Ramified Arithmetic Differential Geometry

In 2017, Buium introduced a remarkable theory where he applied the machinery of arithmetic differential equations to $GL_n$. This theory can be described
as a foundational set of tools building on ideas from Riemannian geometry towards number theory. Specifically, Buium carefully introduced a theory of
curvature and Levi-Civita connections recovering the fundamental theorem in Riemannian geometry. It is important in this theory to note that there is no
manifold on which these connections are taken but rather a set of tools inspired by Riemannian geometry used to study unramified extensions of $Q_p$.
In this theory, the integers considered as a space are curved.

However, this 2017 foundation suffers from the fact that there is only one unique lift of Frobenius in any unramified extension of $Q_p$. In differential geometry
terms these tools behave most similarly to those of cohomogeneity one, where all coefficients of a metric depend only one one coordinate. This limits the theory,
for example, the fundamental concept of geodesics is completely missing. Recently, with Buium, the speaker took up adapting this situation by including ramified
extensions which allow for more than one lift of Frobenius. This is passing from arithmetic ODEs to arithmetic PDEs. This talk will report on progress of this
adaptation and show how systematically including ramification produces new tools and a better theory. This will be shown to be a theory useful to study Galois
groups of totally ramified extensions of local fields.