96 problems

Let $I(a_{1}+m,dots,a_{n}+m)$ denote the toric ideal associated to the sequence $(a_{1}+m,dots,a_{n}+m)$, where $m\geq 0$ is an integer. HerzogSrinivasan conjecture. The Betti numb…

Let $M$ be a compact locally conformally Kähler (LCK) manifold, and let $b_1(M)$ denote its first Betti number. Vaisman's conjecture. The number $b_1(M)$ should be odd. This conjec…

Let $G$ be a closed graph, let $J_G$ be its binomial edge ideal, and let $<$ be a diagonal term order. Ene–Herzog–Hibi conjecture. The graded Betti numbers of $J_G$ agree with thos…

Let $X$ be a smooth projective variety, let $B$ be a line bundle on $X$, let $L_d$ be the asymptotic sequence of line bundles considered in the paper, and define the Betti numbers…

Fix integers $d,m$ with $d\geq m\geq 1$, and suppose that $\alpha$ satisfies … Let $\Delta\sim\Delta_m(n,n^{-\alpha})$ be the corresponding multiparameter random simplicial complex…

Let $(M,\omega)$ be a $2n$-dimensional closed symplectic manifold equipped with a Hamiltonian $S^1$-action whose fixed points are all isolated. Write $b_j$ for the $j$-th Betti num…

Let $G$ be a simple graph on the vertex set $[n]$, let $S=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $J_G$ be its binomial edge ideal. Fix the specified monomial order $<$ and writ…

Let $M^n$ be a connected positively curved manifold, and let $b_{2i}(M)=\dim H_{2i}(M;\mathbb{F})$ denote its even-degree Betti numbers over a coefficient field. Gromov's linear Be…

Virtual infinite Betti number conjecture. A hyperbolic $3$-manifold $M$ has finite covers with arbitrarily large first Betti number; equivalently, for every integer $N>0$, there is…

Miller–Villarreal conjectures. One always has

Let $S\subset\mathbb{R}^k$ be a semi-algebraic set defined by $s\leq k/2$ quadratic inequalities, and suppose the paper's bound on its Betti numbers has the form $(O(k/s))^s$. Opti…

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

Let $I$ be the ideal generated by the entries of $XY-YX$ for generic $n$ by $n$ matrices, and let $J$ be generated by the off-diagonal entries. Let $d_{\min}$ be the quantity from…

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 compact orientable irreducible atoroidal $3$-manifold with infinite fundamental group. A finite cover of $M$ is a finite-sheeted covering space of $M$, and the first B…

Gromov's conjecture. The total Betti number should satisfy

The determinantal Betti numbers conjecture. All KW semigroups have determinantal Betti numbers. Consequently, the Betti numbers of KW semigroups depend only on their embedding dime…

Let $M$ be a closed $n$-dimensional manifold with almost nonnegative Ricci curvature, meaning that its diameter and Ricci curvature satisfy … Here $b_1(M)$ denotes the first Betti…

Let $G$) be a finite simple graph, let $P_n^2$ be the squared path on $[n]$, and for $k\ge 2$ and $n\ge k+3$ let … where $\Delta_k(P_n^2)$ is the $k$-cut complex of $P_n^2$. Fix…

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…

Let $K$ be a field, let $S=K[x_1,\ldots,x_N]$ be standard graded, and let $I\subseteq S$ be a homogeneous ideal containing a regular sequence of degrees…

Let $C_{p,d}(\mathbb{P}^n)$ denote the Chow variety of effective algebraic $p$-cycles of degree $d$ in complex projective space, and let its Betti numbers be the ranks of its singu…

Let $\mathcal{F}$ be a family of sets in $\mathbb{R}^d$ such that for every non-empty finite subfamily $\mathcal{G}$ and every $i=0,1,\ldots,\lceil d/2\rceil-1$, one has … where…