17 problems

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…

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

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 $\Sigma$ 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…

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

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 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…

A shape $(n,\kappa,r,s)$ is called plausible when it satisfies the paper's plausibility criterion for detecting whether sufficiently general ideals of that shape are plausibly gene…

Assume $b_k=a_k^2$, with the integers $a_k$ pairwise relatively prime, and define … Let $\Sigma$ denote the parameter from the preceding construction, and let $\beta$ be the probab…

Let $K$ be the base field, let $n=3$, and let $B,C,D$ be generic $3\times 3$ matrices. Consider Eq., the nonsymmetric generic matrix equation in the unknown matrix…

Generic wickedness conjecture. A generic element of $\mathcal{C}$ is wicked.

Made's weak conjecture. For a generic Lagrangian $L$, there is an open dense subset $U$ of $H^1(M,\mathbb R)$ such that, for every $c\in U$, there is a unique $(L,c)$-minimizing me…

Made's strong conjecture. For a generic Lagrangian, there is a unique minimizing measure, and this measure is supported on a periodic orbit or a fixed point of the Euler–Lagrange f…