Exposes graded and microformal geometry, focusing on Q-manifolds.
problem Describes new geometric structures and their applications.
method Introduces Q-manifolds and microformal geometry. result Establishes connections between Q-manifolds and Lie algebras. Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while pr…
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…
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.
Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.
problem Calculating Lie algebra homology of gauge algebras using cyclic homology.
method Extends proof to bornological Lie algebra homology of Fréchet and LF-algebras, prepares statements about homological algebra of topological vector spaces.
result Constructs a spectral sequence to calculate stable part of bornological Lie algebra homology of gauge algebras.
Homology of partition algebras matches symmetric group homology under certain conditions.
problem Understanding homology of partition algebras and comparing it to symmetric groups.
method Inductive resolution and high acyclicity arguments, parallel to earlier work on Brauer algebras.
result Homology of partition algebras is isomorphic to symmetric group homology under specific conditions.
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
New Lie algebras from knot homology.
problem Defining Lie algebras from knot homology.
method Using group homology, analogous to Goldman Lie algebra.
result Relations among new Lie algebras discussed.
The paper connects Brauer algebra homology to symmetric group homology.
problem Understanding the homology of Brauer algebras.
method Interpreting Brauer algebras as Tor-groups and comparing to symmetric group homology.
result Isomorphism of homology groups under specific conditions.
Generators and relations for knot Floer homology algebras computed.
problem Computing knot Floer homology using algebras defined by Ozsváth and Szabó.
method Generators and relations description, homology computation, formality determination.
result Generators and relations for the algebras are found, and their homology is computed.
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
New equivariant version of Khovanov homology for annuli.
problem Developing a new mathematical framework for annular Khovanov homology.
method Using Frobenius algebra and equivariant cohomology of CP1. result Introduced an equivariant version of the Temperley-Lieb algebra.
We determine the second homology group of the homological Goldman Lie algebra for an oriented surface.
Homologies of Jones and partition algebras match cyclic and symmetric groups.
problem Matching homologies of specific algebras to cyclic and symmetric groups.
method Proving isomorphisms between algebras' homologies and group homologies.
result Homologies of Jones and partition algebras are isomorphic to cyclic and symmetric groups.
New method uses algebras to speed up link Floer homology calculations.
problem Computing link Floer homology efficiently.
method Using bordered algebras to compute link Floer homology.
result Fast computation of the Thuston polytope for links.
Research connects Lie algebras to configuration space (co)homology.
problem Understanding the (co)homology of configuration spaces.
method Identifying Lie algebra (co)homology as a counterpart to configuration space (co)homology.
result Lie algebras and configuration spaces have a deep mathematical relationship.
Homological algebra used to study local equivalence of complex rings.
problem Local equivalence of bounded complexes over polynomial rings.
method Homological algebra approach
result Results have been proved in many places in the literature.
Study of Khovanov homology invariants from U(1)imesU(1)-equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of U(1)imesU(1)-equivariant Khovanov homology using algebraic filtrations. result Extracted two families of concordance invariants.
An L∞-algebra is built on symplectic manifold homology.
problem No specific problem stated; focuses on construction of algebra.
method Construction of an L∞-algebra on symplectic manifold homology. result The constructed L∞-algebra naturally projects to a Lie algebra extension. A central result here is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these…
New algebraic method for knot Floer homology computation.
problem Computing knot Floer homology efficiently.
method Extending bordered Floer homology to partial knot projections and establishing a pairing result.
result Identification of knot Floer homology with its algebraic definition.
Maps Heegaard Floer homology to Hecke algebras for surfaces.
problem Connecting Heegaard Floer homology with Hecke algebras.
method Constructs a map from Heegaard Floer homology to Hecke algebras.
result Shows the map is an isomorphism of algebras.
We give a homological interpretation of the coefficients of the Hilbert series for an algebra associated with a directed graph and its dual algebra. This allows us to obtain necessary conditions for Koszulity of such algebras in terms of homological properties of the graphs. We use our results to construct algebras wit…
We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in Z2 generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras…
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology asso…
New homological results for bordered Floer algebras derived from hypertoric categories.
problem Homological properties of bordered Floer algebras.
method Affine quasi hereditary property of equivariant hypertoric convolution algebras and computation of Ext groups.
result Existence of standard modules and isomorphism of Ext groups to bordered strands dg algebras.
Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self-distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study ass…
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.
Paper studies homology and cohomology of Temperley-Lieb algebra TL_n(a).
problem Homology and cohomology of Temperley-Lieb algebra TL_n(a).
method Homological stability and computation of stable homology. Use of chain complex of 'planar injective words'.
result Vanishing of homology and cohomology up to degree (n-2) under certain conditions.
Study uses superalgebra homology to classify 4D Engel-like Lie algebras.
problem Classifying 4-dimensional Engel-like Lie algebras.
method Applied homology groups of Lie superalgebras.
result Distinguished and classified 4D Engel-like Lie algebras.
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
We determine the minimal number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.
Heegaard Floer homology connects to polynomial representations of Hecke algebras.
problem Understanding polynomial representations of double affine Hecke algebras.
method Using higher-dimensional Heegaard Floer homology and topological interpretations.
result Recovery of polynomial representations from Heegaard Floer homology.
While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied …
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.
New method computes automorphisms of surface groups using skein algebras.
problem Computing automorphisms of surface groups.
method Using skein algebras and Goldman Lie algebra.
result Refined formula for automorphisms of homology cylinders.
Improves an existing algorithm for computing algebraic set homology groups.
problem Efficiently computing the homology groups of algebraic or semi-algebraic sets.
method Adaptive grid algorithm on the unit sphere.
result Practical improvement of an existing algorithm for homology computation.
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
Researchers determined the second homology group of a specific symplectic derivation Lie algebra.
problem Determining the entire homology group of a specific symplectic derivation Lie algebra.
method Used classical representation theory of Sp(2g; Q) and weight decomposition.
result Determined H_2(\mathfrak{c}_g^{+})
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
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.
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …
The paper explains why a specific type of link homology is useful.
problem Link homology theories and their applications.
method Explains the usefulness of gl(N) theories instead of sl(N) for link homology.
result It is useful to consider gl(N) theories for link homology.
We give a short elementary proof that a Khovanov-type link homology constructed from a diagonalisable Frobenius algebra is degenerate.
We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connectio…
Factorization homology theories of topological manifolds, after Beilinson, Drinfeld and Lurie, are homology-type theories for topological n-manifolds whose coefficient systems are n-disk algebras or n-disk stacks. In this work we prove a precise formulation of this idea, giving an axiomatic characterization of fa…
We determine all the ideals of the homological Goldman Lie algebra, which reflects the structure of an oriented surface.
In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theori…