22 problems

Let $R$ be a local valuation ring with fraction field $K$, let $G$ be a finite group, and let $\Lambda$ be a Hopf order in $K[G]$. Write $G_0^R(\Lambda)$ for the relevant Grothendi…

A module over $\mathbf{Gr}(\mathscr{F})$ is called realizable if it is isomorphic to $[\mathcal{M}]_\oplus$ for some transitive subcategory of $\mathrm{add}(\mathscr{F}\cdot L)$, w…

Let $e$ be a distinguished nilpotent element, so that $A_e=Q_e$, and let $c$ be the corresponding two-sided cell. Let $J_c$ be identified with…

Let $\mathcal{A}$ be the algebra generated by the elements $R_{\mathbf d}$ over $\mathbb{Z}[q_1^{\pm1},q_2^{\pm1}]$, and specialize its parameters by … Let…

Symmetric-power criterion conjecture. If the canonical morphism above is an isomorphism, then $\pi$ is an isomorphism.

Let $R$ be a complete intersection of even dimension with an isolated singularity, and let ${\mathbb E}(R)$ denote the real vector space referred to in the source. Dao–Kurano's one…

Let $cmathfrak{M}$ be the set of multisegments, let $cLambda(cmathfrak{m})$ be the RSK-standard representation attached to $cmathfrak{m}cincmathfrak{M}$, and let $cmathcal{R}'$ be…

Let $K^n(\operatorname{\mathcal{U}})$ be the free abelian group generated by isomorphism classes of direct summands of $H^*V_n$, where $V_n=(\operatorname{\mathbb{F}}_p)^n$, and le…

Let $\mathcal{E}$ be an exact category, and let $\operatorname{\mathsf{AR}}(\mathcal{E})$ denote the subgroup generated by its Auslander–Reiten conflations inside the subgroup…

Let $A$ be an abelian variety of dimension $g$, let $K=K_0(A)\otimes\mathbf Q$, and let $\operatorname{Fil}^r_\Gamma$ be the gamma filtration associated with the composed lambda-ri…

Let $A$ be an abelian variety of dimension $g$, let $K=K_0(A)\otimes\mathbf Q$, and write $K^{(p)}$ and $\widehat K^{(r)}$ for the eigenspace components associated with the relevan…

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field of characteristic zero, and let $K_0(A)\otimes\mathbf Q$ carry its usual gamma filtration…

K-theoretic Clifford algebra isomorphism conjecture. The map $\gamma$ is an isomorphism.

The Heisenberg algebra inclusion conjecture. This injection is an isomorphism:

Let $\bD$ be the divided power algebra over $\bk$, let $R$ be the relevant grading ring, and let $\cC_n$ be the category of finitely generated $\bk$-modules supported on…

Let $R$ be a complete intersection isolated singularity of even dimension $d$. The rationalized numerical Grothendieck group is denoted by $\overline{G_0(R)}_{\mathbb Q}$. Finite C…

Khovanov's conjecture. The ring homomorphism $\gamma$ is a ring isomorphism.

Distinct-extension conjecture. Any such extension decategorifies an extension of the ground field as above, and if $\mathcal{A}$ and $\mathcal{A}'$ are distinct sub-Hopf algebras,…

Heisenberg categorification isomorphism conjecture. The injective algebra homomorphism $\mathbf{F}$ is an isomorphism.

The Heisenberg algebra–Grothendieck group conjecture. The map $\gamma$ is an isomorphism.

Let $\mathfrak{g}=\mathfrak{spo}(2n|2m+1)$, and let $\mathfrak{p}$ be the maximal parabolic subalgebra obtained by removing the simple root $\epsilon_m$, with Levi subalgebra…

Caldero–Keller conjecture. The antisymmetric bilinear form is well-defined on $K_0(A)$.