14 problems

Let $\mathbb{K}$ be a non-archimedean local field with ring of integers $\mathscr{O}_\mathbb{K}$, prime ideal $\mathfrak{p}=\textbf{p}\mathscr{O}_\mathbb{K}$, and residue field…

Let $\Sigma$ be a smooth surface with smooth boundary, endowed with a Riemannian metric. Let $\varkappa$ denote the boundary operator and $D_\Sigma$ the Dirichlet-to-Neumann operat…

Let $M$ be a compact manifold and let $\Gamma\subset\mathbb{R}^p$ be a connected cone with nonempty interior and $H^1_{\rm dR}(\Gamma)=0$. Let…

Let $M$ be a compact oriented smooth manifold with smooth boundary $\partial M$, let $E_0$ and $E_1$ be smooth complex Hermitian vector bundles over $M$, and let…

Let $(M,H)$ be a compact Carnot manifold, and let $T\in \Psi^{-d_H}_H(M)$ be a pseudodifferential operator on $(M,H)$ in the van Erp-Yuncken sense. Let $\nu$ be a density on $M$, i…

Assume the conditions denoted by Assumptions. Let $\Pi_{\hbar,\mu}$ be the spectral projector defined in, and let … Let $g:[0,1]\to\mathbb{R}$ be continuous with $g(0)=g(1)=0$, and…

Boutet de Monvel's conjecture. There is a maximal $\epsilon_0>0$ such that, for every $\epsilon\in(0,\epsilon_0)$, the extended flows define a real-analytic diffeomorphism

Let $P=P^*\in \Psi^{-m}(\mathbb{T}^n_\theta)$ with $m>0$ have principal symbol $\rho_{-m}(\xi)$, and set $p=nm^{-1}$. For a selfadjoint compact operator, write…

Let $A^0$ be a scalar operator and let … be a strongly convex surface, meaning that … for all $\xi\in\Sigma^0_\lambda$ and all $\eta$ satisfying … where the sign depends on the con…

Smoothness and invertibility conjecture. For each $k>\frac{d}{2}+1$, the operator $\Psi^L$ depends smoothly on the immersion $f\in\operatorname{Imm}^k(M,N)$ and is invertible as a…

Local cyclic index theorem conjecture. For any local elliptic operator $A$,

Local K-theory index theorem conjecture. For any elliptic operator $A$,

Local K-theory connecting homomorphism conjecture. In local $K$-theory, $\partial^{K, loc}$ is an isomorphism rationally.

Let $\Psi{\rm DO}_0^*$ be the group of invertible zeroth-order classical pseudodifferential operators, and let $\mathcal E$ be a $\Psi{\rm DO}_0^*$-bundle over a base manifold admi…