26 problems

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.

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…

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:

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 $X$ 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…

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

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…

Bowen-type saddle-connection conjecture. Then $G$ exhibits saddle connections similar to the saddle connections in Example 1.

Generic-family conjecture. Smooth vector fields met in a generic finite-parameter family belong to $Vect^*\,S^2$.

Let $X$ be an affine variety with coordinate algebra $A_X$, let $A\mathcal{V}_X$ denote the corresponding algebra of functions and vector fields, and let a gauge module mean an…

Index-vanishing conjecture. Then

Generic-path conjecture. Given $v_0$ and $v_1$, there is a one-parameter family of traversing fields $\{v_t\}_{t\in[0,1]}$ connecting $v_0$ to $v_1$ such that…

Boundary-generic realization conjecture. The constraints

Disk obstruction conjecture. The standard $(n+1)$-disk $D^{n+1}$ does not admit a traversing boundary concave vector field.

Let $M$ be a manifold and let $X$ be a vector field on $M$. Say that $X$ is accumulated by a class of vector fields if it lies in the closure of that class in the relevant topology…

A sectional-hyperbolic set is an invariant set with a partially hyperbolic splitting whose contracted bundle is uniformly contracted and whose complementary bundle is sectional-exp…

Let $M$ be the underlying compact manifold, let ${\cal X}^1(M)$ denote the space of $C^1$ vector fields, and let $X\in{\cal X}^1(M)$. A vector field is approximable by homoclinic t…

Let $X$ be a vector field. A singular Axiom A vector field without cycle has finitely many chain recurrent classes, each singular hyperbolic; a robustly heterodimensional cycle is…

Let $X$ be a generic vector field, and let $C(\sigma)$ be a non-trivial chain recurrent class containing a hyperbolic singularity $\sigma$. A chain recurrent class is non-trivial w…

Let $X$ be a vector field. A singular Axiom A vector field without cycle has only finitely many chain recurrent classes, each singular hyperbolic; a homoclinic tangency is a non-tr…

Let $X$ be a vector field, and let $C^r$ denote the differentiability topology. A horseshoe is the standard hyperbolic chaotic invariant set, and a Morse-Smale vector field has hyp…

Let a vector field be robustly non-hyperbolic if it belongs to the robustly non-hyperbolic class considered in the conjecture, and let $C^r$ denote the relevant differentiability t…

Arbitrary-regularity approximation conjecture. In any $C^r$ neighborhood of $X$, there exists a vector field that coincides with $X$ in a neighborhood of $X$ and satisfies…