84 problems

Let $\beta_i(t)$ be the $i$th Betti number of the $d$-dimensional Eden growth model at time $t$, with $1\leq i\leq d-1$. Almost-sure scaling conjecture. There exists a constant…

Let $\operatorname{\mathbb{R}P}^n$ denote real projective $n$-space, and let its length mean the invariant defined in the paper. Length monotonicity conjecture. The length of … is…

Let $G$ be a Lie group. Regard $BG$ as the classifying space of the underlying discrete topological group, and let $BG^{\rm top}$ be the classifying space with its usual topology.…

Let $(Y,y)$ be a finite aspherical CW-complex which is not acyclic, and let $f\colon (X,x)\to (Y,y)$ be a continuous map. Define the degree of $f$ by letting $\deg(f)$ be the least…

Let a graph property be a partition of the unlabeled graphs on a fixed number of vertices into graphs with and without the property. The property is monotone if it is preserved und…

Finite subset-space cell-structure conjecture. The space $\exp_k X$ has a cell structure obtained from $\exp_{k-1} X$ by adding cells of dimensions $k-c\leq i\leq nk$.

Let $M$ be a closed $(2k-1)$-manifold with torsion-free fundamental group, and let $\alpha$ be a unitary representation used to define the rho-invariant $\rho(M,\alpha)$. Weinberge…

For every $n\geq 1$, let $\operatorname{Res}\,\lambda$ denote the resonance set associated with a number partition $\lambda$ of length $n$. If…

Let $\lambda$, $\mu$, and $\tilde\mu$ be number partitions. For number partitions $\lambda>\mu$, the inclusion $i(\mu,\lambda):\delta_\mu\rightarrow\delta_\lambda$ induces a homomo…

For $n\ge 2$, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and define a partial order $<$ on $MT(n)$ by declaring ${\bf T}<{\bf T}^{\prime}$ when the…

For $n\ge 2$, let $FM_2(n)$ be the Fulton–MacPherson space, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and let $P$ be the dg-operad whose standard…

Let $Y$ be a complex analytic manifold and let … be a projective morphism to the complex disc. A homotopy fiber bundle is a morphism whose fibers are homotopy equivalent in the cor…

Let the BrainiaK crystal space denote the semantic state space equipped with maps induced by semantic composition and tree decomposition. A finite-dimensional quotient or completio…

Let $X$ be a reasonable topological space or a PA-space, and let $Y\subset X$ be a closed subspace. A map is a cofibration if it has the homotopy extension property. Cofibration co…

Wedge-of-spheres conjecture. For all $k\geq 2$, the complex $\mathsf{Cliq}_{k}(G)$ is homotopy equivalent to a wedge sum of $(2k-3)$-dimensional spheres. Moreover, the number of sp…

Higher path-homology vanishing conjecture. The path homology of $\vec{C}_n^S$ vanishes in every degree at least three:

Let $\operatorname{Fl}_{n+1}$ be the flag variety, let $\beta$ be a strictly monotonic class, and write $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ for the corresponding algebrai…

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-ra…

Let $P_h$ be the polynomial algebra on $h$ variables over $\mathbb F_p$, equipped with the action of the Steenrod algebra $\mathscr A_p$. For a positive integer $n$, write its $p$-…

Let $p$ be a prime, let $G=({\Bbb Z}/p)^k$, and let $X$ be a finite free $G$-complex. Carlsson's toral rank conjecture. One has … This conjecture gives a lower bound on the total m…

Mod-8 signature conjecture. The evaluation

Discrete configuration-space conjecture. For all $m\leq n$, the ordered configuration space of $m$ points in $K(n)$ is homotopy equivalent to the $m$th discrete ordered configurati…

Eilenberg–MacLane homology conjecture. The following three groups are equivalent:

Triangulation-invariance conjecture. There is a good definition of $T_p(M,G)$, and

Let $G$ be a planar graph, and let $\mathcal{M}(G)$ denote its matching complex, whose vertices are the edges of $G$ and whose simplices are sets of pairwise disjoint edges. Planar…