Defines tensor products for A-infinity structures using diagonals.
problem No specific problem stated; focuses on new definitions.
method Uses diagonals of associahedra and multiplihedra to define tensor products.
result Defines tensor products for various A-infinity structures.
Extends pillowcase homology for immersed curves in a 3-ball.
problem Constructing algebraic versions of Lagrangian Floer homology.
method Associate algebra A and A-infinity modules M(L) to immersed curves in the pillowcase, proving isomorphism with algebraic module pairings.
result Lagrangian Floer homology is isomorphic to a suitable algebraic pairing of modules.
Invariant for 3-manifolds with torus boundary defined.
problem Defining invariants for 3-manifolds with specific boundaries.
method Module over a weighted A-infinity algebra associated to a torus.
result Invariant constructed for bordered 3-manifolds with torus boundary.
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra an…
New method for Lagrangian Floer homology groups using flow trees.
problem Computing equivariant Lagrangian Floer homology.
method Constructing and exploiting an A-infinity module structure on the Floer complex.
result Established constructions of equivariant Lagrangian Floer homology groups.
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. 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.
This paper extends a category equivalence to an A-infinity quasi-equivalence for compact Lie groups.
problem Equivalence of DG categories for smooth singular chains on Lie groups.
method Construction of A-infinity quasi-isomorphisms and use of Van Est map, De Rham theorem.
result Extension of equivalence to A-infinity quasi-equivalence for compact Lie groups.
Study of Pin(2)-monopole Floer homology under connected sums.
problem Understanding Pin(2)-monopole Floer homology behavior under connected sums.
method Constructing a partially defined A-infinity module structure and using Eilenberg-Moore spectral sequences.
result Identify the Floer chain complex of a connected sum via an A-infinity tensor product of modules.
The paper models and deforms A-infinity structures for bordered knot algebras.
problem Understanding A-infinity structures for bordered knot algebras.
method Combinatorial model and weighted deformation of A-infinity structures.
result Explicit combinatorial model for bordered knot algebras' A-infinity structure.
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.
In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as ant…
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…
Paper proves Koszul duality for weighted A-infinity algebras.
problem Koszul duality for weighted A-infinity algebras.
method Constructs new box tensor product for weighted A-infinity bimodules and verifies correspondence between maps and bimodules.
result Proves Koszul duality result between weighted A-infinity algebras.
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddin…
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.
n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of Mau…
We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-ma…
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric var…
Proves connection between curve moduli and Feynman diagrams.
problem Understanding the relationship between symplectic and elliptic structures.
method Establishes homeomorphism between moduli spaces of curves and Feynman diagrams.
result Moduli spaces determine A∞ structures in both models. This paper provides a rational model for fiberwise THH transfer using A-infinity algebras.
problem Rational models for fiberwise THH transfer of fibrations over a base space.
method Explicit description of Hochschild homology transfer in terms of A-infinity algebras.
result Rational models for Becker-Gottlieb transfer and fiberwise THH-simple structures.
Higher nilpotent analogues of the A−∞-structure are explicitly defined on arbitrary simplicial complexes, generalizing explicit construction of /hep-th/0704.2609. These structures are associated with the higher nilpotent differential dn, satisfying dnn=0, which is naturally defined on triangulated manifo…
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.
Constructs combinatorial 2D topological field theories from cyclic A-infinity algebras.
problem Developing a combinatorial framework for 2D topological field theories.
method Using triangulations and polygonal decompositions, constructing cochains on a CW complex.
result Existence of combinatorial 2D topological field theories based on cyclic A-infinity algebras.
Study Legendrian links using representations and sheaves.
problem Understanding Legendrian links through algebraic and geometric representations.
method Investigate an A∞ category of n-dimensional representations and conjecture equivalence to sheaves. result Established cohomological equivalence for Legendrian (2,m) torus links. This paper categorifies Morse theory for manifolds with boundaries.
problem Categorifying Morse theory for manifolds with boundaries.
method Defining a relative Morse complex using handlebody decomposition and constructing an A∞-category structure. result The homology of the relative Morse complex is isomorphic to the relative singular homology.
We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential cup product on this sequence, satisfying a Leibniz rule up to curvature terms.…
Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;Λ) for one Lagrangian or HF^*((L,b),(L',b');Λ) for two, where b,b' are choices of bounding cochains, and exist if and o…
A new approach to Morse theory using folded ribbon trees.
problem Applying Morse theory on symmetric products of surfaces.
method Introducing an A-infinity category with objects as κ-tuples of Morse functions, and showing conditions for the endomorphism to be a Hecke algebra.
result The endomorphism of a specific type of κ-tuple of Morse functions on T*R^2 is the Hecke algebra associated to the symmetric group.
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.
This paper equips Morse cochain complexes with A∞-algebra structures.
problem Equipping Morse cochain complexes with A∞-algebra structures. method Analogous to K. Fukaya's definition, this paper provides a detailed treatment of Abouzaid's approach.
result Provides a coherent and detailed treatment of Abouzaid's approach to Morse cochain complexes.
New spectral sequence connects link homology to Hochschild homology.
problem Computing Hochschild homology of link invariants.
method Uses spectral sequence with E2-page from Khovanov homology of links in S1imesS2. result Spectral sequence converges to Hochschild homology of bordered Floer invariants.
Introduces symplectic flatness for connections over symplectic manifolds.
problem Flatness conditions for connections over symplectic manifolds.
method Introduces symplectic flatness condition and twisting of differential complexes.
result Symplectic flat connections represent a subclass of Yang-Mills connections.
We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology, over fields of characteristic zero. The key ingredient is the construction of a degree one Hochschild cohomology class on a Floer A-infinity algebra associated to the (k,k)-nilpotent slice Y, ob…
We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The exi…
New invariants for knotted surfaces derived from link homology.
problem Distinguishing knotted surfaces from unknotted ones.
method Link homology and A∞-algebras. result Invariant distinguishes unknotted sphere from certain knotted spheres.
Paper proves Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
problem Proving Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
method Analyzes CR-manifolds with contact structure conformal to Heisenberg group, proving volume form is a strong A_infinity weight.
result Proves Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.
problem Constructing algebraic structures for 3-manifold homology.
method Uses graphical calculus to construct Koszul dual weighted A∞-algebras and dualizing bimodules. result Proves duality of constructed algebras and bimodules.
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered co…
The abstract introduces a new A∞ duality via LSFT algebra.
problem Legendrian knot duality and its A∞ extension. method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of A∞ bimodules over Aug+. result Explicit construction of homotopy inverse for the A∞ Sabloff map. What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …
A new quandle from link modules helps identify link properties.
problem Identifying link properties from their modules.
method Defining quandle operations on multivariate Alexander modules.
result The fundamental multivariate Alexander quandle determines the link module sequence.
Classifies modules of surface-knots in terms of their properties.
problem Characterizing modules of surface-knots in terms of their properties.
method Using homology and covering spaces, the reduced first module is characterized.
result The reduced first module for every genus g is characterized in terms of properties of a finitely generated module.
New method learns both module structure and sequencing in neural networks.
problem Learning only the parameters and order of execution of neural modules.
method Expands the approach to learn the internal structure of modules, including the ordering and combination of arithmetic operators.
result Performance comparable to hand-designed modules achieved without extra supervisory signals.
This paper adds a product structure to generating family cohomology for Legendrian submanifolds.
problem Developing invariants for Legendrian submanifolds in jet spaces.
method Constructing moduli spaces of flow trees to define a product structure on generating family cohomology.
result Generating family cohomology gains a ring structure with an associative product.
This article explains A∞-algebras and Hochschild homology.
problem Understanding A∞-algebras and Hochschild homology. method Elementary construction and detailed proofs of results.
result Unified discussion of algebraic results from various sources.