18 problems

A $C^r$-diffeomorphism is a diffeomorphism of class $C^r$, where $r$ is a real number satisfying $r\geq 2$. A symbolic extension of a dynamical system is a symbolic dynamical syste…

Boundary-determination conjecture. If $\partial X$ and $\partial X'$ are diffeomorphic, then $X$ and $X'$ are diffeomorphic.

Let $M$ be a compact manifold and let $\operatorname{Diff}^r(M)$ be the space of $C^r$-diffeomorphisms, where $r\geq 1$, endowed with the $C^r$-topology. Let $\mathcal{MS}$ denote…

Let $X$ be a $\mathbb Z[\mathbb Z]$-homology $S^1\times S^3$ admitting an orientation-reversing diffeomorphism that induces an orientation-preserving map on $H^1(X;\mathbb R)$. Let…

Genericity conjecture. For each closed manifold $M$ and regularity class $k\geq 1$, the expanding on average pairs $(f,g)$ are open and dense in

Let $M$ be a closed manifold with a volume, let $k\ge 1$, and let $\operatorname{Diff}^k_{\operatorname{vol}}(M)$ denote the space of $C^k$ volume-preserving diffeomorphisms of $M$…

Let $\M=\S^4$ and let $\barbemb_{\W}\colon\barbell\hra \S^4$ be a barbell embedding with both cuffs unknotted. For a bar word $\W=\by\bx^{\pm1}$, write $\bg(\barbemb_{\W})$ for the…

Nontriviality conjecture. The diffeomorphism $f$ is not smoothly isotopic to the identity. This would provide a potentially nontrivial pseudo-isotopy of $S^4$ arising from the disc…

Let $M$ be a compact boundaryless manifold of dimension at least $1$. Write $\operatorname{Diff}^\infty(\mathbb T\times M)$ for the space of smooth diffeomorphisms, and let $id$ de…

Uniqueness conjecture. A unique diffeomorphism can be assured by its Jacobian determinant and curl.

Let $Y$ be a closed, connected, oriented $3$-manifold. A compact, oriented smooth $4$-manifold $W$ with boundary $\partial W=Y$ has mapping class groups of diffeomorphisms and home…

Let $X$ be a 4-manifold with $k_1 X=0$. The subgroups $\Theta(\mathcal{J}(X))$ and $\Theta(\mathcal{J}(X)\cap\ker\Sigma)$ are defined in the target of the pseudo-isotopy invariant…

Ma{n}'e's conjecture. If a diffeomorphism satisfies the star condition, then it is hyperbolic, that is, its chain recurrent set is hyperbolic.

Thin-sandwich conjecture. Initial data is characterized by a metric $g_{ab}$ and the projection of its ADM velocity into $T_g\mathcal{M}/\mathcal{D}_0$. The conjecture expresses th…

For all $b1>1$ and $0<b2<1$, let $D_{b1,b2}$ be the set of diffeomorphisms $f\colon[0;1]\to[0;1]$ whose fixed points are exactly $0$ and $1$, with $Df(0)=b1$ and $Df(1)=b2$. C1-den…

Let $X=(C_0,C_1,A_0,A_1,\varphi)$ be an admissible set such that … and $H_2(\partial C_0)$ is torsion free. Let $(W,z)$ be a null-bordism of $X$, let $\alpha_W$ be the associated i…

Let $f$ be the diffeomorphism under consideration, and let $g_i=h_i f h_i^{-1}$ be tidy perturbations of $f$ converging to $g=hfh^{-1}$, where the diffeomorphisms $h_i$ converge to…

Let $S^1$ be the circle, let $\operatorname{Diff}^1(S^1)$ denote the space of $C^1$ diffeomorphisms of $S^1$, and for $f\in\operatorname{Diff}^1(S^1)$ let $C(f)$ be its centralizer…