63 problems

Arnold's conjecture. Generic finite-parameter families of vector fields on the two-sphere are structurally stable: close families are weakly topologically equivalent.

Let $H(x,y)$ be a smooth function, and let $V_n$ be a planar vector field whose two components are the real and imaginary parts of $(\partial_x+i\partial_y)^nH(x,y)$. A Loewner poi…

Let $M$ be a compact connected manifold, let $X$ be a $C^1$-generic vector field on $M$, and let $f:M\to\mathbb{R}$ be a function that is either $C^1$ or $C^0$. Call $f$ invariant…

Let $M$ be a non-orientable closed surface, and let $r\ge 2$. Write $\mathfrak{X}^r(M)$ for the space of $C^r$ vector fields on $M$. A vector field is Morse–Smale if its singularit…

Wn-module conjecture. Every quotient $A_{n,k}$ for $k>1$ is a $W_n$-module in the class $\mathcal C$; the Lie bracket on $gr^+(A_n)$ is $W_n$-equivariant.

Let $X$ be a nonvanishing vector field on the plane, let $f$ be a positive smooth function, and let $C(X)$ denote the associated Lie algebra or vector space of vector fields. Centr…

Let $X$ be an analytic vector field on the plane or sphere, and let $C(X)$ be the real vector space whose elements are analytic vector fields associated with $X$. Infinite-dimensio…

Let $L$ be the Lie algebra generated by two vector fields on $\mathbb{R}$ in general position. For $x_1,x_2,x_3,x_4,y\in L$, write $[x_1,x_2,x_3,x_4,y]$ for the corresponding itera…

Zero-or-undefined index conjecture. Any pseudo vector field must have either index equal to zero or an undefined index.

Let $\zeta$ be a unit vector field on the round $3$-sphere. A unit vector field is called harmonic when it is a critical point of the energy functional for variations through unit…

Generic two-parameter degeneracy conjecture. In generic two-parameter families of vector fields on a surface, only the following degeneracies may occur: one or two simultaneous deg…

Componentwise inversion conjecture. The inversion formula … .

Bracket-width conjecture. The bracket width of $\operatorname{Vec}(C)$ is exactly two.

Homogeneous-chambar singular-set conjecture. For every $k\geq 1$,

4-chambar classification conjecture. Up to affine conjugacy, every $4$-chambar on an open subset of $\mathbb{C}$ is one of the following types:

Semi-simple vector-field conjecture. Any such $X$ is not almost a $p$-chambar.

Let $M^\star$ be the punctured 2-sphere under consideration, and for each positive integer $k$ let $X_{\mathrm{m},k}$ denote the meridian type vector field whose relative homology…

Let $W^k(\mathbb{R}^n)$ denote the class of smooth vector fields that are $k$-conservative, and let $j$ and $k$ be nonnegative integers. Distinctness conjecture. For every $k$, the…

Let $\psi$ be the spinor … where $\eta\in\varmathbb{C}^2$ is a constant spinor, and let $A$ be the vector field … where $w\in\varmathbb{R}^3$ is a constant vector. Let…

Let $w:U\subset\mathbb{R}^2\to\mathbb{R}$ be a function defined in a neighborhood $U$ of $(0,0)$, and suppose that $(0,0)$ is an isolated zero of the vector field…

Stein's conjecture. Is the operator $H^{\varepsilon_0}_{U}$ bounded on $L^p(\mathbb{R}^2)$ for some $p\in(1,\infty)$? This is the singular-integral variant of Zygmund's conjecture,…

Zygmund's conjecture. Is the operator $M^{\varepsilon_0}_{U}$ bounded on $L^p(\mathbb{R}^2)$ for some $p\in(1,\infty)$? This is a long-standing open problem concerning maximal func…

Let $X$ be a compact manifold and let $v$ be a smooth vector field that is of gradient type in a neighborhood of its finite zero set $Z(v)$. Let $\mathsf{Lyap}(v)$ be the least car…

Continuity invariance conjecture. Under these assumptions,

Arnold's versal deformation conjecture. Any finite-codimension degenerate vector field on $S^2$ has a versal deformation.