25 problems

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be an $A_\infty$-functor, and let $HC^-_*(F)$ be its relative cyclic homology equipped with the relative connection $\nabla^F$. Relative c…

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…

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 $\mathcal{C}$ 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 $C^\alpha(S^1)$ 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 $\mathfrak{M}$ 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…

Hodge-to-de Rham degeneration conjecture. The negative cyclic homology $HC_*^-(A)$ is a locally free $\mathbb{K}[[u]]$-module of finite rank.

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…

Let ${\mathcal A}$ be a smooth and compact DG category, and let $HH_{\bullet}({\mathcal A})$ and $HC^-_{\bullet}({\mathcal A})$ denote its Hochschild and negative cyclic homology.…

Generalized non-commutative degeneration conjecture. For every pair of small DG categories ${\mathcal B}$ and ${\mathcal C}$ over $k$, one has $\varphi_n=0$ for $n\leq 0$. This con…

Kontsevich–Soibelman conjecture. This spectral sequence degenerates at the first sheet. The conjecture generalizes the Hodge-to-de Rham degeneration for smooth projective varieties…

Let $G$ be a compact Lie group acting smoothly on a compact manifold $M$, let $\alpha\in H^3_G(M,\mathbb Z)$, let $H$ be the relevant transformation groupoid, and let $L$ be the as…

Let $\theta\notin\mathbb Q$, let $\mathcal A_\theta$ be the algebraic irrational rotational algebra, and let $\mathcal A_\theta\rtimes\mathbb Z_2$ denote its crossed product by the…

Blanc-Toën conjecture. If $A^{\bullet}$ is smooth and proper, the Chern character map is a quasi-isomorphism.

Local cyclic index theorem conjecture. For any local elliptic operator $A$,

Let $G$ be a reductive $p$-adic group. Choose a minimal parabolic subgroup $P_0\subset G$ with Levi decomposition $P_0=L_0U_0$. For each parabolic subgroup $P\supset P_0$ with Levi…

Generalized Strong Bass conjecture. For each non-elliptic conjugacy class $\langle x\rangle$, the image of

Let $k$ be a ring containing ${\mathbb Q}$, and let $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be a smooth and proper DG algebra over $k$. Write…

Let $X$ be a smooth and compact noncommutative space over $\mathbb{C}$. The periodic cyclic homology $HP_{\bullet}(C_X)$ is defined from the category $C_X$ associated with $X$, and…

Tsygan's cyclic formality conjecture. There exists a natural $\mathbb C[[u]]$-linear quasi-isomorphism of $L_\infty$-modules