48 problems

Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism, and suppose that every orbit of $f$ is bounded. Bounded orbit conjecture. Then $f$ has a fixed point. Boyles disproved…

Fixed-point and spinodal-curve conjecture. (i) If $\lambda>1$, then $G$ has two fixed points, $0$ and $\alpha>0$; moreover, $0$ is unstable and $\alpha$ is stable. (ii) For…

Let $M^{2n}$ be a nonbounding unitary toric manifold, meaning that it does not bound equivariantly, and let $\lceil n/2\rceil$ denote the least integer greater than or equal to…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A point $x$ is a fixed point if $f(x)=x$. The two-fixed-point conjecture. If $f$ has…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map with $f(0)=0$, and let $J(f)$ denote its Jacobian matrix. The $C^1$ fixed point conjecture. If the eigenvalues of $J(f)$ have abs…

Nilpotent fixed-point conjecture. A nilpotent mapping $N:\mathbb C^n\to\mathbb C^n$ possesses at most one fixed point.

Consider the fixed points of the truncator-map dynamics for a system with an increasing number of agents. Fixed-point scarcity conjecture. As the number of agents increases, with o…

Nonelementary fixed-point conjecture. Nonelementary fixed points could correspond to random walks $S_{n}^{(\beta _{1})}$ of both types: oscillating and non-oscillating.

Projective-space generation conjecture. Stably complex $S^1$ actions with isolated fixed points up to bordism are generated by linear actions on projective spaces

Let $\mathfrak{g}$ be a reductive Lie algebra with Weyl group $W$, let $\mathfrak{C}_W$ be the set of two-sided cells in $W$, and let $\mathbf{a}:\mathfrak{C}_W\to\mathbb{N}$ be Lu…

Let $G$ be a complex reductive group with Cartan subalgebra $\mathfrak{t}$, Weyl group $W$, and $A=\mathbb{C}[\mathfrak{t}\oplus\mathfrak{t}^*]^{\mathrm{sgn}}$. Set…

Fixed-point conjecture. There exists a generating set $S$ of $\Gamma$ such that every first-step primitive element with respect to $S$ fixes a point in $I$, while there exists…

Let $\mathbf{f}=[f_1,\ldots,f_K]:\mathbb{R}^K\to\mathbb{R}^K$ be the SUCPA-map. For a fixed point $\boldsymbol{\beta}^*$ of $\mathbf{f}$, let…

Let the SUCPA-map be $\mathbf{f}=[f_1,\ldots,f_K]:\mathbb{R}^K\to\mathbb{R}^K$, and let $\omega(\boldsymbol{\beta}^{[0]})$ denote the $\omega$-limit set of an initial condition…

Finite-or-uncountable fixed-point conjecture. The set $\Lambda$ (or the set $\Theta$) is either finite or uncountable. This question concerns the possible cardinalities of fixed po…

Monotonicity and continuity conjecture. For any $a_1\leq a_2\leq 1$,

Fixed-point conjecture. There is only $1$ fixed point of $f_{4,n}\circ f_{1,n}^{-1}:S_n\to S_n$, namely the identity permutation, corresponding to the word $1...1$.

Unconditional algebraic magnetism conjecture. The conclusion of Theorem 3.?? should remain true without the additional assumption (i), (ii), or (iii).

Spurious fixed-point conjecture. There are only two spurious fixed points of $f(\theta)$ in $\mathbb{R}_{-}$. Moreover, there exists $\delta_{\mathrm{cr}}\in(0,1/2)$ such that the…

Let $W$ be the evolution operator considered in the paper, with parameters $p_1,p_2$, and let ${\rm Fix}(W)$ denote its set of fixed points. Let $S^{2,2}$ be the state space, and c…

A Boolean network is considered whose signed interaction digraph has positive arcs only; for each integer $k$, let $k$-${\textsc{MaxFPP}}$ denote the corresponding fixed-point coun…

Consider the fixed-point set $S$ associated with the branching process, and let $\bm{q}(A)$ denote the extinction probability vector associated with a set $A$, distinct in general…

Least-fixed-point conjecture. These copying and deleting operations force the star to be a single definite value: the least fixed point, rather than an arbitrary fixed point.

Let ${\mathcal F}$ be the self-map defined by the finite system of oper integral equations, and let ${\mathcal X}_\bullet$ denote the conjectural fixed point. Starting from … const…

$C^1$ fixed point conjecture. The origin $0$ is the unique fixed point of $f$.