arXiv research

Calabi surgery modifies Z/2 harmonic 1-forms using 2-valued 1-forms.

Study on 2-valued dynamics on complex plane, showing some dynamics can't be group actions.

Formula for Z_2-valued index of symmetric operators on manifolds.

Study shows singular set of certain graphs has codimension 1.

In this paper we consider a generalization of the Faddeev model for the maps from a closed three-manifold into the two-sphere. We give a novel representation of smooth $S^2$-valued maps based on flat connections. This representation allows us to obtain an analytic description of the homotopy classes of $S^2$-valued m…

Study of spectral flow in symmetric Toeplitz operator families.

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary d…

Study predicts coastal water quality using machine learning, identifying salinity as key factor.

The study characterizes homology 4-manifolds with $g_2\leq 5$ combinatorially.

This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a …

Proves singular set of certain integral hypercurrents has measure zero.

Topological complexity for closed 1-forms

Study on Riemannian Poisson warped product spaces and their properties.

This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed 1-form and a category with a respect to a cohomology class and constructed a Lu…

The paper explores parallel 1-forms on special Finsler manifolds and their properties.

New insights into cohomology of closed 1-forms.

Indices of vector fields and 1-forms studied for singular varieties and actions.

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.

If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…

Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently …

New metrics produce discrete zero sets for nondegenerate harmonic forms.

S.P.Novikov developed an analog of the Morse theory for closed 1-forms. In this paper I suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms.

The study resolves a conjecture about harmonic forms on compact manifolds.

We study the fixed point theory of n-valued maps of a space X using the fixed point theory of maps between X and its configuration spaces. We give some general results to decide whether an n-valued map can be deformed to a fixed point free n-valued map. In the case of surfaces, we provide an algebraic criterion in term…

The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.

Study topological properties of foliations induced by closed 1-forms on orbifolds.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel…

Study Euler obstruction of 1-forms on determinantal singularities.

Vanishing theorem for $L^2$-harmonic forms on Riemannian manifolds with parallel 1-form.

Constructs deformations of Vaisman manifolds preserving foliations.

We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…

Study existence of harmonic 1-forms on Calabi-Yau manifolds.

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.

We prove in this article that given a linearly concave domain $D$ in the projective space $\Bbb{CP}^{n}$, a 1-dimensional comlex analytic set $V$ in $D$, and a meromorphic 1-form $φ$ on $V$, $V$ is a subset of an algebraic variety of $\Bbb{CP}^{n}$ and $φ$ is the restriction to $V$ of an algebraic 1-form on $\Bbb{CP}^{…

Some years ago Moshé Flato pointed up that it could be interesting to develop the Nambu's idea to generalize Hamiltonian mechanic. An interesting new formalism in that direction was proposed by T. Takhtajan. His theory gave new perspectives concerning deformation quantization, and many authors have developed its mathem…

The paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the original problem with a closed 1-form to a traditional problem with a Morse function.…

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal h…

New examples of Z/2 harmonic 1-forms and their branching sets are explored.

Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$…

We consider the closed orbit structure of generic gradient flows of Morse closed 1-forms. The torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. We extend this result to arbitrary Morse closed 1-form…

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski m…

Paper explores connection cochain in abelian extensions and its relation to connection forms.

We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, $J^1π$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian i…

The paper explores how magnetic systems' spectra can identify metrics and 1-forms.

No nontrivial harmonic 1-forms on certain gradient Ricci solitons.