71 problems

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 $\widetilde{p}(t,x,y)$ be the heat kernel associated with the fractional-type Schrödinger operator with singular potential, and let $p(t,x,y)$ be the corresponding free heat ke…

Let $\mathbb{U}=\{x\in\mathbb{R}^n:|x|<1\}$ be the open unit ball, and let $p^N_{\mathbb{U}}(x,y,t)$ denote its Neumann heat kernel. Laugesen–Morpurgo conjecture. For every fixed…

Let $(K,d,bc)$ be a compact metric space with Hausdorff measure $bc$, and let $(\mathcal{E},\mathcal{F})$ be a strongly local regular Dirichlet form whose semigroup has heat kernel…

Let $K$ be the compact gauge quotient group, let $\rho_s$ be the heat-kernel measure on $K$ at time $s$, and let $P_s$ be the regularized measure on the space of connections…

Callias–Taubes conjecture. The small-$t$ heat-kernel trace asymptotic expansion may contain more general powers of $t$ than $t^{(n-m)/d}$, as well as terms involving $\log t$.

Let $K$ denote the Dunkl kernel associated with the multiplicity function $k$. For each $y\in\mathbb R^N$, consider the function $\xi\mapsto K(iy,\xi)$ on $\mathbb R^N$. Dunkl-kern…

Let $(M,g)$ be a connected, oriented, complete Riemannian manifold whose Ricci curvature is bounded from below. For $t>0$ and $x_0\in M$, let $\rho_t(x,x_0)$ be the heat kernel and…

Let $M$ be a Riemannian manifold equipped with a Laplace–Beltrami operator, and let $P_t$ denote the associated heat semigroup. A Poincaré estimate for its heat kernel is an inequa…

Let $M$ be a smooth closed oriented $2n$-dimensional manifold, let $E$ and $F$ be smooth complex vector bundles over $M$, and let $D_+\colon\Gamma(E)\to\Gamma(F)$ be an elliptic di…

Bueler's conjecture. The weighted $L^2$ cohomology of $M$ with respect to $d\mu$ is isomorphic to the de Rham cohomology of $M$.

Let $d\geq 1$, let $U\in L^\infty(\mathbb{T}^d)$, and let $p(t,x,y)$ be the heat kernel associated with the periodic diffusion described by the Dirichlet form in the source. Let…

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

Kontsevich's conjecture. Singularities of the Borel transform of $K(\tau;a,b)$ correspond to relative critical values of the complexified energy functional…

The two-dimensional Sierpinski carpet is a fractal space for which the paper's resistance-form and heat-kernel framework may require modifications. Applicability conjecture. The re…

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 $(X,d,m)$ be a metric measure space satisfying volume doubling, denoted by. Let denote the relevant Poincaré inequality, the uniform capacity condition, and the two-sided heat-…

Let a background field be more general than the electromagnetic background considered earlier, and let its heat kernel or effective action be treated by heat-kernel methods. Genera…

Let $\phi$ be a massive quantum scalar field coupled to a background vector $N_\mu$ through the operator … and let $K_N(x,x';\tau)$ be the corresponding heat kernel. Write…

Let $F_{bc\nu}$ be the field strength of a background vector field in four spacetime dimensions, and define the invariants … where ${}^*F_{\mu\nu}$ is the Hodge dual of…

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 $h_{j,m}$, $j=1,\dots,m$, be the degree-one polynomials in two variables entering the decompositions of the heat kernel. For $m\geq 1$, let $C_+^{(m)}$ denote the corresponding…

Let $D$ be a bounded domain, let $\gamma\in(0,2)$, let $\mathbf{p}^{\gamma,D}_t(x,x)$ be the Liouville heat kernel, and sample $x$ from the Liouville measure. Let $\xi_\gamma>0$ be…