250 problems

Let $G$ be a simple Lie group of real rank $r\ge 2$, let $\Gamma<G$ be a lattice, and let $X$ be an acyclic simplicial complex with $\dim X<r$. Then every simplicial action of…

For every commutative complex Banach algebra $A$, let $\mathfrak{M}(A)$ denote its Gelfand spectrum. Taylor's question asks whether, and how, each higher integral cohomology group…

Cohomology-vanishing conjecture. If

Let $g$ be the genus of a smooth proper curve and let $k$ be a non-negative integer. Write $H^{2g+k}_c(\mathcal{M}_g)$ for the compactly supported cohomology of the moduli space…

Cohomological special effect conjecture. The system ${\mathcal{L}}$ is special if and only if it is cohomologically special.

Tautological generation conjecture. The following assertions hold:

Let $G$ be as above, let $\Pi^{\mathrm{alg}}(G)$ be the finite-dimensional algebraic representations of $G$, and let $\Pi^{\mathrm{coh}}(G_\infty;E)$ denote the irreducible represe…

Let $k$ be a base field and consider the diagram of categories of motives and realization functors described in the paper, with rational coefficients. A morphism of motives is test…

Cohomological complete-reducibility conjecture. If every left $V$-module of finite length is completely reducible, then

Trace formula conjecture. One has

Homological Pisot Conjecture. The tiling has pure point spectrum if its first rational Čech cohomology group has rank equal to the algebraic degree of $\lambda$.

Let $\widetilde{W}$ be an affine Weyl group, and let $\mathcal{T}_{\widetilde{W}}$ be the corresponding toric arrangement. Its complement is the space obtained by removing the arra…

Let $(T,U)$ be a minimal elliptic pair, and let $\chi\colon \mathcal{T}^{+}(\mathcal{O}_k)\to\overline{\mathbb{Q}}_\ell^{\times}$ be a smooth character. Let $s_\chi\geq 0$ denote t…

Let $X$ be a smooth projective surface, and let $C$ be a curve on $X$. Write $h^i(\mathcal O_X(C))=\dim H^i(X,\mathcal O_X(C))$. The bounded cohomology conjecture asserts that ther…

Let $w=sw's$ be an element of the Weyl group with $ll(w)=ll(w')+2$, and let $Z=X(w)\cup X(sw')$. For $i\geq 0$, write … where … and … Let…

Assume that $k$ is a perfect field of characteristic $p>2$, let $m,n,r\in\mathbb{N}$, and write $GL(m|n)_1$ for the first Frobenius kernel of the general linear supergroup scheme.…

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_i$ be trace-preserving associative derivations satisfying conditions (i)--(iii): each $D_j$ is trace-…

Let $\mathrm{Dif}_1$ denote the Lie algebra of differential operators of order at most one on the circle, and let $\Psi_3,\widetilde\Psi_5,\widetilde\Psi_7,\widetilde\Psi_9,\dots$…

Let $G$ be a group, $H$ an abelian group, and $R$ a commutative ring with unit group $R^*$. Let $B_G$ be the classifying space of $G$, and for each cohomology class…

Let $\lambda\in\mathbb C$, let ${\mathfrak{gl}}(\lambda)$ be embedded in $\mathrm{Dif}_1$, and let … be the first page of the associated Feigin Serre–Hochschild spectral sequence.…

Semi-splitting conjecture. The deformation of $H^*(M \times N)$ determined by $H^*(E)$ is semi-split with respect to $H^*(M)$. This conjecture proposes a general cohomological spli…

The smoothness conjecture. If $\operatorname{codim}\mathcal{F}>0$ and the leaves are dense, then

Let an arrangement of finitely many complex central hyperplanes have complement $M$. Let the zero-divisors-cup-length of $H^*(M,\mathbf C)$ mean the largest number of zero-divisors…

Let $M\preccurlyeq N$, let $X=\operatorname{Spec}(A)$ be the o-minimal spectrum under consideration, let $Z\subseteq X$ be definably closed, and let $W:=\theta^{-1}(Z)$, where…

Multiplicative compatibility conjecture. The isomorphism $\operatorname{End}_{\mathfrak{g}}(P)\cong H^*(\psi(P))$ is a ring homomorphism, and the isomorphism