47 problems

Bean's conjecture. The maximum is attained precisely when $F$, up to multiplication by a complex number, is equivalent under $\operatorname{GL}_2(\mathbb R)$ to $F_{n,n}$. In parti…

Let $f$ and $g$ be singular functions such that $f$ is adjacent to $g$, so that a versal unfolding of $g$ is also a versal unfolding for $f$. Let $4mu$ and $tilde\mu$ be the corres…

Tropical discriminant conjecture. For any point configuration $A$, the tropical discriminant $\tau(X_A^*)$ is a union of cones in the secondary fan $\Sigma(A)$.

Let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$, let $\delta(w_i)$ be the unique descent of $w_i$, and let $\Delta_w(t_1,\dotsc,t_m)$ be it…

Let $\alpha=\{\alpha_1<\dotsb<\alpha_k\}$ with $\alpha_k<n$, and let $(w_1,\dotsc,w_m)$ be a Grassmannian Schubert problem for $\mathbb{F}\ell(\alpha;n)$. Let…

For the products $\mathbb P^1\times\mathbb P^{n-1}$ and $\mathbb P^2\times\mathbb P^{n-1}$, consider the corresponding RR discriminants and their nonisotropic parts. Product RR dis…

Let $V$ be the variety under consideration, let its RR discriminant be defined by a polynomial, and call a factor isotropic when it belongs to the isotropic part of that discrimina…

Double-discriminant constant conjecture. The following hold:

Let $p$ be the odd prime and let $\theta$ be the algebraic number considered in the preceding discriminant computation, with $\mathbb{Q}(\theta)$ its generated number field. Discri…

Phase limit set conjecture. The closure of the discriminant coamoeba equals the phase limit set:

Let $f\in\mathbb{Z}[X]$ be a polynomial, respectively a monic polynomial, of degree $n\geq 4$, respectively $n\geq 3$, with discriminant $D\ne 0$. Here $H(g)$ denotes the height of…

Let $u\in S_d$, write $\nu=u^{1/3}$, and let $c=c(u)$ be the product of the distinct rational primes dividing $u$ but not $d$. Let $c'$ be the odd part of $c$. Let…

Let $g_n$ be an isolated bad Gram point, meaning the middle point of a Gram block whose adjacent boundary Gram points $g_{n-1}$ and $g_{n+1}$ are good. In the two-dimensional param…

For $N\in\mathbb{N}$, let $\mathcal{Z}_N$ be the parameter space of functions … with $\overline{a}\in\mathbb{R}^N$. A path $\gamma(r)$ in this space starts at the core function…

$J_{10}^3$ component conjecture. The complement of the discriminant variety has exactly 33 distinct components.

$J_{10}^1$ component conjecture. The complement of the discriminant variety has exactly 11 components, represented by these six original and five transformed topological types.

$X_9^2$ component conjecture. These representatives exhaust the components of the complement of the discriminant variety; in total there are 52 such components.

$+X_9$ component conjecture. Each of the seven virtual components is represented by exactly one connected component of the complement of the discriminant variety of any versal defo…

Let $Q$ be a positive definite integral quadratic form, let $L$ be a rational $k$-dimensional subspace, and let $q_L$ and $q_{L^\perp}$ be the associated integral quadratic forms o…

Let $W\subset L^\vee_\mathbb R$ be a wall, let $\Gamma\subset\Pi$ be a minimal face, and let $H=H_{\Gamma^c}$ be the corresponding relevant subspace. Assume that … Write…

Let $Z$ be the toric variety associated to a wall-crossing, and suppose its derived category has a semi-orthogonal decomposition … where each $D^b(Z_i)$ occurs some number of times…

Bean's conjecture. The maximum value $M_n$ of $Q(F)$ over all such forms is attained precisely when $F$, up to multiplication by a complex number, is equivalent under…

Boyd et al.'s conjecture. The set $S$ has density

Conjectural tail estimates. For every $\epsilon>0$,

Discriminant multiplicity conjecture. For every $D\in\mathbb{N}$, one has