45 problems

Vanishing conjecture. The higher infinitesimal cohomology groups with coefficients in $O/[O,O]$ and $K_1$ were conjectured to vanish.

Let $A$ be an associative algebra, let $\overline{\varphi}$ be the map in the preceding kernel-presentation conjecture, and define $\theta_2(A)=[A,A]/[A,[A,A]]$. Replacing $A$ by a…

Crystalline comparison conjecture. These two complexes are quasi-isomorphic for all $n,m\geq 1$ and all primes $p$; if $A$ is also compact, the homology of either complex is flat o…

Let $k$ be a field of characteristic $0$, and let ${\mathcal A}$ be a smooth proper dg category over $k$. For every integer $n\geq 1$, consider the complex … where…

Let $k$ be a regular $\mathbb{F}_p$-algebra. Consider the natural long exact sequence … and the canonical isomorphism … The element $x-y$ defines a push-forward map…

Bockstein conjecture. For $N$ even,

Let $\mathcal{X}$ be a smooth Deligne--Mumford stack with an étale cover $\pi:U\to\mathcal{X}$ by a scheme, and set $Z=U\times_{\mathcal{X}}U$. The correspondence $Z\times_UZ\to Z$…

Let $X$ be a quasiprojective scheme over a field $k$, let $G$ be a finite group acting on $X$, and assume that $\operatorname{char} k$ does not divide $|G|$. Let…

Let $X$ be an $\mathsf{nc}$-space over $\mathbb C$, with $C_X$ its associated category. A noncommutative Hodge filtration is described by a pair $(\mathcal H,\nabla)$ consisting of…

Let $X$ be an oriented smooth manifold. Let $(\operatorname{coHoch}_\bullet(\mathcal{C}(X)),d,P)$ be the associated mixed complex, and consider its relative cyclic chains. An…

Let $\mathcal{C}$ be a $\mathbb{C}$-linear dg-category. Write $K_*^{\operatorname{top}}(\mathcal{C})$ for its topological $K$-theory groups and $HP_*(\mathcal{C})$ for its periodic…

Let $K^S(A)$ denote stable K-theory and let $\mathrm{HH}(A)$ denote Hochschild homology. Goodwillie's conjecture. Stable K-theory and Hochschild homology should be rationally equiv…

Let $C^\alpha(S^1)$ denote the Hölder algebra of exponent $\alpha$, and let $z$ be the coordinate function on $S^1$. The class $[z\otimes z^{-1}]$ is considered in reduced cyclic h…

Let $M$ be a closed oriented C^-manifold, let $\Omega(M)$ denote its de Rham algebra, and let $\Omega(M)^\vee[-\dim M]$ be the shifted dual complex. Write…

Let $H$ be a Hopf algebra, and let ${}_H\mathcal{M}$ denote the monoidal category of left $H$-modules. Mixed anti-Yetter–Drinfeld contramodules are the coefficients introduced by a…

Let $X$ be a manifold with de Rham differential graded algebra $\Omega^*(X)$, and let $HC^\lambda_*(\Omega^*(X))$ denote the Connes version of cyclic cohomology of this de Rham com…

Let $\mathfrak{M}$ be a quasi-smooth QCA derived stack, let $\mathcal{N}=t_0(\Omega_{\mathfrak{M}}[-1])$ be its associated d-critical locus, and let $\mathcal{N}_{\mathrm{st}}$ be…

Let $M_{X,\sigma}$ be the moduli space under consideration, let $\mathcal{DT}^{\mathbb{C}^{\ast}}(M_{X,\sigma})$ be its $\mathbb{C}^{\ast}$-equivariant categorical DT theory, and l…

For a DG algebra $A$ over a field $K$, the Hochschild-to-cyclic, or Hodge-to-de Rham, spectral sequence … relates Hochschild and periodic cyclic homology, where $u$ has degree $2$.…

Let $A^!=\mathbf\Lambda(\xi_1,\ldots,\xi_n)$ be the exterior algebra, equipped with a Poisson structure $\pi^!$. Let $\mu^!$ be its product and let $\mu^!+\tilde\mu^!$ be the corre…

Work over $k[\![\hbar]\!]$, with $A=k[x_1,\ldots,x_n]$ a Poisson algebra and Poisson structure $\pi$. Let $\mu+\tilde\mu$ be the corresponding deformation-quantized product on…

GJP-program for infinite-dimensional localization. The composition

Burghelea's conjecture. If $K(G,1)$ has the homotopy type of a finite CW-complex, then

Let $G$ be a group. Write $\langle G \rangle^{\infty}$ for the conjugacy classes of infinite-order elements, and for $g\in G$ let $C_g$ be its centralizer and…

Let $M$ be a quasi-conformal closed manifold. Let $\mathcal{A}_{\mu}$ be the algebra of pseudo-differential operators on $M$, localized by the support of operators about the diagon…