12 problems

Let $X$ be a complex manifold of dimension $d$, and let ${\cal M}^\bullet$ be a complex of ${\cal D}_X$-modules with bounded good cohomology. Choose a filtration compatible with th…

Let $X$ be a real analytic manifold, let $F\in D^b_{{\rm w-}\mathbb{R}\text{-c}}(\mathrm{I}[k_X])$, and let $G\in D^b_{{\rm w-}\mathbb{R}\text{-c}}(k_X)$. Closure conjecture. The o…

The technical-loss conjecture. For every $k_0>0$ and $s\in\mathbb R$, there is $C>0$ such that, for all $N>0$, $k>k_0$, and $0<\epsilon<1$,

Let $P=\Delta+V-\lambda^2$ be the Helmholtz operator on $\mathbb{R}^n$, where $\lambda^2>0$ and $V=\sum_{\alpha\in I}\pi_\alpha^*(V^\alpha)$ is a three-body potential with…

Let $\alpha$ be sufficiently close to zero, let $\mathfrak{M}_\alpha$ be a blow-up of the compactification of Minkowski space, and let $W$ denote the soliton profile. A function is…

Let $\cM$ be a module over the sheaf of convergent-exponential differential operators $\cD^{\mathrm{ce}, \Lambda_0}$, with characteristic variety $\mathrm{Char}(\cM)$ and solution…

Let $\rho_1,\rho_2$ be the ranks of the bulk Bloch bundles $\pazocal{E}_+$ and $\pazocal{E}_-$, and assume … Let $\pazocal{I}_e(P)$ denote the edge invariant, let $\mathbb{P}_h$ be…

Let $(M,g)$ be a Riemannian manifold and let $\boldsymbol{\Sigma}$ be an interior hypersurface with positive definite second fundamental form satisfying the transversal geometric c…

Quantum ergodicity conjecture for resonant states. Under these assumptions, there exists a density-one subsequence of resonant states in every strip…

Knot-state microsupport conjecture.

Let $\La$ be a Lagrangian, and let $\Per_\La(\bZ)$ denote the stack of microlocal perverse sheaves on $\La$ with integral coefficients. Codimension-three conjecture. The codimensio…

Let $\Omega$ be the domain, let $\chi(t,x)$ be the control function, let $\Lambda$ be the HUM control operator, and let the optical rays be the associated bicharacteristic rays. As…