Hodge theory is a powerful tool for analyzing the cohomology of compact complex manifolds. I will start with a classic theorem of Grothendieck that the (algebraic) de Rham cohomology of a complex variety can be computed in terms of its (topological) singular cohomology. The result, applied to varieties over $\mathbf{Q}$, leads to a fascinating and mysterious subring $P$ of $\mathbf{C}$---the ring of periods. I will briefly recount what is known and conjectured about $P$, then move on the $p$-adic side of the story. A major theme in arithmetic geometry is that ``all analysis which one usually does over $\mathbf{R}$ can actually be done over $\mathbf{Q}_p$.'' I will quickly introduce some basic notions of the $p$-adics, then move towards a ``$p$-adic $2\pi i$.'' We will see that unlike the ``$2\pi i$ at infinity,'' the $p$-adic incarnation lives inside a large and complicated ring known as $\mathbf{B}_\mathrm{dR}$.