arXiv research

Classifies hexagonal circular 3-webs with cubic polar curves.

New hexagonal circular 3-webs with reducible curves classified.

Classifies symmetries of non-flat 3-webs around a point.

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…

There are two theories describing the linearizability of 3-webs: one is developed in the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654) and another in the article "On the Blaschke conjecture for 3-webs" (J. Geom. Anal. 16, 1 (2006), 69-115). Unfortunately, they cannot be both co…

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 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 …

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…

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 …

We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal wei…

We propose a definition of genericity for singular flat planar 3-webs formed by integral curves of implicit ODEs and give a classification of generic singularities of such webs.

We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…

Extends $sl(3)$ invariant to tangles via webs and functors.

We give a geometric interpretation of weighted homogeneous solutions to the associativity equation in terms of the web theory and construct a massive Frobenius 3-fold germ via a singular 3-web germ satisfying the following conditions: 1) the web germ admits at least one infinitesimal symmetry, 2) the Chern connection f…

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

In the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654), published in 2001, we studied the linearizability problem for 3-webs on a 2-dimensional manifold. Four years after the publication of our article, V.V.Goldberg and V.V.Lychagin in the paper "On linearization of planar three-…

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…

In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we cate…

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 prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local …

Study circular tractrices and pseudospheres in 3D space.

The paper defines circular orderability for quandles and explores their properties.

Paper studies geometric and combinatorial properties of circular snakes.

The study identifies surfaces with Maslovian normal bundles.

Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.

Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

It was shown by Kaup that every origin-preserving automorphism of quasi-circular domains is a polynomial mapping. In this paper, we study how the weight of quasi-circular domains and the degree of such automorphisms are related. By using the Bergman mapping, we prove that every origin-preserving automorphism of normal …

CDFD analyzes circularity and directionality in weighted directed networks.

The sl_3 spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl_3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection v…

Study geometrically characterizes piecewise circular curves with decreasing curvature.

The paper constructs bases for cluster varieties using ${ m SL}_3$-webs and laminations.

We suggest a new, alternative algebraic method for computation of geometrical quantities by means of the embedding of local loops into Lie groups.

Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagona…

Curves become nearly circular over time without initial assumptions.

Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.

We show that circular width is preserved under connected sum of knots for some cases.

The paper provides coordinates for ${ m SL}_3$-web diagrams on surfaces.

The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.

Projected random forests improve circular data prediction with adaptive arc length and finite-sample coverage.

ANGLE tackles circular data regression, improving predictive performance.

Derives conformal parameters of curves using inscribed circular polygons.

Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First w…

We give a complete proof of the fact that the trace of the curvature of the connection associated to a planar d-web (d>3) is the sum of the Blaschke curvatures of its sub 3-webs.

Interprets SL3-web intersections on surfaces.

For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslashη(K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\colon E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critica…

This article provides sufficient conditions for a closed hyperbolic 3-manifold $M$ with non zero first Betti number to fiber over the circle, and to find a fiber in $M$. Those conditions are formulated in terms of the behavior the circular characteristic in finite regular covers of $M$. We define the circular character…

A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular l…