6 problems

Let $(\Omega,h)$ be a quantum disc as above, and let $\mathbf{H}(t)$ denote its heat trace, with $c_\gamma$ the leading Weyl-law constant and $\mu(\Omega)$ its quantum area. Quantu…

Let $M$ be a bounded polytope in $\mathbb{R}^n$, with either the Dirichlet or Neumann boundary condition, and let $(\lambda_k)_{k\geq 0}$ be the corresponding Laplacian eigenvalues…

A Euclidean space form is a compact Riemannian manifold with curvature identically equal to zero. Let $M$ be an $n$-dimensional Euclidean space form, let $(\lambda_k)_{k\geq 0}$ be…

Let $D$ be a convex domain, let $B({\bf x},r)$ be a ball of radius $r$ contained in $D$, and write $\Omega({\bf x})=D\setminus B({\bf x},r)$. Let $\heartsuit(D)$ denote the heart o…

Let $\Omega$ be a bounded planar domain of area $A$, and let $0=\mu_1<\mu_2\leq\mu_3\leq\cdots$ be its Neumann Laplace eigenvalues. For $t>0$, define the area-normalized Neumann he…

Let $\Omega\subset\mathbb{R}^d$ be a domain with volume $|\Omega|$, let $\lambda_i$ be the eigenvalues of its Dirichlet Laplacian, and let $t>0$. Melas' heat-trace conjecture. The…