17 problems

Let ${\mathcal W}$ be a non-parallelizable $3$-web on a real or complex $2$-dimensional manifold. A $3$-web is linearizable if it is equivalent, up to a local diffeomorphism, to a…

Let $M$ and $M'$ carry homogeneous bihamiltonian structures … For small open subsets $U\subset M$ and $U'\subset M'$, let ${\cal B}_{U}$ and ${\cal B}_{U'}$ be the corresponding we…

Let $\mathcal W$ be a web, and let its abelian relations mean the relations among the defining foliations obtained by summing closed one-forms along the leaves. A Liouvillian exten…

A biparacomplex structure is an almost biparacomplex structure in the setting of the paper, and a functorial connection is a connection naturally attached to such a structure. Uniq…

Classification conjecture. The structure theory of maximal-rank webs should yield a complete classification, up to coordinatewise reparametrization, of real-analytic diffeomorphism…

Let $r=6,7$, let $\boldsymbol{Y}_r$ be the relevant del Pezzo variety, let $\boldsymbol{\mathcal W}^{GM}_{\hspace{-0.1cm}\boldsymbol{Y}_r}$ denote its Gelfand–MacPherson web, let…

Let $\mathscr{F}$ be a convex prefolliations of degree $d\geq 3$ on $\mathbb{P}^{2}_{\mathbb{C}}$. Its Legendre web is the $d$-web denoted by $\mathrm{Leg}\mathscr{F}$. Flatness co…

Let $\boldsymbol{\mathcal W}$ be a 6-web satisfying the five listed properties: it is linearizable and skew, has maximal 2-rank equal to $10$, has 1-rank at least $5$, and has a 5-…

Let $n\geq 2$ be even, let $\Pi_n$ be the index set used in the source, and let $AR_\nu(\boldsymbol{\mathcal E}_n)$ be the abelian relation associated with $\nu\in\Pi_n$. Abelian-r…

Let $n\geq 2$ be even, and let $\varepsilon_{n-1}$ be the function defined by the displayed formula in the source for the web $\boldsymbol{\mathcal W}_{0,n+3}$. Euler's even-dimens…

Let $n\geq 3$ be odd, and let $\boldsymbol{\mathcal E}_n^{>0}$ denote Euler's abelian relation for the web $\boldsymbol{\mathcal W}_{0,n+3}$. Euler's abelian-relation conjecture. T…

Let $n\geq 2$ be even, and let $\boldsymbol{\mathcal W}_{0,n+3}$ be the web by rational curves induced by the forgetful maps on $\mathcal M_{0,n+3}$. Burau's conjecture. The web…

Gronwall's conjecture. If two non-flat linear 3-webs are isomorphic, then the diffeomorphism realizing this isomorphism is a homography near any point.

Index bound conjecture. The index of the cubic form is always at most $1$, without assuming that the cubic form is semi-homogeneous.

Legendrian Gronwall conjecture. Every Legendrian 3-web admits at most one distinct local linearization in $\mathbb{P}^3$, except when it is locally equivalent to the dual 3-web of…

Let $\mathcal W$ and $\mathcal W'$ be germs of linear $3$-webs on $(\mathbb C^2,0)$ with non-vanishing curvature, and let $\varphi:(\mathbb C^2,0)\to(\mathbb C^2,0)$ be a germ of b…

Gronwall's conjecture. If the web curvature of $\mathcal{W}$ is not identically zero, these conditions uniquely determine $(h_3,h_2,h_1,h_0)$.