Boundary Value Problems on $p$-Adic Analytic Manifolds

An account is given on newest developments on $p$-adic boundary value problems on $p$-adic analytic manifolds and their relationship with diffusion. In particular, novel coordinate Laplacians on $p$-adic analytic $n$-manifolds constructed with the help of frame bundles, are introduced in this context. These are used to construct elliptic operators. Related Dirichlet problems are formulated and solved, generalising results on compact subdomains of $p$-adic $n$-space. In the end, an outlook towards number-theoretic applications as well as extensions of this theory to ultrametric analytic manifolds is given. This is a substantial upgrade of the presentation given at Branko's 80-th Birthday Conference in Belgrade, May 2025.