Down to characteristic p and then back up again: computations with the p-adic obstruction map

Although extremely simple, Gauss' modular arithmetic is a powerful idea in working with the integers. There is a direct application of this idea in algebraic geometry, where one ""reduces"" a variety modulo an integer. A major undertaking in the second half of the previous century linked topological aspects of the original variety to point counts of its reduction. One can go a little further and carry the analytical properties of the original variety over to its reduction. This gives a refined sense about which subvarieties of the reduction lift back up to the original variety. I will report on joint work with Edgar Costa where we implemented this idea, but mostly, I will give a friendly introduction to the basic concepts.