arXiv research

New geometric system from Hessian operators offers solutions to geometric problems.

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an applicat…

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the eq…

The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.

We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geom…

New field invariant refines real spectrum and relates to absolute Galois group.

Introduces arithmetic analogues of Orr invariants and spaces for absolute Galois groups.

In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector …

We compare the covariant formulation of Quantum Mechanics on a curved spacetime fibred on absolute time with the standard Geometric Quantisation.

Paper studies geometric properties of nonlinear Lebesgue spaces.

New method identifies vanishing arcs for curve singularities.

In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkab…

We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an additive category $\bA$ from an element of $\widetilde{K}^{iso}_1(\bA)$ to an element of the absolute group $K_1^{iso}(\bA)$. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invari…

A Lie group has a unique metric when viewed as a flat absolute parallelism.

The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …

For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…

Classifies financial markets up to financial indistinguishability.

We define an almost--cosymplectic--contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost--coPoisson--Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated …

Study absolute continuity of Wasserstein barycenters on manifolds with singular cost functions.

Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.

We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the cl…

The aim of the present paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism space (GAP-space). The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized Pa…

Real Heegaard Floer homology gets a new grading for certain 3-manifolds.

For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combina…

In this paper we extend and Poincare dualize the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler-Poincare characteristic, to arbitrary manifolds. We use the Poincare dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray-Singer torsion by providing a sli…

Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(F), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism from F to the boundary of Y, a module over A(F). In a previous paper, we defined relative Z/2 differential gradin…

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. T…

We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classic…

The Kreck-Stolz $s$-invariant is a classic path-component invariant for the space and moduli space of positive scalar curvature metrics. It is an absolute (as opposed to relative) invariant, but this strength comes at the expense of being defined only under restrictive topological conditions. The aim of this paper is t…

A new geometric cocycle measures mass of hyperbolic manifolds.

Geometric approach to thermodynamics of chemical reaction networks.

We express the signature modulo 4 of a closed, oriented, $4k$-dimensional $PL$ manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined in Korzeniewski [11]. Let $F \to E \to B$ be a $PL$ fibre bundle, where $F$, $E$ and $B$ are closed, connected, and compatibly orient…

The statistical properties of the multipliers of the absolute returns are investigated using one-minute high-frequency data of financial time series. The multiplier distribution is found to be independent of the box size $s$ when $s$ is larger than some crossover scale, providing direct evidence of the existence of sca…

Study describes isometry groups of specific Lie groups.

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle $TM$ of a manifold $M$. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional argument $y$. Moreover, many new geometric objects, which have n…

New tool: relative Hopf invariant for Poincaré surgery.

New method constructs relative invariants for group actions on extended manifolds.

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated to second order differ…

This paper continues a series of studies devoted to analysis of the bivariate probability distribution P(x,y) of two consecutive price increments x (push) and y (response) at intraday timescales for a group of stocks. Besides the asymmetry properties of P(x,y) such as Market Mill dependence patterns described in preced…

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness requirements often placed on the group actions. In doing so, we allow for groups wi…

The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.

In an earlier paper (math.SG/0110169), we introduced absolute gradings on the three-manifold invariants developed in math.SG/0101206 and math.SG/0105202. Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the…

In a previous preprint we defined an energy associated to every embedding of a surface into $R^n$ or $S^n$. This energy is invariant under Moebius tranformations and the "round" sphere is its only absolute minimum. Here we sketch a proof of the compactness property for a variant of it. The details will appear elsewhere…

Extends Borel invariant to measurable cocycles of 3-manifold groups.

A symplectic manifold $(M,ω)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symple…

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures $(M,I,J',K')$ containing $I$…

We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random 3-manifolds. When the level r is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by the standard Rayleigh distribution and independent of the choice of level. Hence the probability that the qu…