56 problems

Kosniowski's conjecture. If $M^{2n}$ is not a boundary, then the number of fixed points is greater than $f(n)$. Kosniowski conjectured that the candidate function is

Let $G$ be a group with exponential growth, and let $\varphi:G\longrightarrow G$ be an injective group endomorphism. The Reidemeister number $R(\varphi)$ is the number of $\varphi$…

Let $E$ be a Banach space whose current point is … and let $B\subseteq E$ be closed. The following should hold simultaneously: the elements of $B$ satisfy at $t=0$ the initial cond…

Index-bound conjecture. One has

Austin's conjecture. If $\{f_1,\ldots,f_r\}$ is a pairwise commuting family of $\lambda$-contractions on $X$, then the maps have a common fixed point; that is, there exists…

Let $(X,d)$) be a complete metric space, and let $\{f_1,f_2,\dots,f_n\}$ be a contractive family of continuous operators on $X$: for some $\lambda\in(0,1)$, every pair $x,y\in X$ h…

Stein's fixed-point conjecture. Some composition of members of $\mathcal{F}$ has a fixed point.

Let $D\subset\mathbb{C}^n$ be a topologically contractible, strictly pseudoconvex domain, and let $\phi:D\rightarrow D$ be a holomorphic self-map. Its fixed-point set is the set of…

Uniqueness conjecture. Equation (11) has a unique solution.

Reflexive-space fixed point conjecture. Every reflexive Banach space has the fixed point property for nonexpansive maps. Equivalently, if $X$ is reflexive and $C\subset X$ is close…

Malkiewich–Ponto's conjecture. The fiberwise Fuller trace

Let $F$ and $G$ be the continuous self-maps defining the model-based and robot value equations, with positive inverse-temperature parameters $\beta_h$ and $\beta_r$. Since $F$ and…

Let $M$ be a closed smooth manifold, let $f:M\to M$ be a self-map, and let $\ell_n(f)$ lie in the direct sum $\bigoplus_{k\mid n}\mathbb{Z}[\pi_1(M)_{\rho,k}]$, where…

Let $G$ act on a closed smooth manifold $M$, let $f:M\to M$ be a $G$-equivariant map, and let $\ell_G(f)$ be the Klein–Williams invariant. For each conjugacy class of subgroups $H$…

Free perpetuity iteration conjecture. The sequence $(\mathsf f_n,f_n)$ converges to $(\mathsf f,f)$ as $n\to+\infty$ in the topology of uniform convergence on compact subsets of…

In a finite game, consider the set of Nash equilibria in mixed strategies and its connected components. A component is strategically essential if it persists under sufficiently sma…

Let $A[D_5]$ be the Artin group of spherical type $D_5$, and let $Z(A[D_5])$ denote its center. The $D_5$ property $R_\infty$ conjecture. The Artin group $A[D_5]$ and its central q…

Conner–Floyd's conjecture. The fixed point set of the action cannot consist of exactly one isolated point.

Zero displacement conjecture. For every $x\in X$,

Let $\Lambda:[0,1]^N\to[0,1]^N$ be the continuous map defined by simulating the GGDDF policy with availability windows and integrating the energy delivered outside each agent's ava…

Let $P$ be the parameter space and let $F_u:X\to X$ be the map associated with $u\in P$ in Corollary. Assume that assumption (b) guarantees a unique fixed point of $F_u(\cdot)$. Fi…

Fixed-point characterization. The following conditions are equivalent: (i) $F$ has a fixed point in every nonempty spherically complete subspace of $M$; and (ii) the displayed ineq…

Dimension-bound conjecture. The dimension of $M$ is bounded above by a function of $k$; more precisely,

Let ${\mathbb{T}}^n$ be the $n$-torus, let $f:{\mathbb{T}}^n\to{\mathbb{T}}^n$ be a map, and let $\tau$ be a free involution of ${\mathbb{T}}^n$. Nielsen–Borsuk–Ulam dichotomy conj…

The geometry conjecture. There exist vectors $v_1,\ldots,v_m$ in $X$ such that