzing the topological structure of stability conditions and Fukaya type categories using quadratic differentials.
result Spaces of stability conditions can be realized as moduli spaces of quadratic differentials.
We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.
New geometric construction of spectral sequences from Khovanov homology.
problem Understanding spectral sequences from Khovanov homology.
method Cylindrical model to compute Fukaya categories of Hilbert schemes.
result New geometric construction of spectral sequences from Khovanov homology.
Survey on decorated marked surfaces for Calabi-Yau categories.
problem Understanding Calabi-Yau categories and related structures.
method Introducing decorations on marked surfaces to study various categories.
result Exploration of Calabi-Yau-2 and 3 categories, braid groups, quadratic differentials, and stability conditions.
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M i…
This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as mo…
Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.
problem Homotopy rigidity of nearby Lagrangian cocores in Weinstein sectors.
method Spectral wrapped Donaldson-Fukaya category with orthogonal group coefficients.
result Inclusion followed by retract and quotient is null-homotopic.
Study tangle invariants using pillowcase Fukaya category and character varieties.
problem Tangle invariants and their relationships with Khovanov cohomology.
method Define a twisted complex of compact Lagrangians in the pillowcase's Fukaya category, show functoriality, and relate to Khovanov cohomology.
result Hom set with a specific Lagrangian is naturally isomorphic to reduced Khovanov chain complex.
A 2-category categorifies complex Lagrangians in hyperkähler manifolds.
problem Categorify the Fukaya category of hyperkähler manifolds.
method Formalizes morphisms in a 2-category based on Fueter maps.
result Fueter maps correspond to complex gradient trajectories in cotangent bundles.
We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study…
Constructs a cyclic, filtered, strictly unital curved A∞ category for Lagrangian submanifolds and develops Floer theory.
problem Proving that any Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by a given collection of Lagrangian submanifolds.
method Develops a cyclic, filtered, strictly unital curved A∞ category and uses it to prove the above statement. result Any Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by a given collection of Lagrangian submanifolds.
This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.
problem Measuring the distance between filtered A-infinity categories associated with Lagrangian submanifolds.
method Developed a Gromov-Hausdorff distance to measure the difference between these categories.
result Established that the sequence of filtered A-infinity categories forms a Cauchy sequence in Gromov-Hausdorff distance.
Let Ham(M,ω) denote the Frechet Lie group of Hamiltonian symplectomorphisms of a monotone symplectic manifold (M,ω). Let NFuk(M,ω) be the A∞-nerve of the Fukaya category Fuk(M,ω), and let (∣S∣,NFuk(M,ω)) denote the NFuk(M,ω) component of the ``space of ∞-categories''…
The paper studies exact triangles in stable vector bundles on tori.
problem Understanding exact triangles in stable vector bundles on tori.
method Geometric interpretation via Fukaya category and homological mirror symmetry.
result Geometric interpretation of exact triangles in terms of Fukaya category.
The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.
problem Constructing reduced Khovanov homology for links in lens spaces.
method Generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 and constructing cochain complexes for links in S3 and S2imesS1. result The cohomology of the constructed cochain complex for links in S2imesS1 may be a link invariant. Let X be a compact real analytic manifold, and let T∗X be its cotangent bundle. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X. In this paper, we develop a Fukaya A∞-category Fuk(T∗X) whose objects are exact, not necessarily compact Lagrangian branes in…
Given a pointed 4-ended tangle T⊂D3, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and LT from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere (S2,4pt)=∂(D3,T). We prove that …
New algebraic description connects Fukaya category to bordered Floer homology.
problem Connecting Fukaya category to bordered Floer homology.
method Algebraic description using toric varieties and amplituhedron combinatorics.
result Isomorphic Fukaya category to bordered Floer homology algebras.
Study braid group actions on exceptional sequences using branched coverings.
problem Transitivity of braid group action on full exceptional sequences.
method Relate exceptional sequences to branched coverings, apply Birman--Hilden theory.
result Counterexamples to Bondal--Polishchuk conjecture on braid group transitivity.
We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric…
New construction of Fukaya-Seidel categories using complex gradient flow equation.
problem Constructing Fukaya-Seidel categories for specific models.
method Using the complex gradient flow equation and neck-stretching limits.
result Alternative proof of Seidel's spectral sequence for Lagrangian Floer cohomology.
The paper extends topological field theory to noncompact surfaces using symmetric powers.
problem Extending topological field theory to noncompact surfaces without closed boundaries.
method Constructing sectorial covers with combinatorics of the bar resolution.
result Recovering results of Rouquier and Manion on extending Heegaard-Floer theory.
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.
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become A∞-equivalent after a twist by a kind o…
Math verifies Aganagic's proposal for Khovanov homology.
problem Recovering Khovanov homology from braid group action on Fukaya-Seidel categories.
method Direct calculation of containment between combinatorially defined braid group actions.
result Mathematical verification of Aganagic's proposal for Khovanov homology.
New model for Calabi-Yau-X categories using decorated marked surfaces.
problem Constructing models for Calabi-Yau-X categories. method Using graded decorated marked surfaces and string models.
result Isomorphism between braid twist group and spherical twist group.
Constructs exotic Lagrangian tori in Grassmannians using cluster algebra.
problem Finding non-displaceable and non-isotopic Lagrangian tori in Grassmannians.
method Iterative construction based on cluster algebra structure of a mirror Landau-Ginzburg model.
result Examples of exotic Lagrangian tori that support nonzero objects in different summands of the Fukaya category.
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…
New proof of Heegaard Floer surgery formulas using Fukaya category of the torus.
problem Heegaard Floer surgery formulas
method New exact triangle in Fukaya category of the torus
result Extensions of link surgery formula to arbitrary links
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of J-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a s…
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.
This paper constructs a functor preserving unobstructedness in symplectic geometry.
problem Generalizing unobstructedness in symplectic geometry for arbitrary manifolds and Lagrangian submanifolds.
method Using filtered A-infinity categories and Lagrangian Floer theory, the paper constructs a 2-functor.
result The geometric transformation preserves unobstructedness of Lagrangian Floer theory.