arXiv research

A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the …

This paper has been withdrawn by the authors due to the fact that the webs considered in the paper are ``Veronese-like webs'' which are different from Veronese webs.

Study local invariants of divergence-free webs in geometry.

This book was intended to serve as supporting material for a mini-course on web geometry delivered at the 27th Brazilian Mathematical Colloquium which took place at IMPA in the last week of July 2009.

Geodesic flows with specific integrals are linked to special 4-webs.

The curvature of web curves is studied in 3D manifolds.

We exploit the correspondence between the three-dimensional Lorentzian Einstein-Weyl geometries of the hyper-CR type, and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper-CR Einstein--Weyl struc…

We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.

New algebraic geometry and statistical manifold connections proven.

Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web…

The abstract theorem is extended to higher genus surfaces.

We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representatio…

In the complex setting, let $F(x,y,y')=0$ be an analytic or algebraic differential equation with $y'$-degree $d$. We deal with the qualitative study of such equations through the geometry of the planar $d$-web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discu…

Counting tripods on a flat torus using lattice point counting.

In this paper, we study local solutions F=(F1,..,Fn) of a general functional equation of the form F1(U1(x,y))+....+Fn(Un(x,y))=0. A such equation will be called an ``abelian functional equation'' (Afe). We will restrict ourselves to the case when the inner functions Ui's are real rational functions. First we prove that…

In this paper we study a functional equation associated to the Kummer's equation (K) of the trilogarithm. Then we apply our results to web geometry and to characterize the functions solution of (K).

Formula calculates MOY webs and link polynomials.

We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is lineari…

Classifies hexagonal circular 3-webs with cubic polar curves.

We describe a project, called the "Discretization in Geometry and Dynamics Gallery", or DGD Gallery for short, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.

We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are he…

We give various results and applications using the connection $(E,\nabla)$ associated with a $d$-web. Precisely, we exhibit fundamental invariants of the web related to the differential equation of first order which presents the web. They cast some new lights on the connection and its construction, both conceptually an…

We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply a recently found in [AGL] the linearizability conditions for 4-webs in the plane to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is …

In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine struc…

Investigates webs related to cluster algebras and polylogarithms.

We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the …

Much information available on the web is copied, reused or rephrased. The phenomenon that multiple web sources pick up certain information is often called trend. A central problem in the context of web data mining is to detect those web sources that are first to publish information which will give rise to a trend. We p…

For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $Δ$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs W (3, 2,…

In this paper we study the linearizability problem for 3-webs on a 2-dimensional manifold. With an explicit computation based on the theory developed in the paper "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp. 2643-2654), we examine a 3-web whose linearizability was claimed in the same paper. We …

Veronese webs appear as the natural way of passing to the quotient of curves in the projective space. In thi paper, we give the link between classical multidimensionnal webs and veronse webs by mean of interpolation.

New hexagonal circular 3-webs with reducible curves classified.

In this paper, we reconstruct Kuperberg's $G_2$ web space. We introduce a new web (a trivalent diagram) and new relations between Kuperberg's web diagrams and the new diagram. Using the $G_2$ webs, we define crossing formulas corresponding to R-matrices associated to some $G_2$ irreducible representations and calculate…

Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relati…

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conject…

A deformation of the authors' instanton homology for webs is constructed by introducing a local system of coefficients. In the case that the web is planar, the rank of the deformed instanton homology is equal to the number of Tait colorings of the web.

Geometrically interprets integrability of geodesic flow using web theory.

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjectur…

We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symme…

Critical to evaluating the capacity, scalability, and availability of web systems are realistic web traffic generators. Web traffic generation is a classic research problem, no generator accounts for the characteristics of web robots or crawlers that are now the dominant source of traffic to a web server. Administrator…

Proves Nakai webs have rank 0 or 1, provides examples.

Develops invariants for webs and foams using Seiberg-Witten theory.

Extracts main content from web pages using neural sequence labeling.

In the present paper we define Samuelson's webs and their rank. The main result of the paper is the proof that the rank of the Samuelson webs does not exceed 6, as well as finding the conditions under which this rank is maximal for the general Samuelson webs as well as for their singular cases.

Functors from web categories differ despite similar definitions.

Interprets SL3-web intersections on surfaces.

Study of webs in quantum type C, proving equivalence to quantum representations.

Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distri…

The study generalizes Blaschke curvature for higher-dimensional webs and identifies infinite classes of isomorphism.