Uniform sheaves and differential equations

TL;DR

This paper introduces a new uniform sheaf framework based on uniform structures to analyze singularities in complex-analytic differential modules and explores analogous concepts in the p-adic setting.

Contribution

It proposes a novel uniform sheaf approach to cohomology theory of differential modules, extending the analysis to p-adic contexts with real blow-up analogs.

Findings

New uniform sheaf theory for differential modules

Application to p-adic differential equations

Insights into singularity analysis via uniform structures

Abstract

Real blow-ups and more refined "zooms" play a key role in the analysis of singularities of complex-analytic differential modules. They do not change the underlying topology, but the uniform structure. This suggests to revisit the cohomology theory of differential modules with help of a suitable new notion of uniform sheaves based on the uniformity rather than the topology. We also investigate the $p$-adic situation (in particular, an analog of real blow-ups) from this uniform viewpoint.