23 problems

Dalen's conjecture. The function $f$ has an exact fixed point; that is, there exists $x\in S$ such that $x=f(x)$.

Let ${\cal F}$ be a nilpotent subgroup of $\operatorname{Diff}^1_+(S^2)$. The theorem for abelian subgroups should also generalize to this nilpotent setting: there should be a subg…

Existence conjecture. There exists a complex oriented cohomology theory $E^*$ such that every closed oriented manifold is $E^*$-orientable and $E^*$ is $G$-sensitive for every fini…

Let $K$ be a topological space equipped with maps $K\to\{\bullet\}$ lying in both indicated right-orthogonal classes: the class generated by the finite-space maps…

Let $\Delta^+=\{u\in\Delta\mid u(\infty)=1\}\cup\{0\}$ be the subquantale of finitary distance distributions. Non-continuity conjecture. The suplattice $\Delta^+$ is not continuous…

Fixed-point conjecture. When $5cgamma$ is close to $1$, the limits $\bar{\bf u}$ and ${\bf \bar v}$ should be approximately equal; consequently, as $5cgamma\to1$ from above, one sh…

Let $M$ be a compact complex manifold, let $E$ be a holomorphic vector bundle over $M$, and let $\Gamma\subset M\times M$ be a transversal correspondence. Let…

Holomorphic Lefschetz conjecture for correspondences. The displayed formula holds. This is intended as the correspondence analogue of the holomorphic Lefschetz fixed point formula,…

Let $X$ be the function space used for the Poincaré operator, let $\tau$, $\varepsilon$, $x_0$, $\varphi^*$, $\mu$, $\omega$, and $\overline D$ be the parameters and functions defi…

Let ${\mathcal{N}}$ denote Baire space. A partial ordering $\preceq$ on ${\mathcal{N}}$ is $\Pi^0_1$, every $\preceq$-chain has a least upper bound, and…

Coherence-free fixed-point conjecture. Theorem 185-999 should still hold: $T$ has a unique fixed point.

Noncommutative Brouwer retract conjecture. There is no $*$-homomorphism

Noncommutative Brouwer fixed-point conjecture. Every continuous map

Polynomial-growth conjecture. The operators $f_1,\dots,f_n$ have a unique common fixed point. The source motivates this as an algebraic condition intended to rule out the tree-like…

Let $\mathbb{K}$ be a definably complete expansion of an ordered field, and let $f:[0,1]^2\to[0,1]^2$ be a definable continuous function. Then Brower's fixed-point conjecture. Ther…

Converse conjecture. If $T$ and $\mathcal{C}_{L}(T)$ are complete lattices, then $T$ has CLFPP.

Exact fixed-point conjecture. The function $\varphi$ has an exact fixed point; that is, there exists $x\in S$ such that $x=\varphi(x)$.

Let $S$ be a simplex and let $f:S\to S$ be uniformly continuous. Assume that every open set in $S$ contains a point $x$ such that $x\neq f(x)$, equivalently $|x-f(x)|>0$, and that…

Brower and Kakutani fixed point theorem. Brower fixed point theorem. Kakutani fixed point theorem.

Center conjecture. At least one of the following two possibilities holds:

Let $B$ be a spherical building and let $C\subseteq B$ be a closed convex subset. Fixed-point conjecture. Either $C$ is a subbuilding, or the action … has a fixed point. This gener…

Let $B$ be a spherical building and let $K\subseteq B$ be a convex subcomplex. A center is a point fixed by the automorphisms of $B$ preserving $K$. Tits' Center Conjecture. Either…

Let $B$ be a spherical building and let $K\subseteq B$ be a convex subcomplex. Center Conjecture. Then $K$ is a subbuilding or the action … of the automorphisms of $B$ preserving…