40 problems

Let $f$ be a $C^r$ diffeomorphism and let $\mathcal Z^r(f)$ denote its $C^r$-centralizer. Smale's conjecture. For $C^r$-generic $f$, the centralizer is trivial, namely … The source…

Let $\Sigma$ be a surface, let $\operatorname{MCG}(\Sigma)$ denote its mapping class group, and let $S$ be a finite generating set for $\operatorname{MCG}(\Sigma)$. Write $d_S$ for…

Hurley's conjecture. For every $r\geq 1$, there exists a residual subset ${\cal G}$ of $\operatorname{Diff}^r(M)$ such that, for every $f\in {\cal G}$, the union of the basins of t…

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with regular boundary, and let $(\varphi_n)_{n \geq 1}$ be an $L^2$-orthonormal basis of Dirichlet-Laplacian eigenfunctions. F…

Generic simplicity conjecture. We conjecture that the exceptional set $\mathcal{E}$ has measure zero in parameter space.

Let $\mathfrak{G}$ be the class of multi-frequency quasiperiodic Schrödinger operators whose spectrum is an interval, and let the potentials be real trigonometric polynomials with…

Let $\Omega$ be a domain and let $N_{0}$ denote the Newton potential operator, viewed as the inverse of a Laplacian in $L^{2}(\Omega)$ with nonlocal boundary conditions. Let…

Let $U$ be a bounded open set in ${\mathbb R}^d$. Write $C(\overline U)$ for the space of continuous real-valued functions on $\overline U$, and let $G_\delta$ denote a countable i…

Localizability conjecture. There are many spaces of dynamical systems for which any generically localizable property is locally Kolmogorov typical.

Let a Lie algebra of type $(n,r)$ be generated by $n$ degree-one elements with $r$ homogeneous quadratic relations. For a power series, let $[A(z)]$ denote the truncation operation…

Let $B$ be a non-commutative algebra generated by $n$ degree-one elements with $r$ homogeneous quadratic relations; this is an algebra of type $(n,r)$. For a power series…

Let $M$ be a closed manifold, let $r\in \mathbb{N}\cup\\{\infty\\}$, and let $\operatorname{Diff}^r(M)$ denote the $C^r$ diffeomorphism group. The generic trivial centralizer conje…

Generic non-universality conjecture. For any compact computable manifold $M$ (possibly with boundary), there is a $C^\infty$ generic set $\mathcal{S}$ of diffeomorphisms $f:M\to M$…

Generic non-universality conjecture. A $C^\infty$ generic $f:M\to M$ cannot be extended to a robustly Turing-universal CDS.

Let $\Sigma$ be a complete hyperbolic surface of finite type that is not a three-punctured sphere, and let $\operatorname{Mod}(\Sigma)$ be its mapping class group. Let $S$ be a spe…

Let $(X,\mathcal{B},\mathbb{P},T)$ be a measure-preserving system, and let $\mu_N^T f=\sum_{n\in\mathbb{Z}}\mu_N(n)f\circ T^n$ be a dissipative sequence of operators. Assume that ……

Generic finite convergence conjecture. There exists a finite set of polynomials $\varphi_1,\ldots,\varphi_L$ in the coefficients of $f$, $a$, $b$, and $c$ such that, if

Non-arithmetic-singleton conjecture. An arithmetically escaping function computes a set that is not an arithmetic singleton, that is, not a $\boldsymbol{\replace?}^0_n$ singleton f…

Genericity conjecture. For $m\geq 3$, $\mathcal{D}_n^{\operatorname{gen}(m)}$ consists exactly of the $m\times(n+1)$ $\{0,1\}$-matrices such that, for every $k\in[0,n]$, there are…

Let $M$ be a manifold whose dimension is sufficiently large. A property of $\mathrm{Diff}^r(M)$ is generically localizable if it has the generic-localizability property described i…

Guiheneuf's conjecture. There exists a natural topology on the set of billiard tables such that, for every $\epsilon>0$, the set

Degree-of-commutativity conjecture. If $G$ is not virtually abelian, then

Let $\mathbb M$ be a surface with area form $\omega$, and let $C^r_\omega$-generic mean generic in the space of $C^r$ area-preserving surface diffeomorphisms. Generic entropy dicho…

Let $M$ be the manifold under consideration, let $r\in\{1,\ldots,\infty,\omega\}$, and write $\operatorname{Diff}^r(M)$ for the space of $C^r$ diffeomorphisms of $M$. A family…

For a set $X$ and a positive integer $k$, say that $X$ is generalized $\mathrm{low}_k$, or $GL_k$, if … An $n$-generic is a set meeting every dense arithmetical class of the approp…