231 problems

Configuration-space homotopy invariance conjecture. The homotopy type of $\mathrm{Conf}_n(X)$ depends only on the homotopy type of $X$.

Eilenberg–Ganea conjecture. For every group $G$,

Let $\mathcal{X}_{\mathrm{cf}}(3,6)$ be the closure of the variety parametrized by the canonical forms of the real regions of $\mathcal{X}(3,6)$, and let the log canonical compacti…

Atiyah–Sutcliffe Conjecture 2. One has

Atiyah's conjecture. For every $\mathbf{x}\in C_n(\mathbb{R}^3)$, the polynomials $p_1,\ldots,p_n$ are linearly independent over $\mathbb{C}$. This is the original Atiyah problem o…

Lambrechts–Stanley model conjecture. The Lambrechts–Stanley model is a model for the configuration spaces of every simply-connected closed oriented manifold.

Let $K$ be a convex body in $\mathbb R^d$, let $p$ be a prime, and let $k$ be a positive integer, and set $n=p^k$. Nandakumar–Ramana Rao theorem. It is possible to partition $K$ in…

Let $\ell\in\mathbb{R}^n$ be a positive length vector, and let $\mathcal{M}_2(\ell)$ denote the moduli space of closed linkages with length vector $\ell$ in the plane. Length vecto…

Let $G_A$ be a 2D or 3D grid domain with $A$ cells, and let $\mathtt{Cov}_{G_A}(A+1)$ denote the covering configuration space for $A+1$ agents. Write $\chi(G_A)$ for the Euler char…

Sundaram–Welker's conjecture. For every number partition $\lambda$,

Bianchi–Miller–Wilson's conjecture. For $k$ as above, the group $J^k\operatorname{Mod}(U)$ is exactly the kernel of the action of $\operatorname{Mod}(U)$ on…

Let the Bendersky–Gitler spectral sequence be the spectral sequence associated with the nerve of the covering of the fat diagonal in $M^k$, with higher differentials…

Let $M$ be a connected aspherical $2$-dimensional orbifold, meaning that its universal orbifold cover $\widetilde M$ is a contractible orbifold, equivalently that…

Let $D_n$ be the cohomology algebra of the configuration spaces of $n$ ordered points in $SU_2$ up to translations, and let $M_n$ be the intersection cohomology ring of the hyperto…

Let $M$ be a connected orbifold, and let $f(M)$ denote the homomorphism between Lie groupoids associated with the configuration-space construction. A homomorphism of Lie groupoids…

Let $M$ and $N$ be smooth connected manifolds. Write $\operatorname{{\sf{Ran}}}^{\sf{u}}(M)$ for the unital Ran space of $M$, and let $\operatorname{{\sf{Exit}}}(X)$ denote the exi…

Let $X$ be an irreducible smooth complex variety. For a partition $\lambda$ of $k$, let $W_\lambda(X)$ denote the symmetric complement associated with the stratum indexed by…

The periodic-orbit count conjecture. The estimate in case (A) should be improvable to $n+m-2$; that is, the number of distinct $D_n$-orbits of $n$-periodic billiard trajectories in…

Let $\Gamma$ be a graph and let $N$ be a positive integer. Write ${\cal C}^N(\Gamma)$ for the configuration space of $N$ points on $\Gamma$. Torus-gluing conjecture. The configurat…

Let $[T_{\rm poly}^{\bullet}]_{\rm div}$ denote the divergence-compatible polyvector fields, let $\widetilde{\mathcal C}$ be the explicitly constructed $L_{\infty}$-morphism before…

Baez–Dolan's conjecture. The $n-k$-category of endomorphisms of the $k-1$-fold identity of $s$ in $\sigma^k$ is the Poincaré $n-k$-category $\Pi_{n-k}(X_k)$.

Let $q_n$ denote the polynomial equation problem associated with the unordered configuration space of $n$ points, let $g(q_n)$ be its Schwartz genus, and let $L_n$ be the relevant…

In the Bott–Taubes configuration-space construction, let the anomalous term be the contribution from the anomalous face associated with a diagram without chords. This contribution…

Let $Z_2(K)$ be the perturbative configuration-space-integral invariant of a framed knot $K$, and let $Z^{\text{anomaly}}$ denote the framing-dependent renormalization term needed…

Topological complexity conjecture. If $\Gamma$ has at least two essential vertices and $n\geq 2m(\Gamma)$, then