48 problems

Polynomial formula conjecture. For $k=4$ and $k=5$, the Betti numbers are conjectured to satisfy

Polynomial Betti-bound conjecture. All Betti numbers of $S/I$ are bounded, as functions of $d$, by polynomials of degree at most $p-1$.

Miller–Villarreal conjectures. One always has

Kühn­el's conjecture. For $j=0,\dots,\lfloor(d-1)/2\rfloor$,

Let $\Gamma$ be a Buchsbaum simplicial complex, and let $b_{i,j}(\Gamma)$ and $b^e_{i,j}(\Gamma)$ denote its symmetric and exterior iterated Betti numbers. Let $\Delta(\Gamma)$ and…

Let $\Gamma$ be a simplicial complex, and let $b_{i,j}(\Gamma)$ and $b^e_{i,j}(\Gamma)$ denote its symmetric and exterior iterated Betti numbers, respectively. The iterated Betti-n…

Let $\Gamma$ be a simplicial complex, let $\Delta(\Gamma)$ and $\Delta^e(\Gamma)$ denote its symmetric and exterior algebraic shifts, and let $I_{\Delta(\Gamma)}$ and…

Let $m$ be a positive integer, and let $(M^m,g)$ be a compact Riemannian manifold with diameter $\operatorname{diam}(M)=1$. Write $\mathcal{R}m$ for its curvature operator, and let…

Betti-number conjecture. The ideals $I_h$ and $I_e$ have the same Betti numbers if and only if $n=3a$ for some integer $a$.

Asymptotic normality conjecture. The Betti numbers of these three quotient spaces are asymptotically normally distributed, and the associated variances grow linearly at the same ra…

Fix the multiplicity $e$. Betti-number stability conjecture. For every $m\geq 4$, the Betti numbers of $S^e(m,m+1)$ and $S^e(m+1,m+2)$ agree through the indicated range; equivalent…

Fix the multiplicity $e$ and write $\beta_j(a,b):=\beta_j(S^e(a,b))$. Trailing Betti-number conjecture for S^e(2,n). For $n\geq 6$, … and … Moreover, $S^e(2,e-1)$ is conjectured to…

Fix the multiplicity $e$ and write $\beta_j(m):=\beta_j(S^e(m))$. Trailing Betti-number conjecture. For every $m\geq 2$, the Betti numbers satisfy … and … This proposes that the Be…

Wang's maximal-Betti-number conjecture. The variety $W^{(n)}$ has the largest sum of Betti numbers among smooth, projective Calabi-Yau $n$-folds. When $n$ is odd, it also has the s…

Leading Betti-number expansion conjecture. One has

Let … be a non-collapsed sequence of closed Riemannian manifolds with $\mathrm{Ric}\ge -(n-1)$, converging in the Gromov–Hausdorff sense to a compact Ricci limit space $X^n$. Let…

Let $\mathcal{B}$ be a venustus building set, and let $X^\mathbb{R}_{\mathcal{B}}$ be its associated real toric manifold. Write $\beta_k(X^\mathbb{R}_{\mathcal{B}})$ for its ration…

Diagonal recurrence conjecture. For each fixed $r\ge 3$ and $k\ge r+3$, these numbers satisfy

Let $R$ be a $d$-dimensional Noetherian local ring, and let $M$ be a finitely generated nonzero $R$-module. Total rank conjecture. If $M$ has finite length and finite projective di…

Let $R$ be a Cohen–Macaulay local ring with canonical module $\omega$. Jorgensen–Leuschke conjecture. The inequality … implies that $R$ is Gorenstein. The paper states this as an o…

Unimodality conjecture. If $(M,\omega)$ is a closed symplectically rational $8$-manifold admitting a Hamiltonian $S^1$-action, then the even Betti numbers of $M$ are unimodal.

Let $X$ be a smooth projective surface and let $H$ be an ample line bundle. Fix a rank $r>0$ and a first Chern class $c$, and let $M_{X,H}(r,c,\Delta)$ be the corresponding moduli…

Cone conjecture. If the Betti numbers of $I(G)$ depend on the field, then the Betti numbers of $J^2$ also depend on the field.

Let $X$ be a smooth projective variety of dimension $n$, let $B$ be a line bundle, and for sufficiently large $d$ set $L_d:=\mathscr{O}_X(dA+P)$. Write…

Let $M$ be a closed complex manifold of real dimension $2n$, and let the total Betti number of $M$ mean the sum of all its Betti numbers. Sullivan's conjecture. The minimal total B…