29 problems

Let $G$ be a reductive algebraic group acting on affine space $\mathbb{A}^n$ over a field $k$. Linearization conjecture. The action is linear after a suitable polynomial change of…

For $n\geq 1$, let $P_n=\mathbb{K}[z_1,\ldots,z_{2n}]$ be the commutative Poisson algebra with bracket … An action is effective if its kernel is trivial, and regular if it is algeb…

Let $F_n=\mathbb{K}\langle x_1,\ldots,x_n\rangle$ be the free associative algebra, and let $\mathbb{T}_{n-1}$ be the $(n-1)$-dimensional algebraic torus. Free-associative torus-act…

Let $\Phi:\mathbb{C}^n\to\mathbb{C}^n$ be a polynomial automorphism of order two. The Linearization Conjecture states that there exists an automorphism…

Almost action conjecture. If $\sigma:G\to H$ is a map such that

Proper groupoid structure conjecture. If $x$ is a fixed point of a proper groupoid $\Gamma$, then $x$ has an invariant neighborhood on which $\Gamma$ is isomorphic to the action gr…

Poisson linearization conjecture. If the restricted Lie algebroid $T^*_S M$ integrates to a compact, source 1-connected Lie algebroid, then $pi$ is linearizable around $S$.

A Nambu tensor of order $q\geq 3$ has a non-elliptic singularity of Type 1 and class $C^{\infty}$, with signature different from $q-3$. Smooth linearization conjecture. Such singul…

Let $G$ be a linearly reductive algebraic group acting regularly on affine $n$-space $\mathbb{A}^{n}$. A fixed point is a point $P$ fixed by the action of $G$. Kambayashi's lineari…

Let $G\rightrightarrows M$ be a proper Lie groupoid. If $G\rightrightarrows M$ is invariantly linearizable, then it admits a complete $0$-metric $\eta_0$. Conversely, the converse…

Let a control system possess a control admissible symmetry group. EDFL-to-CFL conjecture. If the control system is EDFL, then it must be CFL. The conjecture concerns the relation b…

Let a contact sub-connection have a control admissible symmetry group $G$ with $\dim G=1$. One-dimensional symmetry reduction conjecture. If the contact sub-connection is EDFL by p…

Let a contact sub-connection admit an ESFL reduction by a partial contact curve of codimension $s>1$. Codimension-one reduction conjecture. The contact sub-connection must also adm…

Let $p(z)$ denote the function associated with an ESFL reduction by a codimension $1$ partial contact curve, and let the truncated Euler operator and the higher-order Euler operato…

Let a control system with at least two controls be a Goursat bundle, and suppose it cannot be transformed to Brunovský normal form via an ESFT. DF linearization conjecture. The con…

Arnold's conjecture. The accumulation of special leaves in a neighborhood of the singular point could be the obstruction to the linearizability of the saddle point when $\alpha$ is…

Let $F$ be a diffeomorphism with a hyperbolic fixed point at $0$, and let $\Lambda=DF(0)$. A local conjugation is a smooth transformation $\Phi$ defined near $0$ such that…

Let $F_n$ be the free associative algebra on $n$ generators, and let $\mathbb{T}_{n-1}$ be the $(n-1)$-dimensional algebraic torus. An action is linearizable when it is conjugate t…

Let ${\rm SA}_n(\mathbb{K})$ denote the group of special polynomial automorphisms, and let ${\rm SLIN}_n(\mathbb{K})$ denote the corresponding subgroup satisfying the special linea…

Let $\mathbb{K}$ be a field, and let ${\rm GA}_n(\mathbb{K})$ denote the group of polynomial automorphisms of affine $n$-space over $\mathbb{K}$. Let ${\rm GLIN}_n(\mathbb{K})$ be…

Linearization conjecture. Every $\mathbb{G}_{m}$-action on $\mathbb{A}^{n}$ is linearizable; equivalently, up to conjugation by an automorphism of $\mathbb{A}^{n}$, the action can…

Douady's conjecture. Polynomials and rational functions of degree $d\ge 2$ do not have exotic Siegel disks.

Let $G\rightrightarrows M$ be an $s$-locally trivial proper Lie groupoid. Invariant linearization around a saturated embedded manifold means that a neighborhood of the subgroupoid…

Let $F:\mathbb{C}^n\to\mathbb{C}^n$ be a $p$-periodic polynomial map, meaning that $F^p$ is the identity. A polynomial automorphism is an invertible polynomial map with polynomial…

Algebraic Linearization Conjecture. The group $G$ has a fixed point $p$, and the action of $G$ is linear with respect to a suitable coordinate system of $\mathbb{C}^n$ having $p$ a…