24 problems

Integral Stokes coefficient conjecture. Each $\gamma_i$ determines such a normalized formal heat solution, and there exists a coefficient $S_{ij}^{+}\in\mathbb Z$ such that

Picard–Lefschetz/Alien correspondence. The pointed Alien derivative of $\widetilde K_0$ at $\omega$ is

Let $\textbackslash \varphi$ be the Gaussian free field defining Liouville quantum gravity, let $\textbackslash mu_\textbackslash \varphi$ be the associated Liouville measure, and sa…

Let $U$ be a bounded domain in Euclidean space or in a Riemannian manifold, and let $\varphi_U$ denote its first Dirichlet eigenfunction. A family of domains satisfies…

Let $M$ be a Riemannian manifold with a boundary that need not be compact. The Dirichlet-to-Neumann operator and the associated mass-dependent quantities are as in Assumption … hol…

Let (KE) denote the energy-comparability property, and suppose a metric measure space admits a heat kernel satisfying the heat kernel estimate (hk). Heat-kernel implication questio…

Let $(M_1,d_1,{\Greekmath 0116}_1)$ and $(M_2,d_2,{\Greekmath 0116}_2)$ be metric measure spaces admitting heat kernels satisfying the heat kernel estimate (hk), with the same expo…

The Weyl-invariant Dunkl-kernel estimate. The heat kernel satisfies

The Weyl-invariant heat-kernel estimate. One has

Strong ratio-limit conjecture. Locally uniformly for $(x,y)\in M\times M$,

Let $M$ be a Riemannian manifold, and let $P_+$ and $P_0$ be respectively subcritical and critical operators in $M$. Critical-subcritical decay conjecture. The ratio … locally unif…

Let $Lu=u_t+P(x,\partial_x)u$ be a parabolic operator defined on a noncompact Riemannian manifold $M$, with $\lambda_0(P,M)\geq 0$. Fix a reference point $x_0\in M$. Davies' conjec…

Final conjecture. The estimates from the heat-kernel theorem and the Poisson-kernel theorem hold for all $\nu>-1$.

Let $K$ be the fractal and assume the conditions of Theorems and. Let the results of Strichartz referred to in the source be the stated spectral and fractal-analytic results, norma…

Let $D_1,D_2\subset \mathbb{R}^d$ be smooth bounded convex domains, with $d\geq 1$. For $i=1,2$, let $p_{D_i}(t,x,y)$ denote the Neumann heat kernel in $D_i$. If $D_2\subset D_1$,…

Let $M$ be a Riemannian manifold, and let $P_1$ and $P_2$ be subcritical elliptic operators of the form specified in the paper. Two heat kernels are equivalent when they are compar…

Let $M$ be a connected noncompact Riemannian manifold, and let $k_P^M(x,y,t)$ denote the positive minimal heat kernel of a second-order elliptic operator $P$ on $M$. An operator is…

Let $p_{\mathbb{U}}(t,x,y)$ denote the heat kernel for the Laplacian with Neumann boundary conditions on the unit ball … in $\mathbb{R}^{n}$, where $n\in\mathbb{N}^{\ast}$. The fun…

Let $\nu_t^{s,m_0}$ be the heat kernels associated with the $W^s$ norm on loops, viewed as measures on $\operatorname{Hyp}(S^1,G)$, where $s>1/2$, $t>0$, and $m_0>0$. Let $\mu_l$ b…

Let $\mathbf H\subset\mathbf G$ be an abstract Wiener group with Lie algebra $\mathbf h$, let $\mathbf H_0$ be the identity component of $\mathbf H$, and let…

Let $\mathbf H\subset\mathbf G$ be an abstract Wiener group with Lie algebras $\mathbf h\subset\mathbf g$, and let $\nu_{t/N}^{\mathbf h\subset\mathbf g}$ denote the corresponding…

Equivariant heat coefficient vanishing conjecture. The coefficients satisfy

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^d$, with $d\geq 1$, let $p^D_{\mathbb{B}}(t,x,y)$ be the Dirichlet heat kernel, and let $\varphi_1(x)$ be the ground-state eigenfun…

Let $\Omega$ be a bounded convex domain in the plane symmetric with respect to the $x$-axis, and let $p^N_{\Omega}(t,z,w)$ be its Neumann heat kernel. Let $f:\mathbb{B}\to\Omega$ b…