248 problems

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 $g$ 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…

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

Let $\Gamma$ be a torsion-free group such that $(\Gamma,\sigma)$ satisfies the twisted Bost conjecture, and suppose that the classifying space $B\Gamma$ is a smooth, compact, orien…

Let $(l,m)$ be regular. The endoscopic contribution $e_{\rm endo}({\mathcal A}_2,\mathbb V_{l,m})$ is defined using the cohomology of the moduli space ${\mathcal A}_2$ of principal…

Let $X_0$ be a normal crossing variety with smoothing $X_t$, and let $X$ denote the associated total space. Write $H^k(-,\mathbb Z)_f$ for the torsion-free quotient of integral coh…

Casian–Kodama conjecture. The graph ${\mathcal G}$ is the graph of incidence numbers for the integral cohomology of the real flag manifold ${\mathcal B}$ ($\hat{\mathcal B}$) in te…

The cohomology tensor-product conjecture. There is an equality

Let $W$ be the Weyl group, let $w\in W$, and let $Y_x(\dot w)$ be the stratum defined by … For a $G$-regular linear character $\psi:U_0\to\mathcal{O}^{\times}$, let…

Let $G$ be as above, and let $\pi=A_{\mathfrak q}$ be a cohomological representation of $G$ such that … where $\mathfrak l'_0$ is a compact Lie algebra and $\mathfrak g_1$ is a non…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$ obtained by restriction of scalars from a group of type ${}^3D_4$ or ${}^6D_4$ over a totally real number field, and supp…

Let $G$ be a semisimple group with discrete series. Let $\pi\in\widehat G$ be a cohomological representation, and consider the question of whether there is a lattice $\Gamma$ in…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra, and let $L\subset G$ be its associated Levi subgroup. Let…

Poisson-cohomology conjecture. There is an isomorphism of vector spaces