Activity

Let … and let $S_2$ be the Schur class of analytic functions $f:\mathbb D^2\to\mathbb C$ satisfying … If … define … Determine the exact value of $K_2$.

Open problem. Does every separated Noetherian scheme have the resolution property? Equivalently, can one construct such an $X$ and a coherent sheaf $\mathcal F$ that is not a quoti…

Let $X$, $Y$, and $Z$ be upward closed subsets of $Q_n$. It is conjectured that … Notation: the poset $Q_n$ is the power set of a set with $n$ elements, ordered by subset relation.…

Setting. For a Navier-Stokes velocity field with vorticity $\omega$, direction $\xi = \omega/|\omega|$, and peak set $P(t)$ (the high-vorticity region), define the scale-invariant…

2403.10985 introduced the notion of reversible capacity of a graph and showed that it is at most the Shannon capacity. It is conjectured there that the two values are equal. At the…

Let $U$ and $V$ be finite-dimensional vector spaces over a field $\mathbb{F}$. Let $\mathcal{S}$ be a linear subspace of $\mathrm{Hom}(U,V)$. We say that it is intransitive when…

Consider the real additive white Gaussian noise sequence channel … where the noise coordinates are independent $N(0,N_0/2)$. An $(E,M,\epsilon)$ code has $M$ equiprobable messages,…

Question. If

Let $R$ be a complete local domain containing an algebraically closed field. If the perfection of $R$ , $R_{perf}$ (union of all $p^{e}$ th roots, in the sense of positive characte…

The big quantum cohomology of G/P is generically semisimple.

$M_{23}$ is one of the five Mathieu groups and one of the $26$ sporadic finite simple groups. It has order … and admits a natural action on $23$ points. The problem asks whether th…

Kuznetsov's conjecture. The fourfold $X$ is rational if and only if … for an ordinary, untwisted projective K3 surface $S$.

For a smooth Fano variety $X$ of Hodge-Tate type, the modern form of Dubrovin's conjecture predicts … The refined Gamma conjecture II also identifies categorical and quantum data.…

Conjecture. For every smooth quasi-projective variety $X$, … The lower bound $\operatorname{Rdim}D^b(X)\geq\dim X$ is known quite generally. The difficult direction is to construct…

Let $X$ and $Y$ be smooth projective varieties. Orlov's motivic conjecture predicts … where $h(X)_{\mathbb Q}$ denotes the rational Chow motive. A Fourier-Mukai equivalence produce…

DK hypothesis. K-equivalence should imply an equivalence … More generally, if $p^*K_X\geq q^*K_Y$, one expects a fully faithful embedding from $D^b\operatorname{Coh}(Y)$ into…

Conjecture. Every standard graded Artinian Gorenstein algebra of embedding dimension three over a field of characteristic zero has the weak Lefschetz property.

Open problems. Determine the smallest $d$ for which $H_{d,0}$ is reducible, and the smallest $d$ for which $H_d$ is reducible. The presently known bounds are … Indeed, both spaces…

Four-variable Hessian conjecture ($\mathrm{HC}_4$). Must $\varphi^L$ be a polynomial? Equivalently, must every four-variable gradient polynomial map with constant nonzero Jacobian…

Let $X$ be a smooth projective variety with an algebraic action of a complex torus $T_{\mathbb C}$, and let $Y=X/\!/T_{\mathbb C}$ be a smooth GIT quotient with no orbifold singula…

If $S$ is a surface with ADE singularities, it is known that $\operatorname{Hilb}^n(S)$ is reduced for $n\leq 7$, while reducedness for arbitrary n remains open. Its description th…

It remains open whether a standard graded Artinian Gorenstein algebra of embedding dimension 4 can have a non-unimodal Hilbert function; the analogous phenomenon is impossible in d…