arXiv research

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

The Poincare-Hopf theorem tells us that given a smooth, structurally stable vector field on a surface of genus g, the number of saddles is 2-2g less than the number of sinks and sources. We generalize this result by introducing a more complex combinatorial invariant. Using this tool, we demonstrate that many such struc…

Geometrically interprets cup products and defines combinatorial Pin structures.

We apply the graph complex method to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe t…

In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new definition, most results of [2] become correct without the hypothesis of 2-neighbourli…

A (complete) matching of the cells of a triangulated manifold can be thought as a combinatorial or discrete version of a nonsingular vector field. We give several methods for constructing such matchings.

In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochne…

This paper is a continuation of arXiv:0809.1158, dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operat…

We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in c…

Functors from web categories differ despite similar definitions.

We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from *M to itself, implementing t…

We generalize Turaev's definition of torsion invariants of pairs $(M,ξ)$, where $M$ is a 3-dimensional manifold and $ξ$ is an Euler structure on $M$ (a non-singular vector field up to homotopy relative to the boundary of $M$ and local modifications in the interior of $M$). Namely, we allow $M$ to have arbitrary boundar…

We extend Turaev's theory of Euler structures and torsion invariants on 3-manifolds to the case of vector fields having generic behavior on the boundary. This allows to easily define gluings of Euler structures and to develop a completely general gluing formula for Reidemeister torsion of 3-manifolds. Lastly, we descri…

We provide a combinatorial presentation of the set F of 3-dimensional generic flows, namely the set of pairs (M,v) with M a compact oriented 3-manifold and v a nowhere-zero vector field on M having generic behaviour along the boundary of M, with M viewed up to diffeomorphism and v up to homotopy on M fixed on the bound…

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo--riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,ξ)$ where $M = G/H$ is a Riemannian normal homogeneous space, $G$ is a compact simple Li…

Researchers use discrete Morse theory to improve the topology of matching complexes of complete graphs.

We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M). Namely, we allow M to have arbitrary boundary and x to have simple (convex and/or con…

Constructs combinatorial 2D topological field theories from cyclic A-infinity algebras.

Smooth fractal trees via analytic generators, preserving combinatorial and geometric properties.

The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.

It is shown that if a real value PL-invariant of closed combinatorial manifolds admits a local formula that depends only on the f-vector of the link of each vertex, then the invariant must be a constant times the Euler characteristic.

Formula derived for a magnetic line invariant.

Paper transforms torse-forming vector fields into simpler forms.

The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.

The spaces of Sp(n)-, Sp(n)U(1)- and Sp(n)Sp(1)- invariant, translation invariant, continuous convex valuations on the quaternionic vector space H^n are studied. Combinatorial dimension formulas involving Young diagrams and Schur polynomials are proved.

Graph neural networks improve combinatorial optimization by leveraging inductive bias.

We give a quantum field theoretic derivation of the formula obeyed by the Ray-Singer torsion on product manifolds. Such a derivation has proved elusive up to now. We use a BRST formalism which introduces the idea of an infinite dimensional Universal Gauge Fermion, and is of independent interest being applicable to situ…

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …

We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of $\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topolog…

Conformal vector fields on LCP manifolds are orthogonal and Killing.

For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…

Combining models in appropriate ways to achieve high performance is commonly seen in machine learning fields today. Although a large amount of combinatorial models have been created, little attention is drawn to the commons in different models and their connections. A general modelling technique is thus worth studying …

Study on vector fields on Lie groups reveals surprising algebraic coincidences.

Study biharmonic vector fields and unit vector fields on Riemannian manifolds.

This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…

Defines quaternionic k-vector fields on quaternionic Kähler manifolds.

We revisit the proof by Qin et al. (2014) of bounded regret of the C$^2$UCB contextual combinatorial bandit. We demonstrate an error in the proof of volumetric expansion of the moment matrix, used in upper bounding a function of context vector norms. We prove a relaxed inequality that yields the originally-stated regre…

Study on generalized derivations in polynomial vector fields Lie algebras.

Study on Einstein solitons with specific vector fields and their properties.

Investigates point spectra of vector fields and their properties.

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.

We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…

Examining singularities of commuting vector fields on submanifolds.

Vector fields invariant under Lie group action are finitely generated by polynomial fields.

The paper proves non-existence of torqued and anti-torqued vector fields on hyperbolic spaces.

We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are …

Enhances Hamiltonian systems stability through generalized double bracket vector fields.

Study classifies harmonic vector fields on 3-manifolds.