46 problems

Let $\mathsf{M}$ be a smooth manifold and let $\mathsf{P}$ be a properly supported pseudodifferential operator on $\mathsf{M}$ with complex coefficients. Local solvability means th…

Ivrii's conjecture. If every singular point of $p=0$ is effectively hyperbolic, then the Cauchy problem is $C^{\infty}$ well posed for any lower order term.

Let $X$ be a complex manifold, let $\mathscr{M}$ be a coherent bounded complex of $\mathscr{D}_X$-modules, and let $\Lambda$ contain $\operatorname{char}(\mathscr{M})$. Choose a go…

Let $\phi_i$ be an eigenfunction on the hyperbolic surface and let $Op(a)$ be the operator associated with a fixed symbol $a$. Write the spectral decomposition of $Op(a)\phi_i$ in…

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…

Stability-across-scales conjecture. The connection between dynamical stability conditions at different length scales is a generic feature and extends to physical systems described…

Let two wave fields be measured at a fixed measurement position, and let their time correlation be the correlation obtained by integrating one field against a time-shifted copy of…

The Edwards–Wilkinson equation is … Here $\dot{W}(t,x)$ denotes the space-time white-noise forcing, and the equation is considered in $1+1$ dimensions, where its singularities are…

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…

Frobenius-mass conjectures. The following four assertions are conjectured:

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 $\mathcal{V} \subset \mathbb{C}\mathrm{T}\Omega$ be a real-analytic locally integrable structure of corank $m$ on a $\mathcal{C}^\omega$-smooth manifold, locally generated by a…

Let $M$ be a closed Riemannian manifold, let $\operatorname{\Delta}$ be a non-negative generalized Laplacian on $M$, and let $h_t$ denote the heat kernel of…

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 $x\in\mathbb{R}^{1+3}$ denote the spacetime variable and let $e\in H$ denote a spacelike string direction, where $H$ is the open subset of spacelike vectors. String-localized f…

Let $H\subset\mathbb{R}^{1+3}$ be the open subset of spacelike vectors in Minkowski space, and let $A_{\mu_1\cdots\mu_s}(x,e)$ be the string-localized potential obtained by integra…

Let $(M,g)$ be a real analytic Riemannian manifold of dimension $m$ without boundary, let $\varphi_j$ be an $L^2$-normalized eigenfunction with eigenvalue $\lambda_j^2$, and let it…

The classical and quantum notions involved are completeness of the Hamilton flow associated with a real principal symbol and essential self-adjointness of the corresponding pseudod…

Let $d\epsilon_{it}$ be the micro-local lift of the distribution $\lvert E(z+\lambda j,it)\rvert^2\,dV$ on the frame bundle of the Picard variety, and let $d\epsilon_{it}^F$ be the…

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…

Consider Schrödinger equations whose real-valued potentials have quadratic growth, and let the associated classical orbit depend on the energy. Yajima's conjecture. Singularities o…

Let $P$ be a hyperbolic operator whose principal symbol has triple characteristic points only on the initial plane $t=0$, and suppose that $P$ is effectively hyperbolic at every tr…

Let $S(\sigma)$ be a general downward concave function defining a 2-microlocal frontier at a point $x_0$. Suppose the sufficient conditions constructed in the paper produce a funct…

Let $\boldsymbol{\Sigma}$ be a compact interior hypersurface of a Riemannian manifold $(M,g)$. Let $\Delta_\Sigma$ be the Laplace–Beltrami operator induced on $\Sigma$, and define…