66 problems

Buch's conjecture. The structure constants $c_{u,v}^w$ have alternating signs for all flag varieties.

For a quasi-projective variety $X$ over $K$, let … where $\operatorname{Sym}^n(X)$ is the $n$-fold symmetric power of $X$ and $\operatorname{Sym}^0(X)=\operatorname{Spec}K$. Kapran…

Let Khovanov's Heisenberg category have central charge $-1$, and let its Grothendieck ring be the decategorification obtained from its monoidal structure. Khovanov's conjecture. Th…

Van Straten's exponent conjecture. The generating series of Hilbert schemes of points on a smooth surface can be considered as an exponent in the power structure over the Grothendi…

Let $k$ be a field, let $X$ be a variety over $k$, and let ${\mathcal M}_k$ denote the relevant localization of the Grothendieck ring of varieties. Let $Z_{\rm mot}$ be the motivic…

Let $k$ be a field of characteristic $p$, and let $K_0({\mathcal V}_k)$ be the Grothendieck ring of $k$-varieties: the free abelian group on isomorphism classes of geometrically re…

Let $X$ be a finite-dimensional non-degenerate evolution ${\mathbb K}$-algebra, and write its optimal direct-sum decomposition into indecomposable evolution ideals as … For an evol…

Let ${\mathfrak{g}}$ be a finite-type Lie algebra, let $\Delta_+$ be its set of positive roots, and let $\mu$ be a coweight. For each positive root $\alpha$, write … The normalizat…

Let $X$ and $Y$ be simply connected varieties. Say that they are D-equivalent when their bounded derived categories of coherent sheaves are equivalent, and L-equivalent when their…

Power-structure compatibility conjecture. For every such $k$, $X$, and $n$,

Let $\overline{\mathcal{M}}_{0,n}$ be the fine moduli space of stable $n$-pointed curves of genus zero, and let its class in the Grothendieck ring of varieties be represented by th…

Let $X$ be a smooth variety over $k$, and let $m\geq 1$. The symmetric power $\operatorname{Sym}^m X=X^m/S_m$ is the quotient by the symmetric group. A variety has $\mathbb{D}$-sin…

Let $K_0(\operatorname{Var}_k^{\dim})$ be the graded Grothendieck ring of varieties, equipped with the symmetric-power operations $\operatorname{Sym}^m$ and the involution…

Várilly-Alvarado and Vakil's density-function conjecture. The quantities $\rho^{\textnormal{geo}}(x)$ and $\rho^{\textnormal{gr}}(x)$ exist and are independent of the choice of…

Várilly-Alvarado and Vakil's Grothendieck-ring measure conjecture. For all $B$, the limit $\lim_{g\to\infty}\pi^{\textnormal{gr}}(B,g)$ converges to a measure…

Let $V$ be a finite-dimensional vector space, let $\operatorname{Jord}(V)$ be the free Jordan algebra generated by $V$, and let $a(V)$ and $b(V)$ be the unique elements of the augm…

L-equivalence conjecture for hyperkähler manifolds. If $X$ and $Y$ are $L$-equivalent, then their bounded derived categories of coherent sheaves are equivalent:

Aluffi–Marcolli–Qaisar conjecture. The sequence $a_0,a_1,\ldots,a_n$ is log-concave, meaning that

Shang's character-formula conjecture. In the Grothendieck ring of $GL(V)$,

Let $\mathcal{A}$ be the infinite-rank cluster algebra and let … be the injective ring morphism described above. Let $\mathcal{O}^{\mathrm{sh,f}}_{\mathbb{Z}}$ be the full subcateg…

For a smooth degree-three cover of the projective line, the normalized scrollar invariants lie on the line segment joining $(1/2,1/2)$ and $(1/3,2/3)$. Let $\rho^{\mathrm{geo}}$ be…

Let $X$ and $Y$ be projective hyperkähler manifolds. They are $L$-equivalent if there exists an integer $n\in\mathbb{Z}$ such that … and they are $D$-equivalent if there is a linea…

Let $k$ be a field of characteristic not $2$. Let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of $k$-varieties, let $\operatorname{GW}(k)$ be the Grothendieck–Witt ring, a…

Let $X$ and $Y$ be varieties, with $[X]$ and $[Y]$ denoting their classes in the Grothendieck ring of varieties $K_0(\operatorname{Var}/k)$. Larsen–Lunts conjecture. If … then $X$…

Let $X$ and $Y$ be smooth projective simply connected varieties, and let $D(X)$ and $D(Y)$ denote their bounded derived categories of coherent sheaves. Two varieties are $D$-equiva…