10 problems

Consider equation … with sufficiently large energy should blow up in finite time, in the sense that $w_{rr}(t,0)$ diverges as $t\nearrow T$ for some $T>0$. More precisely, there sh…

Let $A$ be an anti-self-dual connection on a bundle $E\rightarrow T\times\mathbb{C}$, with curvature satisfying $|F_A|\sim O(|w|^{-2})$. A holomorphic structure…

Let $E$ be a rank $r\geq 2n$ locally-free instanton sheaf of trivial splitting type on $\mathbb{P}^{2n}$ and charge $c$. Lifting conjecture. There is a rank $r$ locally-free instan…

Let $\vec{\mathtt{q}}=(\mathtt{q}_0,\mathtt{q}_1,\mathtt{q}_2,\mathtt{q}_3)$ and $\mathtt{Q}=\mathtt{q}_0\mathtt{q}_1\mathtt{q}_2\mathtt{q}_3$. Define … where…

Let $X_M$ be the resolution of the $A_{M-1}$ singularity, and let $\mathscr O_{c,\lambda_1,\dots,\lambda_{M-1}}(X_M)$ be its family of deformation-quantizations. Let…

Let $\Sigma$ be an affine algebraic curve, and let $\operatorname{Diff}(\Sigma)$ be its algebra of polynomial differential operators. The five-dimensional quantum field theory on…

Generic non-jumping-line conjecture. The generic line is not jumping.

Kuznetsov's reconstruction conjecture. The generic line is not jumping, and $E$ can be reconstructed from its theta-sheaf.

Let $Y$ be the Fano threefold under consideration, let $E$ be an instanton on $Y$, and let $D_E$ be its curve of jumping lines. Let $\mathcal L_E$ be the associated line bundle on…

Let $M^{k,c_1}_\phi$ be the smooth completion of the moduli space of ${\rm U}(n)$ Yang--Mills instantons of first Chern class $c_1$, energy $k$, and framing $\phi$ on a four-dimens…