22 problems

Stable fixed point conjecture. A subset $\sigma\subseteq[n]$ is the support of a stable fixed point if and only if $\sigma$ is a target-free clique.

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…

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 $G$ be a finite primitive non-regular permutation group of degree $n$. Write $\operatorname{fix}(g)$ for the number of points fixed by $g\in G$. Neumann's conjecture. There exi…

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…

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 $1...1$.

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

Composite-graph permitted motif conjecture. $G$ is a permitted motif if and only if every component $G_i$ is permitted.

Let $W$ act as a complex reflection group on $V$, let $\tau$ be a finite-order element of the normalizer of $W$ in $\operatorname{GL}_{\mathbb C}(V)$, and let…

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…

In the large-degree regime, let $\mu$ and $\alpha$ be the parameters appearing in the scalar density-evolution recursion, and let $h(v)$ be the function defined by that recursion.…

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…

Fixed-point conjecture. The map $\mathcal S$ has at least one fixed point in $(0,\tilde y)$.

Let $p$ be a polynomial of degree $d\geq 2$, and let $n\geq 2$. Degree-sharp multiplier conjecture. Then $p^n$ has a fixed point $\xi$ satisfying … The source states that this is s…

Let $X$ be a projective variety over $\mathbb{C}$ and let $\varphi:X\to X$ be an endomorphism with $\varphi^*L\cong L^{\otimes q}$ for some ample line bundle $L$ on $X$. Suppose th…

Extended fixed-point index conjecture. The map $\widetilde{\Phi}_d$ is finite, and the analogous results to the main theorems hold for $\widetilde{\Phi}_d$ for every parameter valu…

Extended multiplier-index conjecture. The converse of the emptiness criterion also holds when $d\geq 8$. Moreover, if