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.
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48 results for “abstract algebra”
Survey on algebraic fibers of group extensions and their finiteness properties.
problem Existence and finiteness of algebraic fibers in group extensions.
method Analysis of abstract and pro- groups.
result Finiteness properties of algebraic fibers in group extensions.
Abstract reviews symmetry and reduction in dynamical systems.
problem Understanding symmetries and reductions in dynamical systems.
method Algebraic formulation for dynamics of physical systems.
result Describes a reduction procedure for classical and quantum evolutions.
We define a certain abstract planar algebra by generators and relations, study various aspects of its structure, and then identify it with Jones' spin planar algebra.
Paper reconciles RCM and SCM frameworks for causal inference.
problem Clarifying the relationship between RCM and SCM frameworks.
method Neutral logical perspective, previous work, and abstract representation.
result Every RCM emerges as an abstraction of some representable RCM.
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
Abstract: A new approach to technical indicators without lag.
problem Defining classical technical indicators as bounded operators for lag-free trading.
method Using linear algebra to redefine technical indicators as bounded operators in space.
result Demonstrated the no-lag versions of technical indicators are simpler and more effective.
Abstract: Proves relative versions of group splitting results.
problem Group splittings and invariant sets.
method Relative versions of earlier results about almost invariant sets and splittings of groups.
result Existence and uniqueness of relative versions of algebraic regular neighbourhoods and JSJ decompositions.
A new algebra for Frobenius manifolds solves PDEs and constraints.
problem Understanding the algebraic structure of Frobenius manifolds.
method Constructing a Virasoro-like algebra and deriving PDEs and constraints.
result Solves a family of quadratic PDEs for the genus-zero free energy.
The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we …
Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
We prove the rigidity of presymplectic actions of a compact semisimple Lie algebra on a presymplectic manifold of constant rank in the local and global case. The proof uses an abstract normal form theorem we had stated in a previous work, based on an iterative process of Nash-Moser type. In order to use correctly this …
The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential opera…
We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract …
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.
problem Constructing a Cartan calculus in tangent categories.
method Define scalar multiplication by a commutative ring object to equip tangent bundles with -module structure.
result Every object in tangent categories carries a Cartan calculus of Lie-Rinehart forms.
The abstract discusses connecting quantum mechanics and algebraic index theories.
problem Exploring the connection between quantum mechanics and algebraic index theories.
method Explains how the classical algebraic index theorem can be proved in terms of BV quantization of topological quantum mechanics and 2d chiral CFT.
result Shows how the generating function of all genus Gromov-Witten invariants on elliptic curves is mirror equivalent to an elliptic chiral index.
Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.
problem Homotopy Poisson algebra models for reduced spaces.
method Cattaneo-Zambon compatibility and regularity conditions, equivariant map, homotopy Poisson algebra.
result Derivation of homotopy Poisson algebra generalizing classical BFV algebra.
In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove …
Defines tensor eigenvalues and singular values without basis, simplifying analysis.
problem Defines tensor eigenvalues and singular values without basis.
method Intrinsic definition of tensor eigenvalues and singular values using concepts from pure mathematics.
result Shows the relationship between tensor analysis and pure mathematics.
Revives Vogel's diagrammatic technique for universal Lie algebra computations.
problem The universality of Lie algebra quantities remains open, despite many being described.
method Diagrammatic algebra based on Vogel's Λ-algebra.
result Diagrammatic technique enables truly universal computations in Lie theory.
Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.
problem Understanding and characterizing generalised spin structures and their properties.
method Definitions, basic notions, connections, covariant Lie derivative, covariant Cartan calculus, symmetry algebra.
result Characterization of homogeneous generalised spin structures.
The abstract defines -structures and connects them to octonion algebras.
problem Classifying -structures and understanding their geometric properties.
method Established an isomorphism between -structures and octonion algebras over .
result The classification of -structures agrees with a parametrisation of octonion algebras with isometric norm.
simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of …
The abstract aims to generalize classical curve concepts to uniquely define complex curves.
problem Lack of sufficient information to distinguish between different curves.
method Generalizing classical concepts of curvature and torsion to higher algebraic curvatures.
result Each analytic branch of a complex curve is uniquely defined by higher algebraic curvatures.
From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An important handicap of this approach is the ignorance of topological and geometrical aspe…
We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of rational cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M whose rational cohomology algebra belongs to this class,…
We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and various iterated tangent/cotangent constructions in symplectic and Poisson geometry…
Abstract machinery finds obstructions to uniform positive scalar curvature.
problem Finding obstructions to uniform positive scalar curvature.
method Coarse index theory and embedding submanifolds.
result Abstract machinery constructs wrong way maps on -theory.
Abstract: Deltoid map connects complex dynamics and algebra.
problem Understanding complex dynamics through a specific map.
method Analyzing the geometry and algebra of the deltoid map.
result Illustrates Julia set and iterated monodromy group.
Abstract studies 3-manifolds and vertex algebras, expanding known connections.
problem Understanding the relationship between 3-manifolds and vertex algebras.
method Developed a dictionary between 3-manifolds and vertex algebras using -series.
result Generalized known entries to higher rank Lie groups and 3-manifolds with toral boundaries.
Geometric AD framework simplifies derivative computation in JAX.
problem Efficient and accurate automatic differentiation.
method Jet functors and Weil algebras for geometric analysis.
result Unified view of derivative propagation with algebraic exactness.
The abstract proves the non-existence of certain real algebraic surfaces.
problem The existence of real polynomial functions with specific properties.
method Algebraic solution to a problem proposed by D. A. Panov.
result There does not exist a real polynomial function with the specified properties.
The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.
problem Investigate how the core Lie algebra controls the Lie superalgebra.
method Construct Lie superalgebras from abstract Lie algebras using Schouten-like brackets and analyze Betti numbers of super homology groups.
result For low dimensional non-abelian Lie algebras, the Betti numbers of super homology groups provide insights into the control of the core Lie algebra.
The abstract introduces a new duality via LSFT algebra.
problem Legendrian knot duality and its extension.
method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of bimodules over .
result Explicit construction of homotopy inverse for the Sabloff map.
In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbe…
This note focuses on some properties and uses of filtered deformations in the context of D=11 supergravity. We define the concept of abstract symbol and give a strong version of the Reconstruction Theorem, namely a bijective correspondence from the space of highly supersymmetric supergravity backgrounds to the space of…
Abstract: Generalizes prequantization map for 2-plectic manifolds.
problem Generalizing prequantization map for 2-plectic manifolds.
method Constructs a -bundle and Lie groupoid over the total space of a manifold with an integral closed 3-form.
result Provides a Lie 2-algebra quasi-isomorphism from observables to weak symmetries.
This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.
problem Characterizing vector bundles in smooth manifolds.
method Introducing differential bundles in a tangent category and proving equivalence with vector bundles in smooth manifolds.
result Differential bundles in smooth manifolds are equivalent to vector bundles.
New classification of complex hypersurfaces in 3D.
problem Classifying simply-transitive Levi non-degenerate hypersurfaces in .
method Novel Lie algebraic approach, new coordinate-free formula for quartic tensor.
result Unique non-tubular model with geometric relations to planar equi-affine geometry.
Graph Neural Networks align with dynamic programming, improving algorithmic reasoning.
problem Demonstrate and quantify alignment between GNNs and dynamic programming.
method Category theory and abstract algebra methods to expose intricate connection.
result Showed GNNs align with dynamic programming beyond individual algorithms.
This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…
Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic K-theorymath.AT
The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matric…
This paper updates knot invariants using Hopf algebras and categorifies their structure.
problem Defining and understanding quantum invariants of knots and three-manifolds.
method Abstract description of categorical structures involving Hopf algebras and their centers.
result The Hopf algebraic center of a knot's image is central to the invariant.
Cyclic Objects from Surfacesmath.GT
The abstract introduces a new cyclic structure for surfaces.
problem Defining a cyclic structure for surfaces.
method Endowing surfaces with a structure of a (co)cyclic object in 3D cobordisms.
result 3D TQFTs induce (co)cyclic modules, which are computed algebraically.
Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.
CoLA automates efficient numerical linear algebra for complex matrix structures.
problem Efficiently solving large-scale linear algebra problems with complex matrix structures.
method Combining linear operator abstraction with compositional dispatch rules.
result Automatic and efficient numerical algorithms for various linear algebra operations.
The abstract explores analogues of Hodge theory in Lie algebroids.
problem Exploring analogues of Hodge theory in Lie algebroids.
method Establishing equivalence of conditions and applying algebraic theory to geometric setting.
result Equivalence of analogues of Hodge theory conditions in Lie algebroids.