31 problems

Ivrii's conjecture. The main part of the spectral asymptotics is given by ${\mathcal E}^{\rm {MW}}$, while: (i) for fixed $(g^{jk})$ and general $(V_j),V$, the remainder is…

Let ${\mathcal F}$ be an arbitrary foliation on a compact Riemannian manifold. In the above notation, let $\Delta_F$ and $\Delta_H$ denote the leafwise and transverse Laplacians, r…

On $\mathbb{T}^2$, let $X=\partial_x$, $Y=2(1-\cos x)\partial_y$, and let $\Delta_H=X^2+Y^2$. The singular set is $S=\{0\}\times\mathbb{T}$, and $c>0$ is the constant in the singul…

Let a curvilinear polygon have $n$ vertices with interior angles $\alpha_1,\dots,\alpha_n$. For $\alpha\in(0,2\pi)$, define … and … Form the multiset…

Let $\beta(\gamma,d)$ be the critical Robin parameter from Theorem, and let $\beta_W(\gamma,d-1)$ be the point where the second term in the two-term Weyl formula changes sign. Stri…

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…

Universality conjecture. For every such $\Omega$,

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$, and let $\lambda^{\rm DN}(bA_0,\Omega)$ den…

Let $\Omega$ be a regular domain in $\mathbb{R}^2$, and let $A_0$ be a magnetic potential whose magnetic field is constant and has norm $1$. Let $\Lambda_{bA_0}$ be the associated…

Low-frequency spectral asymptotics.

Let $t>t_b$ and let $\nu_{L,k}(t)$ be the imaginary parts of the lower-branch eigenvalues. Lower-branch imaginary-part conjecture. There is an $\epsilon$-independent function…

Let $S_U$ and $S_L$ denote the upper and lower branches of the bulk eigenvalues, with real parts indexed increasingly by $\mu_{U,k}(t)$ and $\mu_{L,k}(t)$, and let $t_b$ be the bre…

Let $n\geqslant3$, let $y\in[1/C_\infty,1/C_0)$, and let $\widetilde{C}_1\leqslant\cdots\leqslant\widetilde{C}_{n+1}$ be the nondecreasing reordering of the parameters…

Extreme-eigenvalue expansion conjecture. There exists an integer $k\geqslant1$ such that

Let $M^n$ be a closed $n$-dimensional manifold with $n\geq 2$, and let $\omega_p^1(M^n,g)$ denote its $1$-dimensional $p$-width with respect to a Riemannian metric $g$. The volume…

Let $\Omega$ be a bounded domain in $\mathbb R^d$ with smooth boundary. Let $N(z)$ denote the counting function for the eigenvalues of the buckling problem, and let $B_k$ denote th…

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 $s$ be the continuous function appearing in the eigenvalue-counting asymptotics … for a Sturm–Liouville problem with a non-constant weight whose primitive is arithmetically sel…

Strichartz's conjecture. There exists a uniformly almost periodic function $g$ such that

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…

Let $\Gamma$ be a $C^4$-smooth, asymptotically straight planar curve. Let $\Sigma_{\alpha,\Gamma}(k)$ be the on-shell scattering matrix at energy $k^2$, and let $S_\Gamma$ be the o…

Let $\Gamma$ be a graph with some semi-infinite, asymptotically straight edges, and suppose … Consider the renormalized operator $-\Delta_{\alpha,\Gamma}+\frac14\alpha^2$ as…

Let $V\in L^\infty_{\operatorname{comp}}(\mathbb R^n)$ be real-valued and non-zero, with $n$ odd. Let $N_V(r)$ denote the number of resonances in $\{|\lambda|\le r\}$. Optimal reso…

Consider pairwise commuting unitary semiclassical operators $U_1(\hbar), \ldots, U_d(\hbar)$ with joint principal symbol $F=(f_0^1,\ldots,f_0^d)$, and assume that $F(M)$ is closed.…

Modified Weyl–Berry conjecture. There is a positive constant $c_D$ depending only on $D$ such that