17 problems

Let $H$ be a finite edge-weighted graph, possibly with loops, whose adjacency matrix $A_H$ is entrywise nonnegative and has at most one positive eigenvalue, counted with multiplici…

Let $M=(E,\mathcal{I})$ be a matroid with $|E|=n$ and rank $r_M$. For $0\leq k\leq r_M$, define the homogeneous multivariate independent-set polynomial … For polynomials in…

Let $G$ be a graph with vertex set $V$, and for each $v\in V$ let $S(v)$ be its list of available colors. Let $p_\varnothing$ be the polynomial recursively defined from the associa…

Let $d\geq 3$, and let $(p_{ij})$ be a real symmetric $d\times d$ matrix with nonnegative off-diagonal entries and zero diagonal entries. For $i\ne j$, let…

Let $\mathfrak{g} = \oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$ be a direct sum of special linear Lie algebras, with positive roots $\Delta$ as above, and let…

Let $w\in S_n$, let $\mathfrak{S}_w(x)$ be the Schubert polynomial indexed by $w$, and let $N$ be the normalization operator on monomials, defined by … A homogeneous polynomial is…

Let $f=\sum_{\alpha\in\mathbb{N}^n}c_{\alpha}x^{\alpha}$ be a polynomial, and define its normalization by … For $\lambda\in \mathrm{Par}^k(d)$, let $\mathfrak{S}_{\lambda}^{(k)}$ d…

Hodge–Riemann and equality conjecture. If all convex bodies in the tuples $E_i$ are smooth and strictly convex, then the valuation

Let $d$ be a Dyck path, let $G(d)$ be its indifference graph, and let $X_{G(d)}$ be its chromatic symmetric function. For any finite number of variables, restrict $X_{G(d)}$ to tho…

Let $\lambda$ be a partition whose nonzero parts are distinct, and let $P_\lambda$ be its Schur $P$-function. Schur-$P$ conjecture. The normalized function … is Lorentzian for ever…

Let $\lambda,\mu\in\mathbb{Z}_{\geq0}^n$ be partitions satisfying $\lambda_i\geq\mu_i$ for every $i$, and let $s_{\lambda/\mu}$ be the skew Schur polynomial. Skew-Schur conjecture.…

Let $\alpha\in\mathbb{N}^n$ be a composition, and let $\kappa_\alpha(x_1,\ldots,x_n)$ be its key polynomial. Key-polynomial conjecture. The normalized polynomial … is Lorentzian fo…

Homogeneous-Grothendieck conjecture. The normalized polynomial

Let $\lambda$ be a strict partition, meaning a decreasing sequence of positive integers, and let $P_\lambda(x_1,\ldots,x_m)$ be its Schur $P$-polynomial. Schur-$P$ conjecture. The…

For every integer $n\ge 1$ and every pair of partitions $\mu\subseteq\lambda$, let $s_{\lambda/\mu}(x_1,\ldots,x_n)=\sum_{\alpha\in\mathbb{Z}_{\ge 0}^n}c_\alpha x^\alpha$ be the sk…

Let $\Lambda$ be the weight lattice of $\mathfrak{sl}_m(\mathbb{C})$, let $\mathrm{V}(\lambda)$ be the irreducible module of highest weight $\lambda$, and define its shifted charac…

Let $\mathrm{H}^d_n$ be the space of degree-$d$ homogeneous polynomials in $n$ variables, and let $\mathrm{L}^d_n$ be the cone of Lorentzian polynomials in $\mathrm{H}^d_n$. Let…