New examples of tight contact manifolds with algebraic torsion.
problem Constructing tight contact manifolds with algebraic torsion.
method Constructing examples in odd dimensions, proving Giroux torsion implies algebraic 1-torsion.
result Proves a conjecture about algebraic torsion in odd dimensions.
Study connects spectral and algebraic torsion in geometric contexts.
problem Relating different torsion concepts in geometric settings.
method Example of product geometry, focusing on spin manifolds and two-point space.
result Established connection between spectral and algebraic torsion.
Reidemeister torsion is algebraic for most 3-manifolds.
problem Characterizing algebraic properties of Reidemeister torsion.
method Defined algebraic numbers and proved them to be algebraic integers for specific 3-manifolds.
result Reidemeister torsions are algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds.
We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic tors…
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
New theory allows simultaneous block-diagonalization of commuting operator fields.
problem Normal forms of operator fields.
method Generalized Nijenhuis torsions and generalized Haantjes algebra.
result Simultaneous block-diagonalization of commuting operator fields.
Study characterizes G2-structures on specific Lie groups and identifies harmonic conditions.
problem Characterizing and identifying harmonic G2-structures on almost Abelian Lie groups. method Analyzing left-invariant G2-structures, characterizing torsion forms, and using algebraic conditions. result Established algebraic conditions for harmonic G2-structures and identified admissible torsion classes. This work explores algebraic structures from curvature and torsion in affine connections.
problem Understanding algebraic structures from curvature and torsion in affine connections.
method Post-Lie algebra, D-algebra, and special polynomials.
result A particular class of geometrically special polynomials is generated by torsion and curvature.
Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.
problem Computing and enumerating gradings of Lie algebras.
method Constructive approach to torsion-free gradings, computation of maximal grading, enumeration of all gradings.
result Computation of a maximal grading and enumeration of all torsion-free gradings.
Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is Z[x]/(x2), we determine precisely those graphs whose cohomology contains torsion. For a la…
The paper explores algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
problem Investigating algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.
method Analyzing twisted Alexander polynomials and Reidemeister torsions of torus knots associated with irreducible SLn(C)-representations. result Proves that coefficients of twisted Alexander polynomials are locally constant functions on the SLn(C)-character variety. Study on Hermitian metrics on Lie algebras with specific ideals.
problem Classifying Hermitian metrics on Lie algebras with abelian ideals.
method Examined unimodular Lie algebras with abelian ideals of codimension two, classified metrics.
result Classification of Bismut Kähler-like and Bismut torsion-parallel metrics.
Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
problem Index theory and analytic torsion of nonlinear PDEs.
method Microlocal sheaf theory, factorization algebras, Spencer hypercohomology.
result Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
New formula for torsion function in 3-manifolds with torus boundaries.
problem Computing torsion function for 3-manifolds with specific boundary conditions.
method Defined adjoint torsion function on moduli stack of G-local systems, proved regularity condition, provided formula for product of PGL2-torsions.
result Computed adjoint PGSp4-torsions of figure-eight knot complement for boundary-unipotent local systems.
We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion a…
Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.
problem Determining the genus and fibering of double twist knots.
method Uses adjoint hyperbolic torsion polynomial to analyze double twist knots.
result The adjoint hyperbolic torsion polynomial determines the genus and fibering of double twist knots.
Study torsion's impact on 2D affine Killing vectors on homogeneous surfaces.
problem Effects of torsion on affine Killing vectors on homogeneous surfaces.
method Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
result Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
Obstructs 2-torsion in rational knot concordance group.
problem Identifying 2-torsion elements in rational knot concordance group.
method Localized von Neumann ρ-invariant.
result Provides an obstruction for knots of order 2 in algebraic rational concordance group from being of finite order in rational knot concordance group.
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.
We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \mathfrak{so}(n)$, in other words the intersection $Λ^3 \cap (Λ^1 \otimes \g)$.We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean vers…
The paper explores Lorentzian connections with parallel skew torsion.
problem Understanding metric connections with parallel skew-symmetric torsion in Lorentzian signature.
method Analyzing holonomy algebras, torsion, and curvature; constructing examples; classifying homogeneous spaces.
result Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.
We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic k-torsion in the context of ech, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
Explicit relation found between knot torsion and TQFT signatures.
problem Relating Reidemeister torsion of two-bridge knots to TQFT signatures.
method Established an explicit relation using parabolic representations and SU2-TQFT. result Inverse sum of torsions is constant and signatures have similar asymptotic behavior.
Researchers confirm integral formulas for G2-structures in detail.
problem Verifying integral formulas for G2-structures on Riemannian manifolds. method Detailed analysis and explicit expressions of intrinsic torsion using exterior algebra.
result Agreement of integral formulas with intrinsic torsion components.
We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective mor…
The study introduces polarization of generalized Nijenhuis torsions and their relevance in operator fields.
problem Characterization of Haantjes C∞(M)-modules of operator fields. method Introducing polarization of generalized Nijenhuis torsions and proving algebraic identities.
result Polarizations of generalized Nijenhuis torsions are relevant in the characterization of Haantjes C∞(M)-modules of operator fields. Odd Khovanov homology gets a new algebraic action from super foams.
problem Understanding the algebraic structure of odd Khovanov homology.
method Introducing a local gl1∣1-action on odd Khovanov homology via super foams. result The action of gl1∣1 on odd Khovanov homology is shown to arise from super foams. Characterizes almost Abelian Lie algebras with special H-structures.
problem Identifying almost Abelian Lie algebras with torsion-free H-structures. method Using linear maps and endomorphisms, characterizes the subspace of f for which the Lie algebra admits a special H-structure. result Explicitly computes the subspace of f for various linear Lie groups H. We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a "geometric assembly map" defined on stabilized structure spaces in purely homotopy theoretic terms.
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both L2 combinatorial and L2 analytic torsion invariants …
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Mathematical construction of vertex algebra representations from integrable G2 structures.
problem Constructing representations of a specific vertex algebra from geometric input.
method Integrable G2 structures with closed torsion on group manifolds, embedding into superaffine vertex algebra and chiral de Rham complex.
result Embeddings of deformed Shatashvili-Vafa vertex algebra in the chiral algebra of heterotic G2 backgrounds.
Given a fiber bundle Z→M→B and a flat vector bundle E→M with a compatible action of a discrete group G, and regarding B/G as the non-commutative space corresponding to the crossed product algebra, we construct an analytic torsion form as a non-commutative deRham differential form. We show that our…
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
Study Kuperberg invariants for sutured manifolds using Fox calculus and Reidemeister torsion.
problem Computing Kuperberg invariants for sutured manifolds.
method Fox calculus and Reidemeister torsion.
result Kuperberg invariants can be computed via Fox calculus and extended to polynomial invariants.
We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §de…
We study almost Kaehler manifolds whose curvature tensor satisfies the third curvature condition of Gray. We show that the study of manifolds within this class reduces to the study of a subclass having the property that the torsion of the first canonical Hermitian connection has the simplest possible algebraic form. Th…
We prove the logarithmic divergence of equivariant analytic torsion for one-parameter degenerations of projective algebraic manifolds, when the coefficient vector bundle is given by a Nakano semi-positive vector bundle twisted by the relative canonical bundle.
The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension six is of particular interest. In this work, we give a full classification of the algebraic types of the torsion form, and, based on this, undertake a s…
Classifies algebraic concordance for almost classical knots.
problem Classifying algebraic concordance for almost classical knots.
method Defined virtual algebraic concordance group for almost classical knots.
result Embeds GQ into VGQ and contains non-classical finite-order elements. We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahl…
The study describes the free Lie-Yamaguti algebra.
problem None explicitly stated; focus is on the algebra itself.
method Not explicitly detailed in the abstract.
result Description of the free Lie-Yamaguti algebra.
This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.
problem Characterizing LCSKT structures on almost abelian Lie algebras.
method Analyzing the LCSKT condition and its compatibility with other Hermitian structures.
result Classification of LCSKT almost abelian Lie algebras in dimension 6.
For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative tor…
We introduce non-acyclic PGLn(C)-torsion of a 3-manifold with toroidal boundary as an extension of J. Porti's PGL2(C)-torsion, and present an explicit formula of the PGLn(C)-torsion of a mapping torus for a surface with punctures, by using the higher Teichmüler theory due to V. Fock …
The paper develops surgery theories for foliations and solves a problem posed by Weinberger.
problem Classifying leaves of foliations and verifying Borel conjectures.
method Developed two surgery theories and two kinds of Whitehead torsion for foliations.
result Solved a problem posed by Weinberger and verified conjectures in some cases.